methods used in operations research

• Open access
• Published: 18 April 2017

Operations research in global health: a scoping review with a focus on the themes of health equity and impact

• Beverly D. Bradley 1 , 2 ,
• Tiffany Jung 1 , 2 ,
• Ananya Tandon-Verma 3 ,
• Bassem Khoury 3 ,
• Timothy C. Y. Chan 1 , 3 , 4 &
• Yu-Ling Cheng 1 , 2  

Health Research Policy and Systems volume  15 , Article number:  32 ( 2017 ) Cite this article

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Operations research (OR) is a discipline that uses advanced analytical methods (e.g. simulation, optimisation, decision analysis) to better understand complex systems and aid in decision-making.

Herein, we present a scoping review of the use of OR to analyse issues in global health, with an emphasis on health equity and research impact. A systematic search of five databases was designed to identify relevant published literature. A global overview of 1099 studies highlights the geographic distribution of OR and common OR methods used. From this collection of literature, a narrative description of the use of OR across four main application areas of global health – health systems and operations, clinical medicine, public health and health innovation – is also presented. The theme of health equity is then explored in detail through a subset of 44 studies. Health equity is a critical element of global health that cuts across all four application areas, and is an issue particularly amenable to analysis through OR. Finally, we present seven select cases of OR analyses that have been implemented or have influenced decision-making in global health policy or practice. Based on these cases, we identify three key drivers for success in bridging the gap between OR and global health policy, namely international collaboration with stakeholders, use of contextually appropriate data, and varied communication outlets for research findings. Such cases, however, represent a very small proportion of the literature found.

Poor availability of representative and quality data, and a lack of collaboration between those who develop OR models and stakeholders in the contexts where OR analyses are intended to serve, were found to be common challenges for effective OR modelling in global health.

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‘Global health’ broadly refers to “ an area for study, research, and practice that places a priority on improving health and achieving equity in health for all people worldwide ” [ 1 ]. From population-based prevention to individual-level clinical care, global health encompasses health issues and solutions that transcend borders, and involves a collaborative and interdisciplinary effort [ 1 ]. The goal of achieving equity in health, namely the absence of systematic disparities in health or in the major social determinants of health between groups with different levels of underlying social advantage/disadvantage [ 2 ], has become a particularly important part of the post-2015 development agenda [ 3 – 6 ]. Globally, major progress has been made towards certain Millennium Development Goals and targets; however, many low- and middle-income countries (LMICs), especially in sub-Saharan Africa and Asia, continue to experience high health inequities, both within and between countries [ 3 , 4 ]. Further, these countries carry most of the world’s burden of morbidity and mortality; for example, more than 99% of under-5 child deaths in 2010 occurred in LMICs, and although mortality rates fell in most monitored countries, 15 countries experienced increases in the absolute number of deaths, with 12 of these countries being in sub-Saharan Africa [ 7 ].

Operations (or operational) research (OR) is a discipline that uses advanced analytical methods to better understand complex systems and aid in decision-making [ 8 , 9 ]. OR uses a wide range of problem-solving techniques and computational methods, including computer simulation, mathematical optimisation, statistics and decision analyses, to help improve the operations of organisations. With its orientation towards improving efficiency, cost-effectiveness and decision-making, OR is particularly useful for analysing complex global health issues – especially in settings where the burden of disease is high but health systems are weak and resources limited. Despite the growing use of OR in global health, it is unknown how much of an impact OR is having in this important area as publications rarely discuss whether their findings were implemented or were influential in policy- or decision-making [ 10 , 11 ].

The objective of this scoping review is to examine the extent, range and nature of operations research activity in global health, specifically within healthcare settings, health services delivery, and population health in LMICs. Our goal is to highlight the breadth of healthcare applications of OR in global health, both geographically and across application areas, and – through select case studies – discuss the impact such studies can have on improving healthcare and healthcare equity for communities and populations globally. We aim to encourage OR researchers and global health practitioners alike to continue to apply OR in global health, particularly in areas where OR-based studies may currently be lacking, and to consider sharing the impact of OR work more broadly so that others can learn from challenges and successes.

It should be noted that, in the context of global health, the term ‘operations research’ is sometimes synonymous with implementation research [ 12 ] and is used broadly to encompass cross-sectional, case-control, retrospective or prospective cohort analyses [ 13 – 15 ], as well as qualitative research methods [ 12 , 16 ], all of which are valuable in studying the effectiveness of health services and programs within the day-to-day operating environments of routine practice. In this review, however, we focus on studies where analytical methods or modelling are used to explore health research questions with an orientation towards decision-making or the consideration of ‘what-if’ scenarios – in other words, modelling studies that are prescriptive in their recommendations.

The modelling realm of OR is of particular interest because it can help address global health questions not easily answered with other methods. For example, OR is beneficial in situations where conducting a real-world study might be considered impossible, impractical, too costly or unethical, such as when choosing between implementing policy ‘A’ or policy ‘B’, when controlled trials to compare a wide variety of available options would be unreasonable, when the disease or illness of interest takes years or decades to progress and the process of evaluating long-term outcomes would be long and expensive, or when simulating virtual cohorts of patients allows researchers to explore questions without ethical consequences. OR is also useful for framing complex financial evaluations, for example, determining the most cost-effective intervention among many options, establishing the optimal way to allocate a limited budget across multiple competing needs, or deciding whether a new intervention (e.g. a vaccine) can be implemented sustainably with limited funding. In LMICs, such OR analyses, which help narrow down the number of possible options or help inform where to focus efforts for more targeted studies, are even more important due to limited resources.

While OR in healthcare in the developed world has been extensively studied in recent years [ 17 – 22 ], the latest review of OR in healthcare in developing countries was published in 1993 [ 10 ]. A few recent review papers and bibliographies have explored the use of OR in developing countries; however, these did not specifically focus on healthcare and included several other sectors such as agriculture, energy and transport [ 11 , 23 – 25 ]. Others have reviewed the use of OR within a very narrow area of global health (e.g. infectious diseases, particularly HIV) [ 26 – 30 ]. Several survey papers and special issues of journals have recently focused on the use of OR to address global health or humanitarian issues, but these were not based on systematic reviews of the literature [ 16 , 31 – 33 ]. Given that the existing literature on this topic is sporadic, not comprehensive in the search strategy, and lacks depth in the analysis of thematic areas, we have chosen a broad scoping review approach. With this approach, we aim to build upon previous work by providing a systematic and comprehensive landscape overview of the use of OR in global health with a more rigorous analytic framework than has been previously performed.

The results of this scoping review are presented in four main sections. First, we present a global overview of the literature, which includes the distribution of OR studies across countries of different income classifications, over time, and across different methodological approaches. Then, we explore the use of OR in four global health application areas with concrete examples in each category. In this review, we consider the four main application areas of global health to be (1) health systems and operations, (2) clinical medicine, (3) public health, and (4) health innovations – from the local to global level. Next, health equity, which is integral to the concept of global health and transcends all four application areas, is explored as a separate overarching theme using a subset of included studies. Health equity is not only a topic of growing interest globally, but is compelling to explore through an OR lens. For example, when health equity is operationalised and quantified using meaningful and measurable criteria [ 2 ], OR methods can be used to analytically find solutions that improve or maximise equity. To our knowledge, the use of OR to analyse issues of health equity has not yet been explored through a systematic review. Finally, by way of selected cases, we present a discussion on implementation and impact, i.e. how OR has influenced real health policy change or aided in decision-making by stakeholders. We highlight common factors among these studies that likely helped contribute to their effective translation into policy or practice, and discuss barriers and challenges to bridging the gap between OR and health policy. We conclude the paper with a discussion of key insights and implications of this review.

We followed the scoping review framework set out by Arksey and O’Malley [ 34 ] and by others who have proposed refinements to this approach [ 35 – 37 ]. Specifically, we followed these five stages: Footnote 1 (1) identifying the research question; (2) identifying relevant studies; (3) study selection; (4) charting the data; and (5) collating, summarising and reporting results.

Stage 1: Identifying the research question

This scoping review seeks to identify the extent, range and nature of OR in global health (i.e. geographical, over time, methodological and across application areas), with an in-depth exploration of literature addressing questions of health equity and literature having made a specific impact in decision-making or policy change.

Stage 2: Identifying relevant studies

The databases HealthStar (a subset of Ovid Medline focused on health systems research), Scopus, Web of Science, Inspec and Compendex were chosen to capture literature from multidisciplinary sources across health research and engineering. Individualised search strategies were designed for each database. Footnote 2 We searched titles, abstracts and keywords for combinations of search terms in the following categories: OR modelling and methodologies; healthcare settings, health services delivery, and population health; LMICs and regions (e.g. sub-Saharan Africa, South East Asia, etc.), including specific country names in these income categories; and policy- and decision-making. The search strategy including all search terms for the Web of Science database is provided in Box 1 as an illustration, and all others are provided in Additional file 1 : Tables S1 to S4. Only papers published in the year 2000 and later were included. This search strategy was refined and validated by ensuring the search captured a set of 15 ideal target papers [ 38 – 52 ] known to the authors. Librarians specialising in both engineering and health sciences literature were consulted when designing the search strategy. Search results were downloaded in August 2014. We also hand-searched 19 review papers and special issue articles [ 11 , 13 , 16 , 17 , 19 , 23 – 33 , 53 – 55 ] for additional references.

Box 1 Example search strategy for Web of Science database

Search results from each database were combined and duplicates were removed. An initial screen of the remaining 14,518 references eliminated studies that were clearly not relevant to this review. This initial screen was largely based on title and keywords; if additional information was required to judge relevance, the abstract was consulted. A large proportion of papers rejected at this stage fell into one of two categories, either (1) field studies and implementation research that did not have a modelling element, or (2) health-related modelling studies that were purely explanatory or descriptive in nature and did not have an orientation towards policy- or decision-making. There were 1408 abstracts remaining after this initial screening, including 31 articles from hand-searching review papers (Fig.  1 ).

Systematic search results and screening process

Stage 3: Study selection

A second screen was conducted whereby a more rigorous set of inclusion criteria was applied to identify the final set of papers for the review. Two co-authors independently reviewed each abstract against the following four key inclusion criteria: (1) the study clearly used methodologies common to the field of OR; (2) the problem or research question was of an OR nature; (3) the study was related to a healthcare or public health issue; and (4) the geographic focus was on LMICs and/or regions (Box 2). For each main criterion, at least one of the sub-points had to be true in order for a paper to be included in the review. Papers for which both reviewers were in agreement were automatically included. Discrepancies were resolved through discussion (with a third reviewer if necessary), or by downloading and reading the full-text. Co-authors met periodically during this stage to discuss any uncertainties related to study selection and ensure consistency in applying the criteria. After applying the inclusion criteria, 1099 papers remained – these comprised the set of studies for our global overview.

Box 2 Inclusion criteria for second screen of review process

• a This list does not cover the full extent of OR-type problems, but these criteria describe the types of OR problems of interest for this review b According to World Bank classifications as of July 1, 2014. Low-income economies are defined as those with a gross national income (GNI) per capita of $1045 or less in 2013; middle-income economies are those with a GNI per capita of more than $1045 but less than $12,746; lower-middle- and upper-middle-income economies are separated at a GNI per capita of $4125

To identify papers among the 1099 studies in the global overview that explore the specific theme of health equity, we searched for the following keywords within titles, abstracts and author addresses: (in)equit* , (in)equalit* , pro-poor , poorest , socio-economic , marginalized , stigm* , quintile* , disparit* and gender . Two co-authors assessed each abstract and collaboratively decided if they addressed an issue aligned with the definition of health equity as described by Braveman and Gruskin [ 2 ]. Full-texts for health equity-themed papers were downloaded and read.

Identifying studies for the impact theme was less straightforward. As noted by others who have reviewed OR in global health or LMICs [ 10 , 11 , 23 , 24 ], many OR studies are published before any evidence of having influenced policy- or decision-making has been demonstrated. Thus, it would be misleading to assess the impact of OR studies based solely on a review of published literature. We therefore took the approach of providing select case examples of studies where impact was described in the publication in order to gain insight from their experiences, with the caveat that additional OR studies have likely had an impact on improving global health. For our purposes, ‘impact’ implies that study results meaningfully informed a policy decision, or that model recommendations were implemented in a real-world situation. Full-texts for impact-themed papers were downloaded and read.

Stage 4: Charting the data

‘Charting’ is “ a technique for synthesising and interpreting qualitative data by sifting, charting and sorting material according to key issues and themes ” [ 34 ]. For each of the studies included, the following items of information were charted by one co-author and cross-checked by another: (1) country or region of focus; (2) income classification of that country Footnote 3 – low income, lower-middle income, or upper-middle income; (3) OR methodology or type of OR model developed/used for the analysis; (4) health issue studied (HIV/AIDS, malaria, childhood pneumonia, etc.); (5) application area of global health – Health Systems and Operations, Clinical Medicine, Public Health, or Health Innovation; and (6) level at which the study was targeted – local, national, regional, global or general. These characteristics were gathered from the abstract, but in cases where this information was not clearly stated, the full text was consulted.

The Health Systems and Operations category refers to studies that looked at the logistics related to the provision of services, the allocation of resources or the operations of health facilities. Clinical Medicine was distinguished from public health in that these studies focused primarily on disease diagnosis, treatment or care for the individual patient (e.g. treatment regimens, case management, etc.), whereas Public Health studies emphasised disease prevention and health promotion at the community or population level (e.g. vaccination policy, mass screening, etc.). The Health Innovation category was reserved for studies that explored healthcare innovations or technologies in the pre-implementation stages of development (e.g. vaccines still in early clinical trials and not yet accepted for widespread use, hypothetical future discoveries – in diagnostic, treatment or vaccine technologies). The study’s target level refers to the level at which the model recommendations would be or were intended to be implemented. ‘Regional’ refers to global regions (e.g. sub-Saharan Africa, South-East Asia) and not sub-national regions. The ‘general’ category was reserved for those studies that considered ‘low-resource settings’ as the target in a very general sense or where the level of intended implementation was not clear.

For the most part, studies were easily categorised; however, a small fraction fell into grey areas. Where there was overlap, a determination was made based on what was deemed to be the dominant category. For example, studies that were based on a local setting but were intended to inform national policy- or decision-making were counted towards the national category because proposed changes would be made at the national level. Similarly, some studies bridged clinical and public health (e.g. screen and treat programs). We considered any study with broad public health goals, regardless of whether treatment was included, as public health.

Stage 5: Collating, summarising and reporting results

Based on Arksey and O’Malley [ 34 ], we present our findings in two ways. First, through basic numerical analysis of the extent, nature and distribution of studies across various dimensions (i.e. global overview, application areas), and second by organising a subset of the literature thematically (i.e. for the themes of health equity and impact).

Global overview

In this section, we present a global overview of the 1099 studies that met the inclusion criteria, including a breakdown of OR studies according to country income classifications, geographic regions, year of publication and methodology.

Figure  2 shows the breakdown of OR studies by country income level. The majority of studies (74%) were focused on a specific low, lower-middle, or upper-middle income country; however, several studies (20%) were targeted towards LMICs broadly, and a small proportion (6%) looked specifically at a grouping of countries or regions that spanned several income categories (Fig.  2a ). Among the 817 studies that had a single-country focus (Fig.  2b ), low-income countries made up 17% whereas middle-income countries (lower-middle and upper-middle) made up 83%, with the majority being in the upper-middle income category. Using number of countries and population as benchmarks, Footnote 4 our findings suggest that lower-middle-income countries are under-represented in the literature, upper-middle-income countries are over-represented, and the representation of low-income countries is roughly proportional to these benchmarks. Lower-middle-income countries make up about 34% of all LMICs and 44% of the LMIC population but only 18% of the literature, while upper-middle-income countries make up 40% of all LMICs and 41% of the LMIC population but 65% of the literature. For comparison, low-income countries, which make up 17% of the literature, represent approximately 26% of all LMICs globally and 15% of the LMIC population.

Breakdown of operations research studies according to World Bank income classification of the country of focus – low-income (L), lower-middle-income (LM) or upper-middle-income (UM) – for ( a ) all studies ( n = 1099) including studies about low- and middle-income countries in general or some combination of regions and/or L-, LM- and UM-income countries; and ( b ) studies focused on a single country only ( n = 817)

Figure  3 provides a more detailed geographical view of the distribution of OR studies across the developing world. Almost 40% of the literature reviewed was focused on just six LMICs. China, Brazil and South Africa were the most frequently studied, and collectively accounted for 25.4% of the studies reviewed. India, Mexico and Thailand accounted for 14.5%; all were classified as upper-middle-income countries, except India, which was a lower-middle-income country. These countries represent just 4.4% of all LMICs, but account for about 52% of the total LMIC population. The low-income country most studied in the OR literature was Uganda, with 26 studies. More papers were focused on Asia and South America than sub-Saharan Africa (excluding South Africa). Approximately 50 LMICs were not studied in any of the global health OR publications identified; these countries account for approximately 5% of the total LMIC population, or approximately 303 million people.

Number of operations research studies by country. Note that only studies that focused on a single country ( n = 817) or multiple specific countries ( n = 55) are represented in this figure. Studies that considered multiple countries are counted once for each country represented

As Fig.  4 suggests, low- and lower-middle income countries have historically been less frequent targets for global health-related OR compared to upper-middle-income countries. Despite a steady increase in the absolute number of studies focused on low-income countries since 2000, the proportion of such studies relative to all global health-related OR has plateaued at approximately 14% since the year 2006. This figure also suggests a trend towards more country-specific analyses rather than studies that consider LMICs in general or groupings of countries (see Other category in Fig.  4 ). A possible explanation for the drop in number of papers for 2013 is the lag between when a paper is published versus when it has been indexed in databases. The year 2014 was not included in Fig.  4 since our review does not encompass the entire year.

Proportion of operations research (OR) studies per year in different country income classifications (bars, left axis); low income (L), lower-middle income (LM), upper-middle income (UM) and Other (includes studies targeted at LMICs in general or some combination of L-, LM- and UM-income countries). Total number of OR papers per year also displayed (line, right axis). Note that 2014 was excluded as this review covers studies indexed up until August 2014 only

A breakdown of OR studies according to methodology is shown in Fig.  5 . A wide range of OR methods have been used to study global health issues, and no single method appears to be dominant. In the section that follows, examples of different methods are provided within the context of four application areas of global health.

Breakdown of operations research methodologies. The Stochastic category includes Markov models (e.g. state-transition and decision process models) and Monte Carlo methods. The Mathematical category includes deterministic models, epidemiological compartmental models, and other analytical models described by governing equations. The Other category includes all remaining smaller categories including artificial neural networks, inventory models, spreadsheet models with no analytical formulation, etc. See Box 2 Criterion (1) for additional details about the methodologies included. DEA data envelopment analysis

Global health application areas

In this section, we explore the volume and breadth of OR literature found across two dimensions of global health; the global health application area and the level at which the analysis was targeted (Fig.  6 ). These application areas were chosen because we felt they were broad enough to cover the full gamut of global health challenges. At the same time, studies within categories would carry a similar flavour in the types of problems studied. Other categorisations could also have been appropriate [ 10 , 16 ]. Similarly, we felt it important to distinguish between different levels of focus as the types of problems, analytical approaches, and scale of implementation would be different across these levels. Detailed examples of OR studies in the four areas of global health are described in more detail in the sub-sections that follow.

Percentage of operations research studies ( n = 1099) by application area of global health and by analysis target level (local, national, regional, global or general)

The majority (58.3%) of OR literature explores clinical medicine and public health issues at a national level, with locally-focused health systems and operations studies being the next most frequently studied area (11.0%). Since policies affecting clinical medicine and public health are typically mandated by national ministries of health or implemented by national public health programs, it makes sense that the OR studies in these areas have been targeted at the national level. Although very helpful for exploring the impact of interventions on a macro scale or adding to discussions on global priority-setting, fewer studies were targeted at the regional or global level (12.2%) or towards LMICs in general (6.8%).

Health systems and operations

About 20% of the literature was related to health systems and operations, and most of these studies were focused on the local or national level. At a local level, common analyses included improving the day-to-day operations of health facilities (e.g. patient flow and wait times in health facilities [ 49 , 56 – 60 ] and emergency departments [ 61 – 64 ], facilities layout planning [ 65 , 66 ], inventory planning [ 67 , 68 ], nurse rostering [ 69 – 75 ], and surgical scheduling [ 76 – 79 ]) and health services planning (e.g. location-allocation of emergency medical services [ 80 – 82 ] or new health facilities [ 83 – 86 ]). The majority of these types of problems were analysed using simulation (25.0%) or optimisation (35.0%), or a combination of both methods (3.3%). For example, discrete-event simulation (DES) was used to analyse patient wait times in health clinics in Zambia [ 49 ] and Colombia [ 59 ], and a combination DES-optimisation model was used to study ambulance positioning and response time for an urban city in Brazil [ 81 ].

At the national level, questions related to supply chains and logistics were often explored using methods such as agent-based simulation, DES and other types of microsimulation. For example, a series of recent studies used the HERMES (Highly Extensible Resource for Modeling Event-Driven Supply Chains) software to develop DES models of the vaccine supply chain in Niger and Thailand; researchers explored the impact on vaccine availability of introducing new vaccines into the supply chain [ 87 , 88 ], changing vaccine vial size or replacing multi-dose vials with single-dose vials [ 46 , 89 ], removing the regional level of distribution [ 90 ], and trade-offs between augmenting transport versus increasing cold storage capacity [ 47 , 91 ].

The lack of adequate resources represents a major constraint for health systems in LMICs, thus the efficiency with which available resources are being used was another common theme. A small subset of OR literature (2%) used data envelopment analysis at both a local and national level to analyse the efficiency of health facilities and systems in many low-resource settings. ‘Technical efficiency’ is typically defined as a ratio between a weighted sum of outputs (e.g. number immunisations provided, number of antenatal visits, etc.) and a weighted sum of inputs (e.g. human and financial resources, supplies, beds, etc.); a less than ideal efficiency, as indicated by an ‘envelope’, indicates that a health facility can potentially expand their outputs without changing the quantity of inputs used [ 92 ]. Data envelopment analysis studies have been set in India [ 93 – 95 ], Kenya [ 96 , 97 ], Sierra Leone [ 98 ], Angola [ 92 ] and Zambia [ 99 ], and helped identify inefficiencies in health services delivery, as well as opportunities to better use existing resources. Other studies exploring efficient resource allocation included a simulation model used to provide insight into better resource utilisation (e.g. personnel and physical resources) in an emergency department in Malaysia [ 100 ], and an optimisation model used to explore the optimal allocation of resources in a region in Tanzania given different health objectives (e.g. minimise number of deaths, minimise disease incidence, minimise loss of quality of life, etc.) [ 101 ].

One area of health systems and operations that was not often studied using OR was medical equipment and health technology management. In addition to two previously mentioned studies about increasing vaccine cold storage equipment [ 47 , 91 ], we identified just nine OR studies related to medical equipment. Some examples include a cost-utility analysis of introducing PET scanning technology for lung cancer diagnosis in Iran [ 102 ]; a DES model of a mammography clinic in Brazil that took into account equipment failures and maintenance [ 103 ]; a queuing model developed to improve response and turn-around time of equipment repair work orders in a clinical engineering department in Cuba [ 104 ]; and models to help inform general medical equipment purchasing [ 105 ] and replacement schedules [ 106 ] in LMICs.

Clinical medicine

Most clinical medicine studies were focused at the national level. Common themes were assessing the cost-effectiveness of adopting new treatment or diagnostic strategies, comparing outcomes or cost-effectiveness of competing treatment options, and estimating the benefits of scaling-up treatment access.

Almost 45% (168) of studies in this category were related to HIV/AIDS, malaria or tuberculosis (TB). For example, STDSIM [ 107 ] – a microsimulation decision-support model – has been used in several studies to analyse the impact of expanding anti-retroviral (ARV) access [ 108 , 109 ] as well as treating other curable sexually transmitted infections in order to prevent HIV infection [ 110 – 112 ]. Shillcutt et al. [ 113 ] used a decision-tree model to evaluate the relative cost-effectiveness of presumptive treatment, field standard microscopy, or rapid diagnostic tests for malaria diagnosis in different sub-Saharan African settings. A combination decision analysis and Markov model was used by Mandalakas et al. [ 114 ] to compare the cost-effectiveness of different TB prevention strategies using WHO-recommended isoniazid preventive therapy for children in close contact with infectious TB cases.

Stochastic models, such as Markov models, were common methods for clinical studies, representing almost 26% of the studies in this category. Such models are useful for simulating cohorts of patients with a specific illness as they transition from one disease state to another throughout the course of an illness or even their lifetime. For example, the cost-effectiveness of different treatment options for patients with chronic hepatitis B was studied using Markov disease models in China [ 115 , 116 ], Brazil [ 117 , 118 ], Turkey [ 119 ] and India [ 120 ], over time horizons ranging from 20 to 40 years.

Interestingly, 53 of the 70 studies related to the diagnosis or treatment of cancer, cardiovascular disease or diabetes were published in the past 5 years (between 2009 and 2014), consistent with increased global attention on such non-communicable diseases in LMICs [ 121 , 122 ]. For example, a Markov model was developed to compare the cost-utility, in terms of quality-adjusted life years, of four different treatment options for lung cancer in Thailand [ 123 ]. DES models were used to analyse the cost-effectiveness of saxagliptin as a treatment for type II diabetes in both Argentina [ 124 ] and Brazil [ 125 ]. The treatment of mental health issues is one area that has not been studied extensively with OR – we found only 13 studies in the clinical medicine category that focused on mental illnesses in LMICs such as depression and schizophrenia.

Public health

Public health, specifically at the national level, was the most common global health area explored using OR. Vaccination policies, particularly the introduction of vaccines into routine child immunisation programmes, and other disease prevention strategies such as screening programs (e.g. for cervical cancer), were among the most common types of problems explored.

An example vaccination model is the TRIVAC decision-analysis model from the Pan American Health Organization ProVac initiative, which was used to assess the cost-effectiveness of adding vaccines (e.g. pneumococcal conjugate vaccine, Hib and rotavirus) to the routine child immunisation schedule in LMICs, particularly in Latin America [ 126 , 127 ]. Among preventative public health measures, studies exploring screening and/or vaccination combinations were common. For example, Demarteau et al. explored efficient combinations of cervical cancer prevention strategies (e.g. screening and/or vaccination against human papillomavirus) using a combination Markov and optimisation model, in both Brazil [ 128 ] and Nigeria [ 129 ]. The Markov model estimated the costs and outcomes of different strategies, which was used as input to an optimisation model that determined the combination of prevention strategies that minimised cervical cancer cases for a fixed budget.

Similar to the clinical medicine category, HIV/AIDS, malaria and TB were a common focus for public health studies, with approximately 30% of all studies in this category dedicated to these illnesses. Simulation platforms, such as OpenMalaria [ 130 ] and STDSIM [ 107 ], have provided the modelling foundation for several public health-oriented OR studies related to such illnesses, at both a national and regional level. STDSIM was used to analyse focused public health interventions for high risk groups such as commercial sex workers [ 131 , 132 ]. The OpenMalaria model was used to simulate the impact of interventions such as indoor residual spraying in the highlands of western Kenya [ 133 ].

Global-level studies represented only 2% of studies, and most of these (52%) were in the public health category. Examples of such studies include a model to recommend the required size and resulting cost of an international stockpile of cholera vaccine to enhance efforts to mitigate cholera outbreaks in the wake of natural disasters [ 134 ], and a comparison of the potential impact of rotavirus versus human papillomavirus vaccination across 72 countries eligible for support from the Global Alliance for Vaccines and Immunization (GAVI), taking into account affordability, cost-effectiveness and distributional equity [ 135 ].

One area of disease prevention that lay at the intersection of clinical medicine and public health is the prevention of mother-to-child transmission of HIV. Although some prevention strategies are of a clinical nature (e.g. administering ARVs or nevirapine), we considered this a public health issue as there are other behavioural considerations as well (e.g. recommendations for early weaning or avoidance of breast-feeding). Examples of such studies include a DES model used to evaluate relative benefits of ARVs at childbirth and/or bottle-feeding in Tanzania [ 45 ], a mathematical model comparing different feeding recommendations (i.e. exclusive replacement-feeding versus exclusive breast-feeding for durations of 4 or 6 months) at different compliance levels in Uganda and Kenya [ 136 ], and simulation studies exploring the cost-effectiveness of implementing the WHO’s 2010 guidelines for the elimination of mother-to-child transmission in Zimbabwe [ 137 , 138 ].

Health innovation

Innovation was the least studied category of global health-related OR, with only 47 papers. The majority of these studies (89%) were related to vaccines, either in the early phases of clinical trials or yet to be developed, and were predominantly focused on HIV and malaria. Common themes were modelling the potential impact of imperfect or partially effective vaccines [ 139 – 143 ] or vaccines with rapidly waning protection [ 144 , 145 ], modelling changes in behaviour (i.e. adopting riskier or relaxed behaviour) with the introduction of a newly developed vaccine [ 146 – 149 ], modelling the cost-effectiveness or willingness-to-pay thresholds of a new vaccine [ 150 – 157 ], forecasting demand for a new vaccine [ 158 ], or combinations of these issues [ 159 – 165 ].

Some studies explored the best ways to implement or roll-out a new vaccine should it become available (e.g. through the Expanded Programme on Immunization (EPI), school-based programmes, mass vaccination campaigns, targeted high risk groups, planning for follow-up boosters, etc.) particularly in cases where initial supplies are expected to be limited [ 166 – 171 ], as well as how a partially effective vaccine would measure up against existing prevention strategies [ 172 ] (e.g. male circumcision in the case of HIV [ 173 ] or insecticide-treated nets in the case of malaria [ 174 ]). Lee et al. [ 175 ] used a DES model of Niger’s vaccine supply chain to analyse the impact of developing thermostable versions of six currently available EPI vaccines, an innovation that could relieve bottlenecks in the cold chain. They found that thermostable versions of any of the EPI vaccines, either individually or in combination with other vaccines, would decrease cold storage and transport utilisation and increase the availability of all vaccines, even non-thermostable ones. Levin et al. [ 176 ] also explored thermostable vaccine introduction in Cambodia, Ghana and Bangladesh – their model was a spreadsheet-based decision tree and costing analysis.

Other studies examined innovations in drugs and new diagnostic technologies [ 177 – 179 ]. For example, Dowdy et al. [ 178 ] used a decision analysis model to estimate the cost-effectiveness of a novel point-of-care TB diagnostic tool in comparison to existing methods in South Africa, Brazil and Kenya. Cost-effectiveness was sensitive to the specificity and cost of the new test, but its introduction was estimated to avert almost 50% more disability-adjusted life years per 1000 TB suspects [ 178 ].

The examples provided in this section highlight how OR can be a useful tool for informing health policies and decision-making in low-resource settings – from studies with local health facility-level implications to analyses that are global in scope, exploring issues that span all application areas of global health. We have highlighted areas where there has been a strong OR focus; for example, national-level studies focused on clinical and public health and studies about infectious diseases such as HIV/AIDS, malaria and TB. Areas where OR analyses have been lacking include health technologies and non-vaccine-related innovation, and non-communicable diseases such as cancer, diabetes and mental health.

Health equity theme

This section focuses on an important goal of global health – achieving equity in health for all people worldwide. The challenge in studying health equity is that there is no single way to identify or measure it within a community or population. We felt it would be compelling to discuss how issues of health equity have been analysed using an OR approach, especially given this challenge.

Out of the 1099 papers included in this review, we identified 44 studies that considered health equity as an important part of the research question being explored. Due to our review’s focus on healthcare provision and public health, rather than wider social determinants of health, the studies in this section are primarily focused on healthcare equity, specifically as it relates to socially disadvantaged groups. These studies spanned all four application areas of global health (health systems and operations ( n = 16), clinical medicine ( n = 4), public health ( n = 22) and innovation ( n = 2)) and all target levels (local ( n = 6), national ( n = 29), regional ( n = 3), global (n = 4) and general ( n = 2)). Geographically, studies were predominantly focused on South Africa ( n = 10), China ( n = 5) and India ( n = 4); all other locations had just one or two studies.

Studies differed in how they operationalised (i.e. defined the measurement of) healthcare equity. Some studies defined inequity as a quantifiable disparity in a specific health indicator across different social groups (e.g. mortality risk across wealth quintiles [ 180 ], malaria incidence in children and pregnant women vs. adults [ 181 ]) and estimated how this indicator might change with a more equity-centred approach to an intervention. Other studies parameterised equity as a model variable that ranged between two extremes – from least to most equitable (e.g. percent coverage of an intervention [ 182 , 183 ], measures of spatial accessibility [ 184 ] or a modified Gini coefficient [ 185 ]) – allowing researchers to explore the circumstances under which this parameter was less than ideal or even how to maximise it. Some applied a single ethical principle when operationalising equity (e.g. Wilson et al . [ 183 ] took an egalitarian approach), whereas others explored their research question through multiple ethical lenses [ 185 , 186 ].

We also looked at the distribution of healthcare equity studies across groups with different levels of underlying social advantage/disadvantage, including wealth, geographic location, sex or other social status (Fig.  7 ).

Number of equity-themed OR papers by topic area. Note that some studies looked at equity across several categories; these were counted for each relevant category. Marginalised groups include people living with HIV or other stigmatised illnesses. High risk groups include men who have sex with men, commercial sex workers, or people considered to be in high risk age groups for certain diseases

Accessibility of healthcare, for both the financially and geographically disadvantaged, was a common theme among equity-related papers. The impact of health insurance and/or universal coverage [ 185 , 187 – 189 ], user fees [ 190 ] and subsidies [ 191 ] on equitable healthcare accessibility and affordability was one of the most prominent themes. For example, Waters et al. [ 185 ] used a statistical probit model to analyse the potential impact of a health insurance program and various insurance eligibility standards on both overall access to healthcare, as well as equitable access to healthcare across all economic quintiles in Ecuador. Economic status was also considered by Pagel et al. [ 192 ], who explored how different community-based strategies to prevent post-partum haemorrhage affected women of different economic quintiles in Malawi, and Carrera et al. [ 193 ], who showed that an equity-focused approach to child health that prioritises the poorest and most marginalised populations could lead to higher decreases in child mortality while being more cost-effective than traditional approaches.

Geographic accessibility and distribution of health services, and the identification of geographic disparities in health, were explored by a number of resource location-allocation studies [ 183 , 184 , 186 , 194 – 196 ]. For instance, Moore and Stamm [ 184 ] built a location optimisation model for cholera treatment facilities in Haiti, using the Enhanced Two-Step Floating Catchment Area method. They present their model with five unique objective functions, including one that minimises inequitable access, in order to explore the trade-offs between adequate, equitable and efficient coverage of treatment centres. Similarly, a resource allocation model of a Zambian health service delivery program parameterised equity in the objective of their optimisation model for decision-making based on resource efficiency and equity across varying geographic locations [ 186 ]. In India, a location-allocation model was used to propose new health facility locations for improved geographic access to healthcare [ 194 ].

Health equity issues related to sex [ 146 , 197 – 204 ], high-risk groups (e.g. commercial sex workers) [ 205 – 207 ] or marginalised groups (e.g. people living with HIV) [ 182 , 208 – 212 ] were commonly associated with health issues such as HIV and cardiovascular diseases. For instance, upon recognising that women often lack the power to negotiate safe sex in developing countries and can be exposed to HIV against their will, studies have analysed the effects of post-circumcision changes in male condom use [ 198 , 200 ] and women-initiated vaginal microbicides [ 199 ] on gender health equity in Southern Africa. The issue of high HIV burden among sex workers was analysed using a deterministic model that compared the impact of several interventions, including equitable access to ARVs and community empowerment programs that educate female sex workers about preventive measures against HIV [ 207 ]. Two studies applied an equity lens to mathematical optimisation problems exploring optimal HIV treatment strategies in South Africa [ 182 , 183 ]. Wilson et al. [ 183 ] formulated an ‘equity objective function’ to propose ARV allocation strategies that would ensure each individual infected with HIV has an equal chance of receiving ARVs. Cleary et al. [ 182 ] parameterised the concept of health equity as the percent coverage of treatment in HIV/AIDS patients. By placing different constraints on this parameter in the model, they were able to highlight the trade-off between maximising equity versus maximising health outcomes, where the ‘opportunity cost’ is QALY’s forgone in the former scenario, and higher proportions of unmet need in the latter [ 182 ].

These examples highlight the utility of OR for informing equitable health policy decision-making in low-resource settings. Equity is not a concept easily measured, nor will it be possible to achieve consensus on how it should be measured. A major contribution of OR is that it allows for equity to be quantified in different ways, often within the same modelling framework, such that trade-offs and consequences can be explored more systematically, opening up important discussions about how best to reduce systematic disparities in health for all people worldwide.

Impact theme

In this section, we highlight seven OR studies in which the authors described how their work was implemented or was influential to specific health policy changes or decisions. This compilation is not an exhaustive list; however, studies describing implementation or impact represent a very small fraction of all papers in this review (we estimate less than 10% based on our review of abstracts). In the sub-section that follows, we explore several features of these studies that may have helped contribute to the effective translation of model recommendations into policy or practice, and discuss barriers and challenges to bridging the gap between operations research and health policy.

Case examples of OR impact

The first four studies are examples of impact at the national or global level. Dowdy et al. [ 213 ] used a decision tree model to estimate the cost-effectiveness of serological testing for active TB in India. Serological tests are widely used in India and other developing countries because they are fast, simple and readily available; however, no international guidelines recommend their use over other diagnostic tests such as sputum smear microscopy. The study found that serology tests can result in more secondary infections and false-positive diagnoses, and cost more per-patient, compared to sputum smear microscopy. Their findings, which were presented to a WHO Expert Group on TB in 2010, were influential to the WHO’s subsequent policy statement recommending against the use of commercial serological testing for active TB [ 213 ].

Hutton et al. [ 38 ] developed a combination decision tree and Markov model of hepatitis B infection and progression, which compared options for hepatitis B screening, vaccination and treatment in the United States and China. In China, they found that providing catch-up vaccination for children under 19 would improve health outcomes as well as save healthcare costs in the long run due to the number of infections averted. Their modelling work in 2008 was influential in China’s decision to expand free catch-up vaccination to all children under 15 in April 2009 [ 38 ].

In the wake of a global debate to shift the significant resources being used for polio eradication towards effective control [ 214 ], a systems dynamics disease outbreak model for polio developed by Thompson and Tebbens [ 215 , 216 ] demonstrated that shifting to a control strategy would not only be more costly in the long run, but would lead to more cumulative cases as populations become more susceptible to new outbreaks [ 215 ]. After the results of their model were presented to global stakeholders at a WHO-convened consultation in 2007, experts were convinced that efforts towards completing eradication must continue; for example, the director of the global polio-eradication initiative at the WHO in Geneva commented that Thompson’s work put “ a nail in the coffin for the idea that there is a cheap and painless way out ”, and a representative from the global immunisation program at the United States Centers for Disease Control and Prevention commented that this analysis showed there is no viable control option and that we need to intensify eradication efforts [ 217 ]. As eradication efforts continue today, there is hope that complete eradication can be achieved in 2016; in 2015, there were fewer cases in fewer countries than ever before, and in January 2016, India marked its fifth year without a case of polio [ 218 ].

A DES model was developed by Langley et al. [ 39 ] to evaluate the impact of automated nucleic amplification test (aNAAT), a new TB diagnostic test, compared to existing techniques in Tanzania. The model recommended several combinations of TB diagnostic options incorporating aNAAT testing that were cost-effective in both urban and rural settings. At the time of publication, policymakers in Tanzania were considering specific sites for a trial of the new aNAAT technology, and results from the DES model were going to be used to inform the implementation plan for the trial [ 39 ].

The following three studies are examples of implementation on a more local level. Cruz et al. [ 104 ] developed a queuing simulation model to help enhance medical equipment repair service quality for the clinical engineering department of a 600-bed hospital in Cuba. Simulation results showed that service quality enhancements (i.e. reduced work order backlogs and service times) could be achieved without hiring new personnel. Clinical engineering management implemented two proposed strategies and major service improvements were observed over a 2-year period, as predicted by the model [ 104 ]. Perez et al. [ 59 ] used a combination DES-optimisation model to reduce wait times in the admissions centre of a health centre in Colombia with relatively low additional cost. The solutions proposed by the model were subsequently implemented, and although not explicitly measured, experts in the admissions centre noticed relevant improvements in wait times [ 59 ]. Finally, Friedrich et al. [ 72 ] developed a decision support system (DSS) using linear programming to upgrade the nurse scheduling process at a hospital in South Africa in order to improve the quality of healthcare and nursing services. The model’s objective was to minimise nurse dissatisfaction by better taking into account nurse preferences. Although the system had not been fully implemented at the time of publication, feedback provided through user validation was positive and enthusiastic. Staff managers reported that, in just a few seconds, the system performs the same time consuming computations they carry out manually each month, with improved nurse utilisation and reduced overtime [ 72 ].

Factors contributing to success in ‘bridging the gap’ between OR and impact

Based on these cases, three key drivers for bridging the gap between OR and impact have emerged, namely (1) engagement of local or expert stakeholders in model design and validation, particularly those in policy- or decision-making roles; (2) use of contextually representative data; and (3) a concentrated effort on communication of research findings. All selected cases demonstrated all three of these key drivers even if not explicitly cited in the discussion that follows.

(1) Local or expert stakeholders involved in OR model design and validation

Active participation of local stakeholders has been suggested by others as a key to strengthening health research and policy linkages [ 219 , 220 ] and the examples provided in this section are evidence of this in the field of OR. Such collaborations are important for several reasons. First, the engagement of stakeholders facilitates the identification of relevant and appropriate global health research questions. Thompson and Tebbens advise that “ modelers need to focus on working effectively with the people who need and can use the results ” [ 215 ]. Langley et al. [ 39 ], who underwent a comprehensive review of the questions that policymakers need to address when assessing different TB diagnostic strategies, are a great example of this. Situating their work within this ‘Impact Assessment Framework’ [ 221 ] not only helped identify the questions that their model should answer, but also informed the appropriate choice of modelling methodology to achieve their goals. Identifying relevant research questions is perhaps more easily accomplished for OR studies based on local settings. For example, Friedrich et al. [ 72 ] conducted a root cause analysis of challenges faced at the hospital they wanted to help, and developed their decision-support solution in response to several identified problems around nurse scheduling. Cruz et al. [ 104 ] and Perez et al. [ 59 ] also worked with local collaborators to identify and model relevant problems for the health facilities they worked with.

Second, it is critical to involve local or expert collaborators in the model conception and design because they are intimately knowledgeable about the context, and can help ensure that the model and analysis accurately describes the health issue and addresses the policy questions or decisions they face. This type of collaboration was demonstrated by Hutton et al. [ 38 ], who formed a multi-disciplinary team, including the director of Stanford’s Asian Liver Centre, for their research on hepatitis B in China. They also used an iterative approach to model development, beginning with a very simple representation of patient health states, with details added incrementally based on suggestions from experts until they were satisfied that their model appropriately represented the policy problem [ 38 ]. Friedrich et al. [ 72 ] also underwent an iterative design process, whereby their nurse scheduling DSS model was tested with users and continuously improved throughout development. Langley et al. [ 39 ] worked with experts from Tanzania and Malawi to ensure the input parameters and control logic for their DES model were valid.

Third, engaging local stakeholders throughout the development process can also facilitate trust-building and implicitly lead to capacity building, helping to address the lack of technical ability to interpret findings common in LMICs [ 219 ] and empower the policymakers to take ownership of the process. For example, as a result of the work by Langley et al. [ 39 ], policy advisors in Tanzania have requested the ability to be able to use the simulation model themselves to evaluate alternative diagnostic strategies in the future. A pilot study is underway to demonstrate whether the model is sufficiently user friendly for this type of use [ 39 ]. In the case of Friedrich et al. [ 72 ], the users were pleased that their input was used so extensively in the development of key features of the nurse scheduling DSS, including the interface design and the data validation function that prevents them from entering invalid data; users even requested further training in using the DSS [ 72 ].

If collaboration between local policymakers, researchers and implementers is important for impact, then the lack thereof can be a major barrier to impact. Yet, much of the OR literature reviewed did not have a collaborator or partner in the context where their model was intended to serve. Although some studies did express a desire to use their models and results as a basis for further research in partnership with healthcare organisations, in general, few of the studies in this review mentioned showing (or even the intention of showing) their model or results to relevant stakeholders.

(2) Use of contextually representative data

In addition to having relevant stakeholders involved in the model conception and design, contextually representative data is also likely an important factor in generating OR analyses that have impact or are implementable. The featured case studies are examples of the use of appropriate data. The studies by both Thompson and Tebbens [ 216 ] and Dowdy et al. [ 213 ] were focused on India, and to the extent possible, were populated with input data relevant to the Indian context, from state-level statistics on population, polio incidence, etc. [ 216 ] to costing data obtained from local labs [ 213 ]. These nationally-focused studies were compelling enough to capture the attention of global stakeholders, such as the WHO, leading to broader global implications. Hutton et al. [ 38 ] conducted a comprehensive review of over 250 published papers to populate the data for their model on hepatitis B in China. Input data for the TB diagnostics model in Tanzania came from a range of sources, including the National TB and Leprosy Programme, diagnostic centre laboratory records, and local managers [ 39 ].

For locally-focused studies, especially when local collaborators are involved in the research exercise, it is often possible to prospectively collect the data required for modelling. For example, Cruz et al. [ 104 ] used data collected in the hospital’s electronic technology management system over a 3-year period for both the development of their equipment service simulation model and for the validation of their model recommendations post managerial improvements. For the study to reduce wait times at a health centre in Colombia, the time between patient arrivals and service time in the admissions centre were collected for a short period of time in order to build the simulation model. The model was further validated using admissions data provided by health centre managers [ 59 ].

For many of the other studies in this review, approximations and assumptions were required to estimate certain model parameters. Often researchers had to resort to the use of unrepresentative data, for example data from neighbouring regions or developed countries. These data assumptions and compromises, which are often unavoidable, should be taken into consideration when applying model results and recommendations to a given context.

Availability of reliable data is one of the challenges that sets low-income countries apart from middle-income countries with regards to modelling [ 32 ]. In this review, the studies in some middle-income countries (e.g. China [ 38 ] and Brazil [ 43 ]) were able to make use of centralised hospitalisation information systems and national household surveys, enhancing the validity and robustness of their models and analyses. In fact, access to data may be a reason for location selection on the part of researchers, possibly explaining our finding that low-income countries are less often targets for OR. For example, one study stated “ India was selected for this simulation because it is one of the largest developing countries and sufficient data on breast cancer epidemiology to construct a reliable and valid model were available ” [ 222 ].

(3) Emphasis on communication of research findings

Publication is important, but not sufficient, for the effective communication of research findings, whether from OR or other types of analyses. Communication in various forms beyond the journal publication was an important part of the case studies that influenced change. Perhaps the best example of this was Thompson and Tebben’s work on polio eradication [ 215 ], which they had the opportunity to present at a WHO stakeholder meeting in early 2007. For their presentation, they did not focus on explaining the model, equations, and diagrams in detail, but communicated the key insights in the simplest way possible [ 215 ]. For the study on TB serology testing, two of the authors were affiliated with the Stop TB Partnership’s New Diagnostics Working Group and had the opportunity to present their study findings to a WHO expert panel on TB serological testing, an audience that would be receptive to their work [ 213 ]. One study was actually translated into another language [ 38 ], and for another [ 72 ], user validation interviews were conducted in the local language in order to get the most accurate feedback possible from users.

Many of the OR models had user-friendly interfaces or used visual simulation environments as a means of communicating their model applications and results in a more personalised or accessible fashion. For example, Hutton et al. [ 38 ] specifically used Microsoft Excel because their intent was to develop a model that could easily be shared with policymakers. They “ incorporated sufficient detail to capture important characteristics of hepatitis B disease progression and treatment so that the model would be believable to a clinical audience ” [ 38 ] but tried to keep it simple enough so that those who lack modelling expertise could easily understand it. Langley et al. [ 39 ] also stressed the importance of a visual representation of the modelled processes in order to improve engagement and assist in the validation of their model with experts in Tanzania; the simulation software they chose afforded this possibility. The output of the nurse scheduling DSS model was formatted similarly to the hospital’s previous manual scheduling process so that unit managers would more easily adopt and transition to the new system [ 72 ].

Overall, the continued expansion of research reach and influence requires sustained efforts to communicate findings through different channels, with engaged outreach to, and personal connections with, policymakers and public health officials.

“ Scoping studies aim to map the literature on a particular topic or research area and provide an opportunity to identify key concepts; gaps in the research; and types and sources of evidence to inform practice, policy-making, and research ” [ 35 ]. The goal of this scoping review was to provide a broad overview of the use of OR in global health, with several concrete examples showing the breadth and depth of how this field of research is being applied to important global health challenges worldwide. We also explored the theme of health equity, demonstrating the unique opportunities the field of OR can contribute to this increasingly important area of global health. Cases where OR has had an impact on policy- or decision-making were also highlighted, with examples ranging from the implementation of local-level changes related to the day-to-day operations of health facilities, to decisions about national vaccination policies, to influencing international WHO policies and global perceptions about disease eradication. These cases serve as excellent examples of the importance of collaboration, data and communication for affecting change at the local and global level.

Limitations and challenges of the review

We faced some challenges and limitations when conducting this review. Given the broad interpretations of what constitutes OR, a lack of consistent terminology for OR, and the variety of journals where OR literature in healthcare tends to be published, our search terms were broad in reach and scope, with the consequence that a large amount of literature was captured that was not relevant. Additionally, the geographic search tailored to LMICs was not straightforward; country names are not always considered controlled vocabulary, so every individual country name had to be included as keywords.

We also had to be pragmatic at the outset about the coverage of the review. We chose to focus on papers published in the year 2000 and later. As such, there is some overlap with the hand-searched literature [ 11 , 16 , 17 , 23 , 24 , 26 – 28 , 32 ], but not with the review of OR in global health by Datta [ 10 ], which at the time of our review was over 20 years old. To fill this gap would have been an onerous task. Due to rapid advancements in computing technology, it could be argued that OR models developed before 2000 are out-dated, as are any global health data used to populate them. Further, we also had to be selective when setting the inclusion criteria for the types of ‘OR’ and ‘health’ studies explored given these terms have such broad definitions. We restricted OR studies to those where modelling or analytical methods were used with an orientation towards decision-making. Our health focus was largely on healthcare provision and public health, and did not include the wider social determinants of health; this focus was inherently reflected in the subset of studies explored for the health equity-theme as well. As it was, due to the volume of literature included, we could not summarise or cite all of the studies found; however, we hope this review has provided enough of a landscape overview to prompt further exploration of the utility of OR in this context. A complete database of the 1099 studies is provided as Additional file 2 .

Income categories for countries were based on 2014 World Bank classifications, regardless of how a country may have been classified historically. It would have been difficult to track shifts in income classification for every country and every paper included in this review. Furthermore, we felt the interpretation would be simpler knowing that all studies related to a specific country (e.g. Brazil) were consistently counted towards its current income category (e.g. upper-middle) rather than split across multiple categories.

Our criteria for ‘impact’ when selecting case examples was that the study meaningfully informed a policy decision or the recommendations were implemented in a real-world setting. We were unable to make any inferences about the magnitude of improvement in health that may result from these changes. Only Hutton et al. [ 38 ] presented estimates that 170 million children would be vaccinated for hepatitis B in China as a result of their model recommendations, preventing almost 8 million infections and 70,000 deaths, and saving the equivalent of $1.4 billion over the lifetime of these children.

A final limitation of this review was the restricted search for published literature alone. Those papers that did not describe a particular policy change could have indeed influenced decision-making after their publication. Future work could involve searching grey literature for case studies and policy documents suggesting that knowledge gained from OR was influential in decision-making processes. The lack of published evidence that OR is impacting policy change represents a major missed opportunity for the academic community to learn and better engage in impactful OR work. It has been mentioned that fora are needed where these findings can be discussed [ 219 ] and success stories of policy transfer shared with a broader community [ 14 ], if not in peer-reviewed literature, then elsewhere.

In general, scoping reviews take a considerable amount of time and skill. Balancing feasibility, breadth and comprehensiveness can be a challenge given available time, funding and resources [ 35 , 37 ]. Although not explicitly tracked, we thought it would be informative to provide an estimate of the amount of time that it took to conduct our review. The most time intensive stages were stages 2 to 4 – designing the search, identifying and selecting studies and charting data. Stage 2 was conducted over a period of approximately 4 months by one author on a near full-time basis (estimated 500 person-hours), and stages 3 and 4 were carried out by four co-authors over a period of approximately 8 months, all on a part-time basis (estimated 340 person-hours). This is consistent with the findings of Pham et al. [ 36 ], who reported scoping reviews have taken anywhere from 2 weeks to 20 months to complete.

Global overview and global health application areas

Despite these limitations and challenges, this scoping review, consisting of 1099 studies, is to our knowledge the most comprehensive review of OR in global health to date. Our overview highlighted that low-income countries are less frequently studied using OR compared to middle-income countries, a trend that does not seem to be improving with time. Furthermore, a large proportion of healthcare-related OR in LMICs has focused on just six middle-income countries. If population is an appropriate yardstick for research focus, then perhaps this representation is reasonable; however, the disparity between volume of literature and number of countries in each income category is much more pronounced. That being said, 84 LMICs around the globe have been the focus of at least one OR study since 2000; hopefully, increased global coverage will continue.

Although the aggregate data did not show that any particular OR method was dominant, we found that certain types of research questions were more amenable to specific OR methods than others (e.g. local and national-level health systems problems were commonly studied with simulation or optimisation methods). This highlights the importance of a collaborative and interdisciplinary approach to applying OR in global health, such that those with modelling expertise in specific methods can apply their expertise where it can make the most impact.

We found that the majority of OR literature explores clinical medicine and public health issues at a national level, and that, although very helpful for exploring the impact of interventions on a macro scale or adding to discussions on global priority-setting, fewer studies were targeted at the regional or global level. Since OR models tend to describe the dynamics and interdependencies of actual systems, it likely gets more difficult to develop accurate models as complexity increases (i.e. from local or national systems to regional or global systems). OR models are also highly dependent on input data, which is likely easier to obtain at the local and/or national level. Studies targeted at several countries or a whole region have themselves cautioned that more specific country-level analyses with more representative data are needed for country-level decision-making, which is perhaps why few studies are targeted at these levels [ 135 , 193 ].

One area of global health that OR has made a unique contribution to is the analysis of new health innovations. This small subset of studies highlights the value of OR for analysing global health interventions that cannot yet be trialled or implemented on the ground because the scientific breakthroughs have yet to be achieved. Understanding the potential impact or possible implementation challenges of new innovations is important to their successful roll-out when they do become available. OR can also help highlight important design criteria targets for the development of new innovations, in terms of both cost and minimum levels of efficacy or specificity required to achieve desired outcomes.

Infectious diseases (e.g. HIV/AIDS, malaria, TB) continue to be a major focus of OR globally; however, there has been an increase in the number of OR studies about non-communicable diseases over the past 5 years. Neglected areas representing an opportunity for future OR include analyses focused on low- and lower-middle income countries, non-communicable diseases (particularly mental health), and medical equipment and technology planning.

Additional study characteristics that could have been analysed include the funding sources, the academic institutions of the lead investigators and the quality of the studies themselves. Funding sources have been identified as a potential external influence on both research and policy agendas, but in some cases resources dedicated to an OR study can lead to positive change. For example, two studies from this review [ 43 , 223 ] were funded by the Brazilian Ministry of Health [ 223 ], indicating interest on the part of the government to use OR as a tool to answer a key health systems-related questions. Exploring funding sources in more detail could be an area of future consideration. The quality of the studies themselves could also be highly relevant to successful research uptake, but we did not undertake critical appraisal for this review as quality assessment does not typically form part of the scoping study remit [ 34 ]. Others have reported on the quality of simulation models in healthcare by applying strict quality criteria during the review process [ 18 ].

Health equity: an opportunity for future OR

The healthcare equity-themed studies featured in this review demonstrate the utility of OR for informing equitable health policy decision-making in low-resource settings. These studies, however, represented a relatively minor proportion (4%) of all global health-related OR in the time period studied. Some have argued that a major shortcoming of the Millennium Development Goals was a failure to address equity [ 6 ], and that “ looking forward, equity analyses and actions need to be an integral part of programme strategies rather than an afterthought ” [ 7 ]. We believe a huge opportunity exists to apply the tools and techniques of OR to study health equity in the post-2015 era.

First, OR is extremely versatile in how the concept of health equity can be operationalised. The examples in this review demonstrate a variety of different ways to quantify equity - from measured disparities in specific health indicators to parameterised model variables. Although beyond the scope of this review, these principles can be used to explore the broader social determinants of health as well. As no single measure is sufficient to assess inequities, those applying OR to health equity could benefit from integrating established health equity frameworks into their approach (e.g. the PROGRESS [ 224 , 225 ] and PROGRESS-Plus [ 226 ] frameworks) to help ensure the explicit consideration of important equity factors in the design of OR models and analyses. Second, OR allows for the comparison of different equity goals (e.g. using different ethical principles or comparing efficiency versus equity), often within the same modelling framework. For example, a utilitarian perspective aims to maximise overall societal benefit, whereas an egalitarian approach would strive to achieve equal distribution of, or access to, resources for every person [ 183 ]. OR allows for a more systematic analysis of the trade-offs and consequences of viewing equity from these different perspectives. Third, OR models can study the effect that different policies or decisions might have on marginalised populations without the ethical implications of a real-world study. Arguments could be made that, to achieve health equity, certain groups should be valued over others; for example, perhaps high-risk groups or the least advantaged should be prioritised, rather than treat all cases alike regardless of social standing. OR can aid in testing such sensitive policy hypotheses a priori without unintentional consequences.

OR can be applied at national and sub-national levels, across different socioeconomic groups and marginalised populations, and through different ethical lenses, in order to inform interventions and health policy decisions that will promote better equity in health going forward. We hope the equity-themed literature highlighted in this review can help open up important discussions about how best to model and analyse systemic disparities in health for all people worldwide.

OR impact: recommendations for bridging the gap between OR and policy or practice

Studies describing OR implementation or impact represent a small proportion of the literature reviewed – there is still a far way to go for OR to reach its full potential in global health. Our finding that few papers present details of implementation or impact is consistent with the experience of others who have reviewed OR in developing countries [ 10 , 11 , 23 , 24 ]; for example, Datta [ 10 ] reported that less than 5% of studies reviewed discussed implementation. From the numerous studies included in this review that did not appear to have any influence on policy or decision-making, several persistent challenges emerged as common themes, including lack of local or expert stakeholder engagement in model conception and design, challenges in acquiring reliable and representative data, and a lack of communication strategy beyond the journal publication.

The lack of appropriate packaging of research findings or exclusive dissemination within academic circles was also found by others to be barriers to research uptake in LMICs [ 219 , 227 ]. In recent years, several theories, frameworks and practical handbooks or ‘toolkits’ have been developed by agencies such as the Overseas Development Institute [ 228 – 230 ], the International Development Research Council [ 231 ], the WHO [ 232 ], and the Institute of Development Studies [ 233 ], to help guide and make more effective the translation of research into policy. Specifically, there is a focus in this literature on the effective communication of research findings, which extends beyond just communication products (e.g. policy brief, stories of change, etc.) to a whole body of research on knowledge sharing, knowledge transfer and knowledge translation. The use of packaging and language that are more appropriate and targeted towards implementation can help enhance the impact of OR.

Yet, there have been many success stories of global health research effectively bridging the research-policy gap [ 13 , 234 ]. For example, Zachariah et al. [ 13 ] identified several impactful operational research studies that had implications for policy and practice; however, all of these studies were field studies that did not involve modelling and thus did not meet the inclusion criteria for this review. Additional guidance to be gleaned from these studies for the OR modelling community include (1) the generation of research questions from within existing programs, which are focused and of simple design; (2) working with partners to ensure that sufficient resources (human and financial) are available for an engaged and motivated research process that extends beyond models and analyses; (3) setting realistic expectations of research impact; (4) investing in long-term research and policymaker relationships; and (5) helping build capacity of end-users to use research to demand policy change [ 13 , 234 ].

Promisingly, within the OR community, there is a growing movement towards impact-driven research and publication. The “Doing Good with Good OR” paper series and research award, offered by the Institute for Operations Research and the Management Sciences, and the “OR in Development” prize, offered by the International Federation of Operational Research Societies, are efforts to recognise OR for the impact of the analysis, in addition to its analytical rigor. In general, there is a need for incentivising the engagement of researchers in problems that are relevant and timely to important policy issues [ 219 ]. Hopefully, these efforts, paired with efforts within developing countries to increase end-user capacity to use OR [ 11 , 25 ] will help bridge the gap between OR and impact in LMICs.

There is a tremendous opportunity for OR researchers and global health practitioners alike to continue to apply OR in global health, particularly in areas where such studies may currently be lacking. We hope the findings of this scoping review, which represents the most comprehensive compilation of OR literature in global health to date, are of interest to a wide-ranging group of stakeholders engaged in global health policy and practice. For government bodies and administrators of health programs and services, we hope to have showcased the utility of the OR approach in modelling policy and programme changes to improve efficiency, particularly when resources are limited. We also hope funders of international development research see value in allocating funding to operations research within broader global health programs. We hope those currently engaged in OR can benefit from the impactful studies highlighted in this review, and we encourage them to share the impact of their work more broadly so that others can learn from challenges and successes.

The optional ‘Consultation Exercise’ of the framework was not conducted.

Individualised search strategies were based on whether the database was indexed by subject headings, controlled vocabulary or keywords.

World Bank classification as of July 1, 2014, not the year the study was published. Note that some countries may have shifted categories since 2000.

Based on World Bank country and population listings [ 235 ]. There were 136 low- and middle-income countries as of 2014.

Abbreviations

automated nucleic amplification test

anti-retroviral

discrete-event simulation

decision support system

Expanded Programme on Immunization

low- and middle-income countries

operations research

tuberculosis.

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BDB conceived the study idea, designed the systematic search, conducted the initial screen of search results, reviewed papers for the second screen of search results, categorised papers, contributed to the interpretation of data, and drafted the first manuscript. TJ, AT-V, BK reviewed papers for the second screen of search results and categorised papers, and TJ contributed to writing the paper. TJ, TCYC and Y-LC contributed to the study design and interpretation of data, and critically revised the draft. AT-V and BK provided comments on the draft. All authors read and approved the final manuscript.

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Bradley, B.D., Jung, T., Tandon-Verma, A. et al. Operations research in global health: a scoping review with a focus on the themes of health equity and impact. Health Res Policy Sys 15 , 32 (2017). https://doi.org/10.1186/s12961-017-0187-7

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Management Notes

Operation Research Models – 8 Common Models Explained in Detail | Operations Management

Operation research models.

Operational Research (OR) Models, also known as Management Science Models and Decision Science Models, are mathematical and analytical methods used to answer complex questions and make informed decisions in many fields, including business, engineering, healthcare, logistics, and finance.

By formulating real-world problems as mathematical equations or algorithms, OR models allow decision-makers to find the best solutions under given constraints, optimizing processes, resources, and outcomes.

It is the main objective of OR models to maximize profits, minimize costs, improve efficiency, and maximize overall performance. Decision-making situations involving multiple variables, uncertainties, and constraints need to be considered simultaneously using these models. There are several types of OR models, each suited for a different type of problem. Here are some of the most common types of OR models:

1. Linear Programming (LP) Model:

Linear Programming (LP) is one of the most widely used and prominent OR models. A linear equation represents the relationship between a decision variable and an objective/constraint when the objective function and constraints are all linear.

Profit, cost, utility, or any other relevant metric is typically represented by a linear function, and the objective of LP is to maximize or minimize it. Constraints limit the possible values of these variables, reflecting real-world limitations on resources and capacity, while decision variables represent the quantities to be determined.

A variety of fields utilize LP, including production planning, supply chain optimization, portfolio optimization, resource allocation, and transportation planning. In 1947, George Dantzig developed the Simplex Method, a popular algorithm for solving linear programming problems.

2. Integer Programming (IP) Model:

The concept of integer programming is an extension of linear programming that deals with problems where the decision variables must have integer values, i.e., solutions must be whole numbers, not fractions.

If a decision involves discrete choices, such as selecting a facility location, assigning workers tasks, or determining the number of units to be produced, IP models are particularly useful. Among the applications are project selection, workforce scheduling, and routing.

Since IP problems have discrete variables, solving them is more challenging and computationally intensive than solving LP problems. Algorithms such as Branch and Bound and Cutting Plane are often used to find optimal or near-optimal solutions.

3. Non-Linear Programming Model:

The concept of nonlinear programming refers to problems in which the objective function or constraints are nonlinear. Unlike linear relationships, these problems involve nonlinear equations that may not be easily solved analytically.

There are many applications for non-linear programming models, including engineering design, portfolio optimization, financial planning, and resource management. Iterative methods like Gradient Descent or Newton’s method are often used to solve non-linear programming problems, where successive approximations lead to the optimal solution.

4. Network Models:

A network model is a type of OR model that focuses on problems involving interconnected elements or networks. These models are widely used in the transportation industry, project scheduling, and supply chain logistics, among other applications.

The following are common network models:

a. Shortest Path Problem:

The shortest path problem aims to find the shortest path between two nodes in a network, taking into account distances, costs, or transit times.

b. Max Flow-Min Cut Problem:

This is a problem that determines the maximum flow that can be sent through a network from a source node to a sink node while minimizing the cut (the minimum capacity of edges to disconnect source and sink).

c. Critical Path Method (CPM):

The CPM method is used to determine the critical path, i.e. the sequence of tasks that must be completed in order to avoid project delays.

It is possible to optimize resource utilization, routing, and scheduling in complex systems by using network models.

5. Queuing Models:

A queueing model analyzes the lines or queues in various systems, including customer service centers, manufacturing facilities, and healthcare facilities. Using these models, service levels can be optimized, waiting times minimized, and resources allocated more efficiently.

When organizations understand the dynamics of queueing systems, they can enhance customer satisfaction and operational efficiency. Queuing models consider factors such as arrival rates, service rates, and the number of servers.

6. Simulation Models:

A simulation model is another group of OR models used to reproduce real-world processes through computer-based models. Simulations allow decision-makers to see how systems behave under different circumstances.

In product design, risk analysis, financial planning, and supply chain optimization, simulation models are particularly useful when real-world experiments would be either too expensive, risky, or time-consuming.

7. Markov Decision Process (MDP) Models:

The MDP model is used for decision-making in uncertain environments. In such situations, the outcomes are probabilistic, and the decision-maker aims to select actions that maximize long-term rewards or minimize long-term costs.

Artificial intelligence and reinforcement learning applications use MDPs to teach agents how to interact with environments and optimize their decisions.

8. Heuristic Models:

In a heuristic model, the solution is not guaranteed to be optimal, but it is good and efficient and can be completed in a reasonable amount of time. For large-scale and complex problems, where finding exact solutions is computationally infeasible, these models are particularly useful.

As a rule-of-thumb strategy, heuristics help narrow down the search space to find satisfactory solutions by guiding the search. Although heuristics do not guarantee optimality, they are an effective tool for tackling real-world problems and delivering practical results.

Operation Research (OR) Models have become indispensable tools for modern decision-making. In complex, dynamic environments, OR models assist organizations in optimizing resources, improving efficiency, and making informed choices by leveraging mathematical and analytical techniques.

In addition to linear programming and integer programming, non-linear programming, network models, queueing models, simulation models, and more, each type of OR model offers unique insights into specific types of problems.

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Approaches combining methods of Operational Research with Business Process Model and Notation: A systematic review

Hana tomaskova.

1 University of Hradec Kralove, Faculty of Informatics and Management, Hradec Kralove, Czech Republic

Gerhard-Wilhelm Weber

2 Faculty of Engineering Management, Poznan University of Technology, Poznan, Poland

3 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

Associated Data

The following information was supplied regarding data availability:

This article does not contain data or code because it is a literature review.

Business process modelling is increasingly used not only by the companies’ management but also by scientists dealing with process models. Process modeling is seldom done without decision-making nodes, which is why operational research methods are increasingly included in the process analyses.

This systematic literature review aimed to provide a detailed and comprehensive description of the relevant aspects of used operational research techniques in Business Process Model and Notation (BPMN) model.

The Web Of Science of Clarivate Analytics was searched for 128 studies of that used operation research techniques and business process model and notation, published in English between 1 January 2004 and 18 May 2020. The inclusion criteria were as follows: Use of Operational Research methods in conjunction with the BPMN, and is available in full-text format. Articles were not excluded based on methodological quality. The background information of the included studies, as well as specific information on the used approaches, were extracted.

In this research, thirty-six studies were included and considered. A total of 11 specific methods falling into the field of Operations Research have been identified, and their use in connection with the process model was described.

Operational research methods are a useful complement to BPMN process analysis. It serves not only to analyze the probability of the process, its economic and personnel demands but also for process reengineering.

Introduction

It has been more than 15 years since ‘Business Process Model and Notation’ or ‘Business Process Modelling Notation’ (BPMN) became the official notation for process modelling. During its lifetime, this notation has gained many users and, thanks to its user-friendliness, it is used in many areas. This wide usage has led to the interconnection and use of other technologies and methods. The fundamental problem of any complex process is decision making. Operational Research as a popular scientific approach is so often associated with procedural issues, making its connection to BPMN is more than natural. This article focuses on the analysis of the relationship between the Business Process Model and Notation (BPMN) process modelling and specific methods of Operational Research.

Business Process Modelling Notation was created by the Business Process Management Initiative (BPMI) as an open standards. It is very similar to flowcharts and Petri nets but offers much more sophisticated tools to describe and simulate behaviour. Silver (2009) stated that this approach is an ‘event-triggered behaviour,’ a description of the ‘something happened’ mode. Business Process Modelling is used to describe, recognize, re-engineer, or improve processes or practices, Tomaskova (2017) . Business Process Model and Notation (BPMN) is the language that is used to model business process steps from start to end. The notation was explicitly designed for wide-ranging use in process analysis, The Object Management Group (2011) . BPMN is both intelligible to non-specialists and allows a complicated processes between different participants to be represented. Another, very significant feature of BPMN is its ‘business-friendly’ orientation, which is essential for the company’s business and knowledge.

Operational Research (OR) is concerned with formulating, modelling, and solving a variety of decision-making situations to find the optimal solutions. The company’s philosophy and decide over business data are the most crucial management actions. The task of the manager is to select in the real system the problem to be analyzed and to formulate it precisely. The standard way of doing this involves the expression of the economic model and then the formulation of a mathematical model. It is necessary to build a simplified model of the real financial systems that only includes the essential elements that describe the formulated problem. The manager has to set the goal of the analysis and subsequent optimization. It is important to define all operations and processes that influence this goal, to describe all the factors, and to verbally express the relationships between the stated purpose and the mentioned processes and factors.

The article is divided into the following parts. The “Related works and background” section lists research articles that are relevant to a given combination of BPMN and OR areas and briefly. That part briefly provides essential information regarding the approaches that are fundamental to this systematic review. The “Research methodology” section describes a systematic search, i.e. entry conditions, exclusion criteria and limitations. The “Results” section presents the results of the analysis of articles fulfilling the requirements of the systematic review. We analyzed publications according to when they were published, their citations, the scientific areas covered, the cooperation of the authors and their keywords. Subsequently, we examined selected articles in terms of methodology, approach and research areas. In the “Discussion”, we focus on scientific gaps and future research. We present a research area where we expect an increase in publications, including their specific components. We also discuss the future development of applied methods and approaches. Finally, the “Conclusion” section summarizes the results and benefits of this study.

Related Works and Background

The background information and related works are listed in the paragraphs below. We first focused on process modelling and BPMN and then on OR and its essential methods and approaches.

Organizational processes and decision support can be captured in many ways, and for many areas, we can mention, for example: strategic management by: Maltz & Kohli (1996) , Certo (2003) , Tomaskova (2009) , Maresova (2010) , Tsakalidis et al. (2019) ; product development research and innovation implementation, see Repenning, 2002 , Garcia (2005) ; IT and economic analyzes see Shane & Cable (2002) , Dedrick, Gurbaxani & Kraemer (2003) , Krenek et al. (2014) , Tomaskova, Kuhnova & Kuca (2015) , Maresova, Tomaskova & Kuca (2016) , Tomaskova et al. (2016) , Maresova, Sobeslav & Krejcar (2017) , Cheng et al. (2019) , Tomaskova, Kopecky & Maresova (2019) , Tomaskova et al. (2019) , Kopecky & Tomaskova (2019) , Kopecky & Tomaskova (2020) ; different simulation approaches analysis, see Sterman (1994) , Kozlowski et al. (2013) , Cimler et al. (2018) or non-standard optimization techniques by: Gavalec & Tomaskova (2010) , Bacovsky, Gavalec & Tomaskova (2013) , Tomaskova & Gavalec (2013 , 2014 ), Gavalec, Plavka & Tomaskova (2014) , Gavalec, Mls & Tomaskova (2015) , Cimler et al. (2017) , Oudah, Jabeen & Dixon (2018) .

Some authors have attempted to provide a solution for process model analysis. For example Melao & Pidd (2000) discussed the strengths and limitations of the various modelling approaches used in business process transformation. The article by Glassey (2008) compares three process modelling processes used in case studies. The article by Sadiq & Orlowska (2000) analyze process models using graph reduction techniques. Other authors like Van der Aalst et al. (2007) , Krogstie, Sindre & Jorgensen (2006) use specific tools, frameworks and methods for process analysis and modelling.

Business process modelling

Today, process modelling and business process management (BPMN) have a significant impact. Process modelling is currently a mainly graphical representation of processes, e.g. in what order particular activities should be implemented and what inputs and outputs the processes require for proper functioning. The primary goal of process modelling is to increase the efficiency and effectiveness of the entire process as well as partial activities. Many business process modelling techniques have been proposed over the last decades, so the article Recker et al. (2009) comparatively assesses representational analyses of 12 popular process modelling techniques to provide insights into the extent to which they differ from each other. The review business process modelling literature and describe the leading process modelling techniques falling to and before 2004 are published in the articl Aguilar-Saven (2004) . The topic of visualization of business process model has been investigated in publication Dani, Freitas & Thom (2019) , where the authors performed a systematic literature review of the topic “visualization of business process models”. Kalogirou (2003) is a particularly fascinating article that illustrates how AI techniques might play an essential role in the modelling and prediction of the performance and control of the combustion process. Although BPM initially focused mainly on the industrial, service and business sectors, it has also appeared in other sectors in recent years. The popularity of BPMN has been confirmed by articles such as Zarour et al. (2019) , which presents the current state-of-the-art of BPN extensions. Publication De Ramon Fernandez, Ruiz Fernandez & Sabuco Garcia (2019) deals with the optimization of clinical processes.

Business process model and notation

Business process model and notation is a language for creating business process models Silver (2009) . Under the auspices of the Object Management Group (OMG), the Business Process Management Initiative (BPMI) created the BPMN as an open standard in 2004 by the first version 1.0. In 2005, BPMI merged with the Object Management Group (OMG), and the following year, the latter issued the BPMN specification document. In 2010, BPMN version 2.0 was developed, and the current version of BPMN 2.0.2 was released in December 2013. History of BPMN and notation development is a frequent topic of BPMN publications, we can mention Nisler & Tomaskova (2017) , Kocbek et al. (2015) , Chinosi & Trombetta (2012) , White (2008) , Van der Aalst, Adriansyah & Van Dongen (2012) and Recker (2012) . BPMN is similar to flowcharts and is based on the concept of Petri nets, but it is a more sophisticated and user-friendly language. The graphic form of BPMN makes it understandable even for non-experts. In BPMN, we distinguish several types of elements that we can use in modelling. The specific standards link these elements. In the base classification, we define four groups of items. These are Flow Objects, Connecting Objects, Swimlanes and Artifacts, see The Object Management Group (2011) .

Operational Reserach

Operational Research (OR) is the well-known approach of using analytical and advanced methods to help make the best possible decisions. As early as 1980, Article by authors Shannon, Long & Buckles (1980) presented the results of a survey of the perception of the usefulness and knowledge of the 12 OR methodologies commonly used in the practice of industrial engineering. The article by Dubey (2010) defines the relationship between OR and another branch of sciences. The article Gu, Goetschalckx & McGinnis (2010) presents a detailed survey of the research on warehouse design, performance evaluation, practical case studies, and computational support tools. The article Negahban & Smith (2014) provided a review of discrete event simulation publications with a particular focus on applications in manufacturing.

OR methods are often associated with new technologies. In article Sarac, Absi & Dauzère-Pérès (2010) , a state-of-the-art on RFID technology deployments in supply chains was given to analyze the impact on the supply chain performance. Xu, Wang & Newman (2011) , in their article, tries to identify future trends of computer-aided process planning (CAPP). Dynamic ride-share systems is investigated in the article Agatz et al. (2012) .

Linear programming

One of the most popular areas of OR in practice is linear programming (LP). The mathematical model of linear programming tasks contains a single linear purpose function, and the actual constraints of the problem are described only by linear equations and inequalities. These tasks are most often encountered in economic practices. Linear programming has been described in several books: Dantzig (1998) , Schrijver (1998) , Dorfman, Samuelson & Solow (1987) .

Multicriterial decision making

The solving of multi-criteria decision-making (MCDM) tasks comprises the search for optimal values of the unknowns, which are simultaneously assessed according to several often contradictory criteria. Thus, the mathematical model of multi-criteria decision problems contains several purpose functions. Depending on how the sets of decision variants are defined, we are talking about the tasks of multi-criteria linear programming or multi-criteria evaluation of options. A review of applications of Analytic Hierarchy Process in operational management is inverstigated in Subramanian & Ramanathan (2012) . The article Velasquez & Hester (2013) performs a literature review of common Multi-Criteria Decision Making methods. The authors present the results of a bibliometric-based survey on AHP and TOPSIS techniques in publication Zyoud & Fuchs-Hanusch (2017) .

Project planning

Project management tasks consist of several separate activities that are interdependent and may be run simultaneously. The most commonly used method is the so-called network analysis, where a network graph is created from the left chronologically arranged project activities representing the project life cycle. The longest possible path from the beginning to the end of the project is recorded by “the critical path”. The non-observance of this path will lead to a slowing down of the whole project, whose time duration is to be optimized. The optimistic, pessimistic, and most probable estimate of the implementation of the entire project is determined. The article Nutt (1983) relates the project planning process and implementation. Critical Path Method (CPM) is found in the article Jaafari (1984) , to be equally useful as a planning tool for linear or repetitive projects.

The Resource-Constrained Project Scheduling Problem (RCPSP) is a general problem in scheduling. The article Pellerin, Perrier & Berthaut (2020) examines the general tendency of moving from pure metaheuristic methods to solving the RCPSP to hybrid methods that rely on different metaheuristic strategies ( Cimr, Cimler & Tomaskova, 2018 ).

Nonlinear programming

Nonlinear programming is the case when the purpose function is not linear. Tasks then often have a large number of local extremes and often also have great difficulty finding them.

Dynamic programming

If constraints are functions of some parameter, which is most often time, we are talking about dynamic programming. This approach deals with the modelling of more complex multi-stage optimization problems divisible into related sub-problems. Depending on the time parameter, the system is always in one of the acceptable states during the process. At certain times it is necessary to choose from a set of possible decisions, which again results in the transition to the next state. We call the strategy a sequence of these states of the system and choices, looking for the course with the best valuation. Simulations are often used to model and analyze the operation of complex systems without realization and in less than real-time.

• Queuing theory is a type of dynamic programming task. It deals with streamlining the functioning of systems in which it is necessary to gradually serve all units whose requirements are continuously met on so-called service lines. The challenge is to find the most effective way to handle these requirements.
• Inventory management models address the issue of optimizing the supply process and the volume of inventory stored. Costs associated with ordering, issuing, and keeping stocks in stock should be minimized.

Stochastic programming

Stochastic programming deals with optimization problems in which they act as parameters of their constraints of random variables. Probabilistic calculus methods solve these problems, and their results have the character of random variables. Stochastic processes can also be ranked among tasks with the input data uncertainties. This approach is used to describe the behavior of systems evolving. We are talking about stochastic processes, a special case is the so-called Markov chains and Markov processes. Basic books on this topic are, for example: Kall, Wallace & Kall (1994) , Birge & Louveaux (2011) , Shapiro, Dentcheva & Ruszczyński (2014) .

Research Methodology

Kitchenham & Charters (2007) highlighted three essential elements for a systematic literary review: the determination of the research question(s), the organisation of an unbiased and extensive analysis of related publications, and the determination of precise criteria of inclusion and exclusion.

We identified three research questions:

• Research question 1 (R1): Greater adaptability of BPMN elements causes greater application of this notation in publications.
• Research question 2 (R2): The connection between BPMN and OR methods is most often applied to the business and economics areas.
• Research question 3 (R3): The queue theory is the most widely used method in BPMN processes.

The analysis process and criteria are given in the following relevant subsections.

Eligibility criteria

This study included publications listed in the Web Of Science (WOS) database of Clarivate Analytics that were published between 1 January 2004 and 18 May 2020. The year 2004 was selected as this is when BPMN was created by BPMI.

Exclusion criteria (EC) are:

• EC1 = The publication was published in a language other than English.
• EC2 = The full text of the publications was not available.
• EC3 = The publication did not coincide with the topic of systematic research.
• EC4 = BPMN was used only as a presentation tool and not as part of the research.

Information sources and search

The primary source of information for the study was the database Web Of Science (WOS) of Clarivate Analytics. An advanced search was performed for the search query mentiones below. The search was performed in the Topics (TS) section.

Especially, the CORE database with the indexes listed in Table 1 was selected. The search was performed for ‘All document types,’ ‘All languages’ and the years 2004–2020.

Study selection

The first step of the review process involved title and abstract screening, followed by a full-text review of the remaining articles. Two independent assessors verified the results of the title and abstract screening and the full-text review. One assessed the suitability of the results from the perspective of OR and the other from an IT perspective, i.e. whether it was BPMN notation and its use. Articles were included if they met all the following criteria: (i) they used an OR method, (ii) a BPMN model was used and (iii) the complete text was available in English (abstracts, commentaries, letters and unpublished data were excluded). Studies were not excluded based on their methodological quality.

The selected publications were examined from many perspectives, and each contribution was coded according to different criteria. This study aimed to enhance the discipline’s fundamental progress in understanding the link between OR methods and BPMN. The results of this study could encourage scientists to use OR methods for process analysis.

A limitation of this review was restricting the included articles to English-language publications that looked at process analysis using OR and BPMN published between 1 January 2004 and 18 May 2020. Relevant studies in other languages or published after 18 May 2020 may have been omitted.

Data collection process

Data was collected based on keywords selected from the article Lane, Mansour & Harpell (1993) , which analyzed the quantitative techniques of Operation Research. From this document, the 18 Operation Research methods were selected and listed in the Table 2 .

The results were further categorized as to whether they corresponded to the given keywords and their meaning. The main results of the systematic literature review were obtained by analyzing by the two main guidelines of PRISMA: Moher et al. (2009) and MECIR: Higgins et al. (2018) .

Synthesis of results

The individual studies were subjected to bibliometric analysis and then the studies were assessed according to the content and methods used. The bibliometric analysis describes and analyses up to date research. It aims at summarizing the latest progress in the field by quantitatively investigating the literature. This method provides a vast canvas of knowledge from the micro-level (institutes, researchers, and campuses) to the macro-level (countries and continents) Mryglod et al. (2013) . Frequency analysis was used to find the most common scientific areas, the countries with the most publications and the most common keywords. Science mapping was performing using the VOS viewer, Venn diagrams and bar and bubble graphs, Van Eck et al. (2010) , Cobo et al. (2011) .

The Venn/Euler diagram graphically represents the relationships of the largest set of keywords. Euler diagrams are considered to be an effective means of visualizing containment, intersection, and exclusion. The goal of this type of graph is to communicate scientific results visually. Leonhard Euler first popularized the principle of labeled closed curves in the article Euler (1775) Alternative names for Euler diagrams include ‘Euler circles.’ They can also be incorrectly called Venn diagrams. Venn diagrams require all possible curve intersections to be present, so can be seen as a subset of Euler diagrams, that is, every Venn diagram is a Euler diagram, but not every Euler diagram is a Venn diagram. John Venn introduced Venn diagrams a hundred years after Euler in the article Venn (1880) . Venn diagram is a schematic graph used in logic theory to depict collections of sets and represent their relationships.

The initial search resulted in 128 articles. After removing duplicates, 107 were left that underwent title and abstract screening. After screening, 61 articles remained that underwent full-text review. The final number of included articles for information abstraction was 36. Overview of the number of publications according to exclusion criteria is shown in Fig. 1 .

Eighteen keywords selected from the article by Lane, Mansour & Harpell (1993) were involved in the study. These keywords have been classified according to whether a publication meeting a study condition has been found for them. Only for 13 keywords were found a publication suitable for this study, as can be seen in Table 2

Categorization of publications based on the clarivate analytics

Journals and books covered by the Web of Science Core Collection were assigned to at least one Web of Science category. Each Web of Science category was mapped to one research area Clarivate Analytics (2019) . The research areas for the selected publications were:

• COMPUTER SCIENCE (CS)
• ENGINEERING (En)
• OPERATIONAL RESEARCH MANAGEMENT SCIENCE (OR)
• BUSINESS ECONOMICS (BE)
• ROBOTICS (Ro)
• AUTOMATION CONTROL SYSTEMS (ACS)
• TELECOMMUNICATIONS (Te)
• TRANSPORTATION (Tr)

We selected four main groups, for which we compiled a bar graph and a Venn diagram after analysis. We chose the number of four research areas for representation in the Venn diagram; four sets are still well arranged. Another argument was the number of publications in other areas, where the set "ROBOTICS" contains two documents and the sets ‘AUTOMATION CONTROL SYSTEMS,’ ‘TELECOMMUNICATIONS’ and ‘TRANSPORTATION’ each one document.

Bar graph on Fig. 2 is based on frequency analysis and contains the total number of publications in a given research area, their average number of citations, and the corresponding average number of pages per article. The graph shows the results by type of purpose. The first part shows the frequency of documents for each research areas. The second part focuses on the average number of citations, and the third shows the average number of pages per article.

The Venn diagram, in Fig. 3 , shows selected four research areas as sets, including their intersection areas. In a specific area, we also stated the relevant number of documents and their average number of citations.

This part of the bibliometric analysis showed us the answer to the research question R2. Although BPMN was explicitly designed for corporate analysis and economic analysis, and Operational Research focuses primarily on addressing managerial decisions, most publications were not in the field of business economics (BE). Surprisingly, this area actually has the fewest publications. The field of computer science had the most papers, and papers in the field of OR had the most citations. The field of BE had the most extended publications, however, i.e. the average number of pages per paper.

Result1: Research question R2—not confirmed.

Year of publication

Figure 4 illustrates the distribution over time of the selected publications with BPMN milestones. The BPMN versions adoption dates, taken from OMG.org (2018) , complements this figure.

The different BPMN versions brought more or fewer changes in notation. While the changes between BPMN 1.0 and BPMN 1.2 were rather consmetics, e.g. renaming ‘Rule’ elements to ‘Conditional’ or slight increasing the number of elements from 48 to 55. The arrival of BPMN 2.0 was a major breakthrough and represented the largest revision of BPMN since its inception. In this version, it is possible to create a new ‘Choreography model,’ ‘Collaborations model’ and ‘Conversation model’ in BPMN in addition to collaborative processes and internal (private) business processes. Events are now divided into ‘interrupted’ and ‘non-interrupted’ and ‘catching’ and ‘throwing.’ The message type is newly introduced, and the data object has three specifications. BPMN 2.0 contains 116 elements. BPMN 2.0.2 included only minor modifications in terms of typos.

Given the magnitude of changes between the different versions of the BPMN notation, the sharp increase in publications following the introduction of the BPMN 2.0 notation can be considered a confirmation of research question R1. It is very interesting that publications in the field of BE did not appear until 2017.

Result: Research question R1—confirmed.

The average number of citations of the analysed documents was 2.22. The first quartile was 0, and the third quartile was 3.75. The median was equal to 1 and data variability above the third quartile was limited to seven citations. We identified two outliers values: 12 citations for Hasic, De Smedt & Vanthienen (2018) and 15 citations for article Wu et al. (2015) .

Author analyses

Bibliometric analysis cannot be done without review by the authors. We focused on illustrating co-authorship. The total number of authors of publications selected for this study was 84: al achhab, m (1), aouina, zk (1), ayani, r (1), aysolmaz, b (1), bahaweres, rb (1), batoulis, k (1), ben ayed, ne (1), ben said, l (1), ben-abdallah, h (3), bisogno, s (1), bocciarelli, p (1), boukadi, k (1), braghetto, kr (1), burattin, a (1), calabrese, a (1), ceballos, hg (2), chien, cf (1), cho, sy (1), creese, s (1), cunha, p (1), d’ambrogio, a (1), d’ambrogio, sa (1), de lara, j (1), de smedt, j (2), demirors, o (1), duran, f (2), el hichami, o (1), el mohajir, b (1), ferreira, je (1), figl, k (1), fitriyah, a (1), flores-solorio, v (2), fookes, c (1), garcia-vazquez, jp (1), ghiron, nl (1), ghlala, r (1), gomez-martinez, e (1), hansen, z (1), hansen, znl (3), happa, j (1), hasic, f (2), herbert, lt (8), holm, g (1), iren, d (1), jacobsen, p (3), jobczyk, k (1), kamrani, f (1), khlif, w (2), kluza, k (1), ligeza, a (1), manuel vara, j (1), marcos, e (1), mazhar, s (1), mendling, j (1), mendoza morales, le (1), mengersen, k (1), monsalve, c (1), moradi, f (1), naoum, m (1), onggo, bss (1), pablo garcia, j (1), perez-blanco, f (1), pitchforth, j (1), proudlove, nc (1), rekik, m (1), rocha, c (2), rosemann, m (1), rozy, nf (1), salaun, g (2), sharp, r (4), sperduti, a (1), suchenia, a (1), tang, rz (1), tokdemir, g (1), tomaskova, h (1), vanden broucke, sklm (1), vanthienen, j (3), veluscek, m (1), villavicencio, m (1), vincent, jm (1), weske, m (1), wisniewski, p (1), wu, ppy (2), xie, y (1).

These authors formed different sized groups, as can be seen in Fig. 5 . We grouped the authors according to their co-authors’ collaborations with a curve connecting the co-authors. The size of the node of this connection corresponds to the number of documents by the given author. The colours used to distinguish the authors were created using the average years of the publication of their papers.

For the authors’ average publication years, the first quartile was 2015, the third quartile was 2018.5 and the median was 2017. The variability outside the lower and upper quartiles was given by 2011 and 2020. We identified one outlier value corresponding to the year 2009.

The most prominent groups were around the authors listed in Fig. 6 . This figure also contains the number of documents by the authors, their total number of citations and their average value.

According to this analysis Wu, P. Y. had the highest number of citations (7.5), followed by De Smedt, J. (7) and Hasic, F. (7). Herbert, L.T. had the most documents (8) and Tomaskova, H. had no co-author connections.

The authors were also analyzed in terms of their country or region affiliations. A total of 25 countries were identified and their location, including the number of relevant publications, are shown in Fig. 7 . The countries with the highest number of affiliated publications were Denmark (8) and Tunisia (4), followed by Belgium, France, Saudi Arabia, Italy and Spain, who all had three.

Keywords analysis

The keywords were categorized according to those identified by the published authors and the keywords PLUS assigned by Clarivate Analytics databases. The data in KeyWords Plus are words or phrases that frequently appear in the titles of an article’s references but do not appear in the title of the item itself. Based upon a special algorithm that is unique to Clarivate Analytics databases, KeyWords Plus enhances the power of cited-reference searching by searching across disciplines for all the articles that have cited references in common, more information is on the web link Clarivate Analytics (2018) . A total of 130 unique keywords and 46 unique KeyWords Plus keywords were found for selected publications.

A total of 130 author keywords were mentioned in the publications and a general view of their interconnection can be seen in Fig. 8 .

Below is a list of all author keywords with the number of the weight-link to other keywords: activity theory (4), affiliation (6), agent based model (4), agent-based systems engineering (3), airport passenger facilitation (8), atl (5), automated verification (4), bayesian network (4), bayesian networks (4), bpm (6), bpmn (60), bpmn business processes (4), bpmn extension (3), bpmn model restructuring (5), business process (18), business process automation (3), business process management (13), business process model (5), business process model measures (3), business process modelling notation (4), business process optimisation (5), business process outsourcing (3), business processes (3), cloud computing (3), clustering (5), communication theory (11), configurable reference model (8), consequence modelling and management (10), contextual factors (8), cycle time (4), decision making (15), decision mining (3), decision model and notation (3), decision modeling (4), decision modelling (5), dikw (11), discrete-event simulation (4), dmn (15), effort prediction model (3), engineering agent-based systems (3), engineering systems (6), enterprise risk management (4), eqn (5), evolutionary algorithm (2), evolutionary algorithms (5), facilitated modelling (4), fault tree analysis (6), fault tree generation (6), flow (8), formal risk analysis (6), genetic algorithm (3), healthcare (4), hierarchical clustering (6), incident response (11), integrated modelling (5), interviews (11), jeqn (5), knowledge discovery (6), knowledge management (11), knowledge rediscovery (6), licenses (11), maude (7), mc-dmn (5), mcdm (5), mda (5), metrics (5), model checking (4), model transformations (5), model-driven architecture (4), model-driven engineering (5), modelling (4), object modeling (4), optimisation (6), organizational mining (3), performance (5), performance evaluation (3), petri nets (5), pproduction optimisation (2), preference to criteria (5), prism (8), probability (2), process configuration (8), process enhancement (3), process chain network (5), process merging (8), process mining (6), process modeling (4), process modelling (5), project management (3), qualitative analysis (4), quantitative model checking (10), quantitative service analysis (6), quantitative workflow analysis (4), queues (3), queuing theory (4), reliability analysis and risk assessment methods (4), resource allocation (4), restructuring (6), rewriting logic (7), rules (5), safety assessment software tools (4), safety management and decision making (4), security (11), security operation center (11), sense-making (11), separation of concerns (5), service engineering (6), scheduling (4), simulation (4), simulations (3), social network (5), social network analysis (3), social network model (6), socio-technical systems (sts) (4), soundness (4), space-sensitive process model (8), statistical model checking (4), stocastic bpmn (2), stochastic automata network (3), stochastic bpmn (11), stochastic model checking (13), stochastic modeling and analysis (4), structural and semantic aspects (5), tacit knowledge (11), task analysis (11), task assignment (4), task model (4), timed automata (4), topsis (5), verification (2).

As you can see in the figure, most of the author’s keywords are directly or indirectly linked with the term ‘BPMN’, but there are also isolated groups. In the following text, we’ve listed separate keyword groups. We’ve added a year of publication, a number of citations, and a specific document to which the keywords belong.

• 2013; citations; (business process automation; business process model measures; effort prediction model; project management) Aysolmaz, Iren & Demirors (2013) .
• 2014; citation; (evolutionary algorithm; pproduction optimisation; stocastic bpmn) Herbert et al. (2014) ,
• 2015; citations; (agent based model; bayesian network; business process modelling notation; modelling; socio-technical systems (sts)) Wu et al. (2015) ,
• 2015; citation; (affiliation; bpm; hierarchical clustering; knowledge discovery; knowledge rediscovery; restructuring; social network model) Khlif & Ben-Abdallah (2015) ,
• 2016; citations; (bpmn extension; business process outsourcing; cloud computing; genetic algorithm) Rekik, Boukadi & Ben-Abdallah (2016) .
• 2017; citations; (bpmn model restructuring; clustering; metrics; rules; social network; structural and semantic aspects) Khlif, Ben-Abdallah & Ben Ayed (2017) .
• 2019; citations; (atl; business process model; model transformations; model-driven engineering; petri nets; process chain network) Gómez-Martnez et al. (2019) .

As mentioned above, there were only 46 KeyWords Plus keywords (the number of links to other keywords is given in parentheses after the keyword): accuracy (6), ambiguity (6), automation (3), bpmn (20), business process models (6), checking (6), cognitive effectiveness (7), communities (2), complex (0), confidence (6), context (9), critical path (9), decision-making (7), design (7), dimensions (7), distributed simulation (1), framework (8), functional size (2), group creativity (6), identification (9), implementation (5), information (6), integration (2), model (7), neural-network (7), organizational knowledge (1), patterns (6), performance (9), process execution (9), process models (9), productivity (2), quality (2), reality (2), reference models (2), resources (9), risk (6), science research (2), semantics (9), sensemaking (1), simulation (9), strategy (0), systems (6), tables (7), verification (15), web (1), workflow (9).

As can be seen in Fig. 9 , these keywords are far more separate from each other compared to the author’s keywords.

Classification of articles by methodology

Based on the expert assessment, we examined the documents regarding the methods and approaches used. We created seven groups corresponding to a method or approach that was an essential part of the publication: probabilistic models, Decision Model and Notation (DMN), dynamic task assignment problem, evolutionary and genetics algorithms, queuing theory, social networks and others. These groups were also based on keyword analysis, as some separate groups of copyright keywords belong to highly unique articles. We assigned each document to just one group. That is in contradiction to research areas, where one article can be attributed to more than one research area. The individual documents and their division between research areas and methodological groups can be seen in Table 3 . We further analyzed the documents regarding their years of publication and plotted a bubble graph ( Fig. 10 ) with the publication years on the x .axis and the methodological groups on the y -axis. The appropriate number of publications corresponding to the given year and the group is indicated in the respective bubble. This quantity is also graphically represented by the size of the given bubble.

The largest group consisted of 10 publications on DMN and BPMN. Given the initiate year of DMN, this is the most significant approach serving with BPMN. DMN 1.0 was made available to the public in September 2015, the OMG group released DMN 1.1 in June 2016, DMN 1.2 was released in January 2019 and the latest version of DMN 1.3 was released in March 2020. The latest version did not affect this systematic search; however, the growth of publications since 2017 (see Fig. 10 , for example, was undoubtedly be affected by the DMN update.

We only assigned four documents to the methodological group focused on queue theory (See Table 3 and Fig. 10 ). The specific articles are listed in the following section under the appropriate heading. As the largest group was the DMN and BPMN group, we can thus rule out research question R3.

Result: Research question R3—not confirmed.

The methods, techniques and approaches used in the included publications are listed in the following section.

Probabilistic models

The probabilistic model can be used to make decisions when the activity reaches an exclusive splitting gateway and the activity’s subject must decide between alternative actions. They can be used for predicting or deciding between alternative works based on desirable outcomes. Probabilistic models were presented in the following publications:

• Herbert & Sharp (2012) : Quantitative analysis of probabilistic BPMN workflows;
• Herbert & Sharp (2013) : Precise quantitative analysis of probabilistic business process model and notation workflows;
• Ceballos, Flores-Solorio & Garcia-Vazquez (2015) : Towards Probabilistic Decision Making on Human Activities modeled with Business Process Diagrams;
• Ceballos, Flores-Solorio & Pablo Garcia (2015) : A Probabilistic BPMN Normal Form to Model and Advise Human Activities;
• Naoum et al. (2016) : A probabilistic method for business process verification: Reachability, Liveness and deadlock detection,

there the (Causal) Bayesian Network or Markov Decision processes were used.

DMN and decision analysis

Decision Model and Notation (DMN) is an industry standard for modeling and executing decisions that are determined by business rules. The association of DMN and BPMN is now common practice:

• Batoulis & Weske (2017) : Soundness of decision-aware business processes,
• De Smedt et al. (2019) : Holistic discovery of decision models from process execution data,
• Durán, Rocha & Salaün (2019) : A rewriting logic approach to resource allocation analysis in business process models,
• Figl et al. (2018) : What we know and what we do not know about DMN,
• Ghlala, Aouina & Ben Said (2017) : MC-DMN: Meeting MCDM with DMN Involving Multi-criteria Decision-Making in Business Process
• Hasic, De Smedt & Vanthienen (2018) : Augmenting processes with decision intelligence: Principles for integrated modelling
• Cho, Happa & Creese (2020) : Capturing Tacit Knowledge in Security Operation Centers,
• Mazhar, Wu & Rosemann (2018) : Designing complex socio-technical process systems - the airport example,
• Suchenia et al. (2019) : Towards knowledge interoperability between the UML, DMN, BPMN and CMMN models
• Tomaskova (2018) : Modeling Business Processes for Decision-Making.

Both standards fall under OMG.

Dynamic task assignment approach

The study : A dynamic task assignment approach based on individual worklists for minimizing the cycle time of business processes by Xie, Chien & Tang (2016) develop a dynamic task assignment approach for minimizing the cycle time of business processes. The contribution of this article lies in developing a dynamic task assignment approach based on queuing theory, individual worklist model, and stochastic theory.

Evolutionary and genetic algorithms

The evolutionary algorithm was applied in the following publications:

• Herbert & Sharp (2014b) : Optimisation of BPMN business models via model checking;
• Herbert et al. (2014) : Evolutionary optimization of production materials workflow processes;
• Herbert, Hansen & Jacobsen (2015) : Using quantitative stochastic model checking tool to increase safety and improve efficiency in production processes;
• Herbert & Hansen (2016) : Restructuring of workflows to minimise errors via stochastic model checking: An automated evolutionary approach;

to optimize the BP diagram, thus looking for a more efficient process. Especially the publication: Specifying business process outsourcing requirements, Rekik, Boukadi & Ben-Abdallah (2016) , presented a genetic algorithm to identify most appropriate activities of a business process that should be outsourced.

Queuing theory

In the article: Comparative analysis of business process litigation using queue theory and simulation (case study: Religious courts of South Jakarta) Bahaweres, Fitriyah & Rozy (2017) , Onggo et al. (2018) . A BPMN extension to support discrete-event simulation for healthcare applications: an explicit representation of queues, attributes and data-driven decision points Onggo et al. (2018) and Gómez-Martnez et al. (2019) . Formal Support of Process Chain Networks using Model-driven Engineering and Petri nets Gómez-Martnez et al. (2019) , the authors use queuing theory and simulation to compare processes modeled in BPMN. In the article: Automated performance analysis of business processes Bocciarelli & D’Ambrogio (2012) , authors presented a BP performance model of EQN (Extended Queueing Network) type.

Social network

The publications below focus on the application of social network analysis metrics (SNA) to studies of biological interaction networks in informatics.

• Khlif & Ben-Abdallah (2015) : Semantic and structural performer clustering in BPMN models transformed into social network models;
• Khlif, Ben-Abdallah & Ben Ayed (2017) : A methodology for the semantic and structural restructuring of BPMN models.

Other approaches

The following publications were unique in their approaches. We can mention for example: Workflow fault tree generation through model checking by Herbert & Sharp (2014a) with FMEA analysis. An effort prediction model based on BPM measures for process by Aysolmaz, Iren & Demirors (2013) with Linear multiple regression analysis. Performance evaluation of business processes through a formal transformation to SAN by Braghetto, Ferreira & Vincent (2011) using Stochastic Automata Network. Estimating performance of a business process model by Kamrani et al. (2009) using a Task assignment approach. Formal verification of business processes as timed automata by Mendoza Morales, Monsalve & Villavicencio (2017) convert BPMN to Timed Automata and then perform standard Queuing analysis. Business models enhancement through discovery of roles by Burattin, Sperduti & Veluscek (2013) , there the authors have extended the process model to roles, specifically designed role-sharing algorithm. Stochastic analysis of BPMN with time in rewriting logic by Duran, Rocha & Salaun (2018) presents a rewriting logic executable specification of BPMN with time and extended with probabilities. SBAT: A STOCHASTIC BPMN ANALYSIS TOOL by Herbert, Hansen & Jacobsen (2014) presents SBAT, a tool framework for the modelling and analysis of complex business workflows and A framework for model integration and holistic modelling of socio-technical systems by Wu et al. (2015) presents a layered framework for the purposes of integrating different socio-technical systems (STS) models and perspectives into a whole-of-systems model.

We have identified several gaps in the research and issues that need to be addressed in future research. The main gaps concern the research area of business economics. We assumed that this area would be the main and most frequent for the combination of BPMN and OR methods. However, we found that this area could be affected by the absence of specific notation. The relevant publications were written only after the release of version DMN 1.1. The effect of DMN notation will be addressed in future research.

An unexpected gap was a solution to finance and human resources management through OR. We would like to introduce publications Savku & Weber (2018) and Graczyk-Kucharska et al. (2020) as the pioneering works. The first article added the problem of optimal consumption problem from cash flow with delay and regimes. The authors developed the general analytic model setting and methods for the solution by studying a stochastic optimal control problem using the tools of the maximum principle. They proved the necessary and sufficient maximum principles for a delayed jump-diffusion with regimes under full and partial information. The second publication focused on transversal competencies, which are sets of knowledge, skills and attitudes required for different positions and in different professions. The authors used the method of multivariate additive regression spline together with artificial neural networks to create a model describing the influence of various variables on the acceleration of the acquisition of transverse competencies.

We assume that future research will be influenced by simulation and prediction methods. This study showed the use of Agent-based modelling methods and discrete-event simulations, or probabilistic models and social networks, but neural networks or artificial intelligence methods appeared in any publication. Based on this study, we further expect the use of more sophisticated approaches and the effect of new techniques. At the same time, it is possible to extend process modelling to inaccurate data using Fuzzy methods.

This paper presented a systematic overview of publications using BPMN and OR methods in process analysis. We analyzed 108 articles, that were selected using the appropriate strings in the advanced search option of in the WOS database. The papers that met the conditions of the study were subjected to various analyzes and were briefly described.

The review showed that the processes modelled by BPMN can be extended or analyzed as probabilistic processes, queue theory, or role and task assignments. Alternatively, processes can be optimized using evolutionary or genetic algorithms. The research also highlighted the need to identify keywords in publications correctly. For example, less than two-thirds of the selected articles contained the keyword BPMN, even though all the documents used this notation. Most of the articles were so-called one-off publications. Only a small number of author teams developed their topic in further continuing publications. Due to this, the average number of citations is relatively low. Due to the average number of citations to the total number of publications in all research areas, documents falling into the field of Operational Research are outstanding; there is an average of seven citations per article.

We analyzed the publications by research area and found that there is great potential for the research area of business economics (BE). Only a few papers were associated with this area (five in total) but all of them had a higher than average number of citations. The first document we included in this research area was published in 2017, that is only in the last quarter of the examined publication years. This focus on BE may have been initiated by the introduction of DMN notation.

Among the authors, smaller collaborating groups around the world were been identified. That groups co-work within the framework of co-authorship and co-citations. We only identified one single-author publication.

The analysis of keywords showed a significant difference between the keywords assigned by the authors and the so-called KeyWords Plus keywords. While the former were almost completely connected across publications, the latter were significantly diversified.

We have pointed out that the introduction of BPMN 2.0 led to an increase in publications using this notation.

Acknowledgments

The authors thank the student M. Kopecký for support in the field of BPMN modeling.

Funding Statement

The research has been supported by a GACR 18-01246S and by the Faculty of Informatics and Management UHK Specific Research Project. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Additional Information and Declarations

The author declares that they have no competing interests.

Hana Tomaskova conceived and designed the experiments, performed the experiments, analyzed the data, performed the computation work, prepared figures and/or tables, authored or reviewed drafts of the paper, and approved the final draft.

Gerhard-Wilhelm Weber analyzed the data, authored or reviewed drafts of the paper, and approved the final draft.

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Top 6 Methods Used in Operation Research

This article throws light upon the top six methods used in operation research. The methods are: 1. Linear Programming 2. Transportation Problems 3. Waiting Line or Queuing Theory 4. Game Theory 5. Simulation and Monte Carlo Technique 6. Dynamic Programming.

Method # 1. Linear Programming :

Linear Programming is a mathematical technique for finding the best use of limited re­sources of a concern. This is a technique to allocate scarce available resources under conditions of certainty in an optimum manner.

By using linear programming technique, a production manager can allocate the limited amount of machine time, labour hours and raw material available with him to the different activities so as to maximise the output/profit.

For solving a problem by linear programming, following conditions must be fulfilled:

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i. Objective i.e., reduction in cost or to maximise the profit, be stated mathematically.

ii. Resources can be measured as quantities i.e., in number, weight, volume or Rupees etc.

iii. There may be many alternative solutions.

iv. Relationship between factors should be linear.

v. Restrictions of the resources must be fully spelt out.

There are several methods to solve linear programming problems such as graphi­cal, index distribution, simplex and modified distribution (MODI) methods. But the graphical method is quite easy and simple. The other method commonly used is simplex method.

A. Graphical Method :

To understand graphical method, let us take following example:

Objective of a firm is to maximise profit by producing product A and / or product B both, of which have to be processed on machines 1 and 2. Product A requires 2 hours on both machines 1 end 2, while product B needs 3 hours on machine 1 and only 1 hour on machine 2. There are only 12 and 8 hours available on machine 1 and 2 respectively. The profit per unit is estimated at Rs. 600 and Rs. 700 in case of A and B respectively. Find out number of units of product A and B, he should produce to maximise the profit?

First Step:

Formulation of Linear programming problems. Let Z is the profit. The maxi­mum profit by manufacturing A and B.

600 A + 700 B = Max. profit Z …(1)

This equation is called objective function.

Now set up the equations for the process times.

Machine I – 2A + 3B ≤ 12 …(2)

Machine II – 2A + B ≤ 8 …(3)

Second Step:

Plotting of the equations on graph (Fig. 30.1). Suppose product A is shown on X-axis and product B is shown on the Y-axis of the graph. Now to plot the equation 2A + 3B ≤ 12>, first find the two terminal points, and then joining these points by a straight line.

Now the question is how to find the two terminal points:

(a) Suppose all the time available on machine 1 is used for making product A, this means the production of product B is zero, this means 6 units of product A would be made, then the first terminal point is (6, 0).

(b) Now supposing that all the time available on machine 1 is utilised for making product B then the production of product A is zero, and only 4 units of product B would be made. The second terminal point is (0, 4).

By joining (6, 0) and (0, 4), we get a straight line BC.

In the same way, the other equation 2A + B ≤ 8 can also be drawn by a line EF by terminal points (4, 0) and (0, 8).

Third Step:

Identify Feasibility Region and ascertain co-ordinates of its corner points. Under this step we have to identify the cross shaded area BOFD (Fig. 30.1), generally known as Feasibility Region and then finding on the co-ordinates of its corner points. These co-ordinates can directly be read with the help of accurately drawn graph or we can also find the co-ordinate with the help of mathematics such as;

and for point D, solve the two equations

2A + 3B = 12

On solving, we get A = 3, B = 2, so the co-ordinate of point D is (3, 2).

Fourth Step:

Test which corner point is most profitable. Putting different co-ordinates of corner points in the objective function Eq. (1), amount of profit in hundred Rupees.

Corner point O (0, 0) = (6 × 0) + (7 × 0) = 0

Corner point B (0, 5) = (6 × 0) + (4 × 4) = 16

Corner point F (4, 0) = (6 × 4) + (7 × 0) = 24

Corner point D (3, 2) = (6 × 3) + (7 × 2) = 32

Thus corner point D is yielding maximum profit of Rs. 3200 and the firm must produce 3 units of product A, and 2 units of product B.

B. Simplex Method:

To solve the linear programming problem, simplex method is quite common. There are several ways to solve the problems by simplex method, but the most easy one is with the help of “Gauss-Jorden reduction process”.

Let us first understand the Clauss Jordan reduction process with the help of following example:

Let the following three equations are in five unknown, find solution for C, D, E in terms of A and B.

2A – B + 2C – D + 3E = 14

A + 2B + 3C + D = 5

A – 2C – 2E = -10

Let us rewrite the given equations in the matrix form.

Now multiply the new first row by negative coefficient of second row against C (i.e. by -3) and add the result with the second row.

In the same way, multiply the new first row with (+2) and add the resultant with third row.

Thus the final matrix will be:

In the same way, eliminate coefficient of D except second equation.

Divide second row by 5/2, we get

Multiply new second row by — and add the resultant with 1st row

Multiply new second row by 1 and add the resultant with third row we get

Thus the final matrix will be

The transportation costs in Rupees from plants to warehouses are shown in Table 2.

There is no other condition; any plant can transport the product to any site up to its require­ment. Now the problem is to find most economical shipment, i.e., to minimise the total transpor­tation costs.

First prepare a table showing the requirements and capacities of sites and plants respec­tively. This type of table is known as Matrix. (See Table 3).

Now put the transport cost at the small square at right corner of large square, as shown in table 4 and apply the northwest corner method (but remember that it is not necessary to solve the problem by this method but for systematic approach this is helpful).

Using this north-west corner method, in the west corner square put the smaller of the two values between the capacity and requirement for the required column. As per example, we have two values 100 and 80, thus putting 80, in the corner as shown in Table 4. This means that transport 80 parts from plant A to site 1. This will fulfill the requirement of site 1, but the entire capacity of plant A is not utilized. Before proceeding to plant B, the remaining capacity of plant A must be utilized.

To do this, place 20 parts still available under site 2 which needs 30 parts. This utilizes the plant A capacity, so now move to plant B. As already 20 parts are shipped to site 2, only 10 parts are needed. These parts may be transported from plant B. Now for site 3, we first ship the remaining capacity (25 – 10 = 15) of plant B and remaining needed by site 3 from point C.

This solution of problem now known is called “initial solution” because it is not possible to say that this distribution is most economical. Total transportation cost is now computed as per Table 5.

The next step is to know whether this solution is the best one i.e. whether the transporta­tion cost cannot further reduce.

To do this, transfer-one part from plant A to site 3. Now balance the Table 4, and only 19 parts are to be shipped to site 2.

Now for this solution, cost table is computed as per Table 7.

Now this shows that solution in Table 7 is more economical than the initial solution which causes a saving of rupees (1045 – 1039 = Rs. 6).

To get the final solution of problem, repetition of the above process of moving one unit from one centre to another at a time until optimal solution is found. This type of method requires much labour. Thus to avoid this cumbersome method, another method is utilised to get final solution with less labour. This method is known as ‘Stepping-stone’ method.

Stepping-Stone Method :

Refer Table 4 to select a square which is empty. Starting from the square under site 3 and opposite Plant A, let us call 3 A square. Next trace a closed loop path from this empty square moving horizontally and vertically only back to this empty square.

Put the positive sign in the starting square 3A and to 2A square giving it a minus sign. Next we move 2B giving positive sign then to 3B which minus sign, then back to 3A. Table 8 shows this path on the original Table 4 with signs.

By applying this plus and minus values in this dosed loop path to the transportation cost in each square we get the amount of improvement by shifting one unit of production from SB to 3A and so on. As we started from 3A, square, let this path be known as 3A path. Thus the saving is given by due to this = 3A – 2A + 2B – SB, substituting the value from right hand corner of each square.

3A path = 2-10 + 7- 5 = – Rs. 6

This shows that each unit from Plant A to site 3 have a saving of Rs. 6 which is similar to the result obtained by north-west corner method, i.e., Rs. 1045 – 1039 = Rs. 6.

After this, all the possible closed loops are found and calculations are done, as shown in Table 8(a).

From the above table, it is clear that only 3 A path is suitable because only in this path there is reduction in the transportation cost (because minus sign is obtained in the result). All other paths cause an increase in the cost.

By trying to transfer maximum 15 parts from SB to 3A (least number of parts having minus square). Now Table 8 is changed to new Table 9.

Now computing shipping cost as per Table 10.

This shows a reduction in the cost. Now again we choose another unused paths from Table 9.

There are two paths:

2C = 2C – 3C + 3A – 2A = 8 – 4 + 2 – 10 = Rs. – 4

C = C – 3C + 3A – 1A = 6 – 4 + 2 – 5 = Re. -1.

This shows that in both paths, there is reduction in cost, but to gain maximum, the higher negative value path is chosen. Then Table 9 changes to Table 11.

In this Table, we shifted 5 parts from 2 A because 5 is the least value in negative squares. Now shipping cost is as per Table 12.

Thus there is further reduction in the transportation cost. But this is not the final solution.

Analysing the solution of Table 11 by loop method, the paths are:

1B = 1B – 2B + 2C – 3C + 3A – 1A = 3 – 7 + 8 – 4 + 2 – 5 = Rs. -3

1C = 1C – 3C + 3A – 1A = 6 – 4 + 2 – 5 = Re. – 1.

Thus 1B path is more economical. Shifting 25 parts (least value in negative squares), the new solution is shown in Table 13.

The transportation cost will be according to Table 14.

Thus a further reduction is there in the transportation cost. But to get optimal solution we have to test new unused paths.

1C = 1C – 3C + 3A – 1A = 6 – 4 + 2 – 5 = Rs. – 1

2A = 2A – 2C + 3C – 3C – 3A = 10 – 8 + 4 – 2 = Rs. 4

3B = 3B – 3A + 1A – 1B = 5 – 2 + 5 – 3 = Rs. 5

Thus a further improvement is possible by adopting path 1C. Transferring 45 parts (least value in negative square), the solution is as per Table 15.

The transportation cost is calculated in Table 16.

Now to improve the solution further we have paths:

3C = 3C – 3A + 1A – 1C = 4 – 2 + 5 – 6 = Re. 1

2A = 2A – 1A + 1C – 2C = 10 – 5 + 6 – 8 = Rs. 3

3A = 2B – 3A + 1A – 1B = 5 – 2 + 5 – 3 = Rs. 5

2B = 2B – 1B + 1C – 2C = 7 – 3 + 6 – 8 = Rs. 2.

We have seen that all possible paths give positive values. Thus the solution of Table 16 is a final result, as now it is not possible to further improve that result, this solution is optimal solution.

Though this method looks a lengthy one, yet by doing this a saving of Rs. 320 (Rs. 1045 – Rs. 815) is realized with the help of operation research over the initial solution of Table 4.

Method # 3. Waiting Line or Queuing Theory :

The object of queuing theory is to examine the problem of waiting and minimise the waiting period or in other words by solving such waiting line problems, we can adjust the waiting time or can reduce the queue to have economical balance between the costs of equipment or people standing idle and cost of providing better service.

The theory can be applied wherever queues are visible may it be bank or post office counter, rail or airline booking window, raw material or semi-finished product waiting for next operation on shop floor or material waiting for inspection or for moving to another place or turner waiting for getting tools for tool room or vehicle waiting for its turn on a petrol pump or service station. Such delays add to the production cost or cause inconvenience during service.

Waiting line or queuing theory is used to solve queue formation situations by analysing the feasibility of adding facilities (manpower or equipment) and assessing the amount and cost of waiting time. This theory helps in determining the optimum amount of facilities (manpower, equipment etc.).

Assumptions in Queuing Theory :

i. The principle of first come, first served is followed.

ii. Length of queue i.e., number of items in the queue remains the same with the passage of time.

iii. Number of items in the queue as well as waiting time by a particular item is random variables, and is not functionally dependent on time.

iv. Arrival rates follow Poisson distribution and service times follow exponential distri­bution.

Applications :

Waiting line theory can be used for efficient decision making in the following fields:

i. To find optimum number of tool crib clerks.

ii. Selection of material handling equipment.

iii. Selection of the size of maintenance crew.

iv. Distribution of service facilities like rest room, first aid centres, drinking water booths etc.

v. Machine interference problem to find out the work load of a single repair man.

vi. Traffic congestion studies.

vii. Job shop scheduling.

viii. Matching construction equipment capacities e.g., Loading of dump trucks by a shovel, pusher dozer feeding a scraper, feeding a crushing plant, asphalt plant hopper by a dumper or a wheel loader.

The method can also be applied to other similar problems, where a team work of equipment is involved.

Let us take a waiting line problem. The main problem of waiting line generally occurs in maintenance department, where there is always a formation of queues because by nature the maintenance operation is an inefficient operation. To avoid the waiting line, if we employ many attendants then cost of keeping them is expensive, if we have few, then cost of idle equipment’s is again considerable.

This can easily be judged from Fig. 30.2.

If in a firm from the past records, it is found that 50 machines are to be repaired per day, but the problem is that how many attendants must be employed in the maintenance depart­ment economically.

An analysis is done by finding the cost and idle times when 1, 2, 3, 4 attendants are employed, the computation is as follows:

Clerk’s idle time = 32 minutes

Mechanics’ waiting time = 39 minutes.

We see that for this problem, the clerk had 32 minutes idle time and the various mechanics waited for total 39 minutes. Now change the service time from 3 to 4 minutes. Now the Table 18 gives the schedule of servicing.

Clerk’s idle time = 22 minutes

Mechanics’ waiting time= 64 minutes.

This results in less idle time for clerk, but the waiting time for the mechanics is almost twice of that of previous one. Further we see that by increasing the service time, the waiting time and length of waiting line increase manifold. If the rate of arrivals is equal to the service time then we can assume that the waiting time and length of waiting line will increase and become infinitely as calculated latter.

Now to have more realistic problem, let us assume that the service time is also random, and that the service time may also vary, depending upon the number of tools to be delivered, their sizes, weights and locations. Table 19 shows a record in this situation.

Clerk’s idle time = 26 minutes

Mechanics’ waiting time = 56 minutes.

On analysing above table, the clerk’s idle time and mechanics’ waiting time will depend on how the arrivals of mechanics’ will match up with the length of service timings. In general, mechanic arrives almost simultaneously with the request of requirements of long service times, and then waiting line will be relatively longer, while clerk is continuously busy.

Thus waiting time can forecast the probable waiting time and probable length of the line, and by this management can decide, what is optimum allocation of personnel and equipment to minimise cost. To do this, the nature of the distribution of arrivals and the service times must be known.

From Table 19 we can calculate, supposing that the clerk is paid at the rate of Rs. 2 per hour and he works for 8 hours per days, then

Cost of clerk’s idle time = 26/60 × 2 roman Re 0.87

This cost is for only 72 minutes, i.e., for 11.00 to 11.12 O’clock. Then cost of idle time/day

= 480/72 × 0.87 = Rs.5.80 per day.

Suppose the mechanic’s average wages are Rs. 3 50 per hour, then

Cost of mechanic’s idle time = 56/60 × 3.50 = Rs. 3.27 for 72 minutes.

Cost of mechanic’s idle time/day = 480/72 × 3.50 = Rs. 21.80 per day.

Thus total cost of idle waiting time = Rs. 21.80 + 5.80 = Rs. 27.60/day.

Now the problem is—would it be economical to have a second clerk? If the second clerk is to be added, then mechanics waiting time can be reduced, but total idle time for clerks would increased. If the net effect on the cost of idle plus waiting time is negative, the second clerk would be needed.

Mathematical Method :

In table 17, 18 and 19, we have explained simple waiting problem. With the use of math­ematical formulae, we can directly get the data, in which we are interested, such as average length of waiting line and average waiting time, etc.

For solving queuing problems mathematically, an expression known as traffic Intensity is used. This is denoted by a letter ‘T’. This is the demand divided by the capacity or more clearly— the mean service time divided by the mean interval between successive arrivals.

The calcula­tion of traffic intensity for any system is shown below:

Suppose the arrival intervals are:

5, 4, 10, 6, 3, 2, 6, 4, 8, 15, 2, 5, 4, 6, 8, 7, 3, 5, 2, 5, 8, 7, 4, 3, 2, 6, 7, 4, 3, 4, 7, 5, 8, 4.

The cumulative totals, obtained by adding the first to intervals and then the first three and so on, are:

9, 19, 25, 28, 30, 36, 40, 48, 63, 65, 70, 74, 80, 88, 95, 98, 103, 105, 111, 119, 126, 130, 133, 135, 141, 148, 153, 165, 160, 166, 171, 179 and 183.

Now cumulative averages obtained by dividing all the cumulative totals by the number of intervals making up each total are as follows:

9/2, 19/3, 25/4, 28/5, 30/6, 36/7, 40/8, 48/9, 63/10, 63/11

70/12, 74/13, 80/14, 88/15, 95/16, 98/17, 103/18, 105/19, 111/20, 119/21

126/22, 130/23, 133/24, 135/25, 141/26, 148/27, 153/28, 158/29, 160/30

166/ 171/179/ and 183/34 or in decimal

4.5, 6.3, 6.2, 5.6, 5.0, 5.1, 5.0, 5.3, 6.3, 5.9, 5.8, 5.7, 5.7, 5.9, 5.9, 5.8, 5.7, 5.5, 5.6, 5.7, 5.7, 5.5, 5.4, 5.4, 5.5, 5.5, 5.4, 5.3, 5.3, 5.3, 5.3, 5.4, 5.4.

When these cumulative averages are plotted against the number of intervals, the curve will be as of Fig. 30.3. It shows that the curve fluctuates a little at the beginning, but then settles down to fairly steady figure of 5.4 minutes. Thus this value is taken as the average interval between arrivals.

The average service time also can be found in a similar way. Let us assume that it will be 4.5 minutes. Then the traffic intensity would be

Traffic intensity T = 4.5/5.4 = 0.833 . . . T = A/S

where A = average arrival rate and S = roman the service rate.

Thus one waiting line serviced by one individual.

1. Mean number in waiting line, i.e. average length of waiting line (L a )

= T 2 /1 – T = A 2 /S(S-A)

2. Mean or average number in line, including the one being serviced (L)

= L a + A/S = A/1 – T = A/S – A

3. Average waiting time of an arrival

W a = L a /A = T/S(1 – T) = A/S(S – A)

4. Average time which an arrival spends in the system or say mean time in system including service

= W = L/A = 1/S – A = 1/S(1 – T)

5. Idle time = (1 – T) = idle time of service man.

In the Table 19, we have come across a distribution with Random Arrival time of mechanics and Random Service Time. With average arrival rate (A) as 12 per hour and average service rate (S) as 20 per hour, thus calculating as per given formulae.

A = 12 and S = 20

T = 12/20 = 0.6

Average length of waiting line La = T 2 /1 – T = 0.36/1 – 0.6 – 0.36/0.4 = 0.9

Average number in line, including the one being serviced

L = T/1 – T = 0.6/1 – 0.6 – 6/4 = 1.5

Average waiting time of an arrival W a = T/S(1 – T) = 0.6/20(1 – 0.6) – 6/20 × 4

= 3/40 = 0.075 hr. = 4.5 minutes.

Average time which an arrival spend in the system

= 1/S(1 – T) = 1/20 × 0.4 = 1/8 hour

Idle time = (1 – T) = 40% of total attended time = 72 × 0.4 = 28.8 roman minutes.

The data computed above are approximately equal to the same value obtained from the simple simulation done in Table 19. The more correct and nearer value can be obtained by simulating average sample.

Probability that queue size exceeds n = (A/s) n

Probability that queue size is zero = 1-(A/S)/1-(A/S) c+1

where C = limiting capacity

when a man has to pass through more than one queue.

This means when a service is completed in phases and at each phase queue is available.

In such cases, following formulae are used:

(i) Average waiting time = (K +1) (A/S)/2KS (1 – [A /S])

(ii) Average queue length = (K + 1) (A/S) 2 /2KS (1 – [A /S])  

Method # 4. Game Theory :

Suppose, a manufacturer who is faced with the problem of choosing a price for his product has the necessity to examine the reaction of his competitors, due to this decision about the price. Suppose, he is trying to decide whether it is worthwhile for him to cut his price, the answer will depend on what his opponent manufacturer will do.

There are four possible outcomes:

1. He cuts the price; the opponent keeps his price constant.

2. He cuts the price; the opponent also cuts the price.

3. He keeps his prices constant, the opponent cuts his price.

4. He keeps his price constant; the opponent keeps the ‘price’ constant.

Thus the manufacturer has to analyse all these different outcomes with the help of game theory which shows the profitability under each situation from which the manufacturer can make a final choice.

In game theory, the decision makers are known as ‘players’, the choices are called strategies and the preferences of the decision makers called “payoffs”.

If the sum of payoff is zero, the same is said to be zero sum, but where the sum of payoff is not zero, these are called ‘non-zero sum game’.

The game theory is a technique which introduces a table known as ‘payoffs matrix’, show­ing the expected values for various outcomes to determine the best way to ‘play’ against the opponent. The object is not to find the best answer, but to minimise the maximum (known as minimax) risk, or reduce your chance of losing.

The use of a ‘pay-off-matrix’ for expressing the problem and evaluating various decisions can have important implications for business or industries. But to make theory really practi­cable, the analyser should have lot of imagination and know new development of the science.

The theory can easily be more understandable with the help of following examples:

Consider the following table:

The choices α, β, ү, δ are the possible strategies, for you and A, B, C, D are the possible strategies of your opponent. How should you play the game? If you play strategy β, you gain 2 points, if the opponent plays A or C, but lose 4 if he plays B. You cannot be sure when you play β what choice your opponent has made.

Here, a minimax rule exists. If you play strategy α, you win a minimum 2. If you play any other strategy, one might not win as much. If the opponent plays B, he does not lose more than 2. If he plays any other strategy, he could lose more.

Thus yours own minimum again is the same as the opponent’s maximum loss. The square αB is unique in this respect and is called the saddle point or the solution of the game when playing against a skillful opponent in the kind of situation one cannot do better in the long run than play this optimum strategy.

Suppose we are setting a plant to produce a new product. We have five alterna­tive methods of manufacturing from which selection may be done. But profit depends on one of the experimental automatic packing systems which will be successful. There are 50-50 chances that either one or the other will be successful but definitely not both.

Depending on which one is successful to prepare the matrix for the five methods. The main problem is that we have to select and start installing the method of manufacture before we know which packing system will work about. So which method will be selected to assure the firm maximum profit?

The answer is simple, we select method 3, because it assures us the greatest profit, in case machine B is successful and even we can earn, more in comparison of machine B, if the machine A is successful.

In game theory usually we assume that the opponent has perfect intelligence and always plays his best strategy to make us loose. There is our example, we are really playing against nature (such problems come under the games against nature) where opponent is nature which is trying to win with perfect intelligence, the nature will certainly pick machine B.

But this is unrealistic, because in competitive situations we take advantage of the fact that our opponent does not have perfect intelligence. This improves our chances of earning more profit; we can easily introduce more probabilities of each outcome to defeat our opponent.

Two Person Zero Sum Game :

This type of game can easily be understandable with the help of following example:

Let two candidates of different political parties say Congress and BJP have agreed to hold public meetings somewhere in their constituencies and are negotiating over the location.

Each wishes to have as friendly audience as possible, and the 12 possible locations vary considerably in the balance between Congress and BJP. Each of the 12 possible sites lies on the intersection of an east-west and a north- south highway, as shown in Fig. 30.4.

Now we analyse the cost: Cost Analysis:

This shows that 5 teams give minimum cost. Table shows that total cost decreases when number of teams increase till it reaches to 5 teams, thereafter cost again increases when more number of teams are engaged.

Therefore, we should employ 5 teams, i.e., 10 workers to get optimum results.

Method # 6. Dynamic Programming:

Dynamic programming is a mathematical technique for solving problems where a sequence of decisions are involved. In such problems, there are number of stages and at each stage there are several alternatives available.

The decisions taken at stage one, act as conditions of the problem for stage two and so on, i.e. the decision taken at stage one affects the choice of decision at the stage two and so on. The basis of this is to select the best amongst the final possible alternative decisions, ignoring all other alternatives, which do not lead to the best (i.e. opti­mum).

Dynamic programming thus attempt to break large, complex problems into a series of smaller problems that are easier to solve separately. In this way, dynamic programming divides the problem into a number of sub-problems or decision stages. This technique is advan­tageous for solving problems, even when incorrect or less-than optimal decisions might have been taken in the past, and enables manager to make decision for future periods.

Dynamic programming is used in production scheduling, maintenance and repair, financial balancing, inventory, equipment replacement etc.

This can be better understood by following example:

Example 11:

A pipe line is to be laid between stations A and E passing successively through one node of each B, C and D as shown in the figure below. The costs for lying from A and B and from D to E are shown in the figure while the costs between B and C and between C and D are given in the table below. We are required to find out the path which will require minimum costs of laying the pipe line from A to E.

Costs from B to C and C to D:

Now we shift C to D, and observe that many more possibilities open. For example, from C 1 , path may pass from D 1 , D 2 and D 3 on its way to E.

The cost of each of these paths is calculated and the optimum is noted as shown in the table below:

This shows that, in passing from C 1 E, the path of minimum cost is through D 3 and similarly minimum cost path from C 2 E and C 3 E is through D 2 and D 3 respectively. From this point, we start taking benefit of dynamic programming. From here, whenever path from C 1 is required, we may follow it through D 3 . (Since we now know that optimum path from C 1 to E is through D 3 ).

In the next stage, we further proceed backwards to point B, and starts examining the optimum costs from various nodes of point B, as shown in the table. In this table, instead of taking all the paths, we take use of optimum paths calculated above from C 1 , C 2 and C 3 .

This shows that out of 3 possibilities from B 1 to E, the path pass through C 2 is optimum and similarly optimum cost path for B 3 E and B 2 E are through C 2 and C 3 respectively.

Now in the last stage, we start from point A and compute the cost for the three possibilities below:

This shows that minimum cost from A to if is 50. And the optimum cost paths is A – B 1 – C 2 – D 2 – E.

In this problem, total numbers of possible paths are 3 × 3 × 3 × 3 = 81. But by adopting dynamic programming, we have to make only 24 calculations. This shows that through this method, lot of labour could be saved.

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