Elsevier

Computers & Chemical Engineering

Volume 82, 2 November 2015, Pages 115-128
Computers & Chemical Engineering

A robust possibilistic programming approach for pharmaceutical supply chain network design

https://doi.org/10.1016/j.compchemeng.2015.06.008Get rights and content

Highlights

  • Developing a bi-objective MILP for a pharmaceutical supply chain network design problem.

  • Adopting a robust possibilistic programming approach to handle epistemic uncertainty of uncertain parameters.

  • Validating the model using a real case study and providing managerial insights.

Abstract

In this paper, a bi-objective mixed integer linear programming (BOMILP) model is developed for a pharmaceutical supply chain network design (PSCND) problem. The model helps to make several decisions about the strategic issues such as opening of pharmaceutical manufacturing centers and main/local distribution centers along with optimal material flows over a mid-term planning horizon as the tactical decisions. It aims to concurrently minimize the total costs and unfulfilled demands as the first and second objective functions. Since the critical parameters are tainted with great degree of epistemic uncertainty, a robust possibilistic programming approach is used to handle uncertain parameters. In order to verify and analyze the proposed model, it is tested on a real case study and managerial insights are provided.

Introduction

A supply chain is usually considered as an integrated process in which a group of organizations such as suppliers, producers, distributors and retailers work together to convert the raw materials into the end products, and distribute them to the end customers. Supply chain network design (SCND) as one of the key decision problems in supply chain management, has an important role in the performance of supply chains (Beamon, 1998, Zahiri et al., 2014a).

Among the considerable published research works on SCND, only a small proportion of these studies directly deal with the pharmaceutical sector. The pharmaceutical industry, which could be considered as an immense global industry, can be defined as a complex of processes, operations and organizations involved in the discovery, development and manufacturing of medications and drugs (Shah, 2004).

In 2009, the pharmaceutical section included approximately 10% of annual healthcare expenditures in the United States and about $600 billion globally (Kelle et al., 2012). In spite of all advances and improvements in the manufacturing, storage, and distribution methods, several pharmaceutical companies are still significantly far from effectively satisfying market demands in a consistent manner. Thus, these pharmaceutical supply chains are quite ready for receiving help from efficient optimization techniques (Masoumi et al., 2012, Papageorgiou, 2009, Shah, 2004).

Challenges and different methodologies in pharmaceutical supply chains have been widely surveyed by Laínez et al. (2012), Papageorgiou (2009), and Yu et al. (2010). Narayana et al. (2012) reviewed the literature on managerial issues in the pharmaceutical industry. Alnaji and Ridha (2013) addressed the role of supply chain management (SCM) applications in the pharmaceutical industry. Rossetti et al. (2011) focused on identification and examination of the major forces in the pharmaceutical supply chains, which may change the way biopharmaceutical medications are purchased, distributed, and sold throughout. Jetly et al. (2012) developed a multi-agent simulation model to analyze pharmaceutical supply chains. Sousa et al. (2005) presented a mixed-integer linear programming (MILP) model as a location-allocation problem to design a supply chain network involving primary (active) ingredients production sites and final product distribution centers. The model aims to maximize company's net profit value (NPV). They developed two decomposition solution algorithms including a heuristic and a Lagrangean decomposition method. In another paper, Sousa et al. (2011) proposed a dynamic allocation/planning problem aiming to maximize the NPV of a company working in a global pharmaceutical supply chain. Two decomposition algorithms were also developed to solve large-scaled problem instances. Susarla and Karimi (2012) proposed a MILP model for multi-period planning of a pharmaceutical firm, which integrates procurement, production, and distribution operations while accounting for tax differentials, material shelf-lives, inventory holding cost, and waste treatment. Abdelkafi et al. (2009) proposed a method for balancing the cost and risk of supply shortage by utilizing Bayesian principle for re-evaluation of supply strategies over time. Rotstein et al. (1999) presented an optimization based approach for selecting product development and introduction strategies, as well as capacity planning and investment strategy for a pharmaceutical supply chain in which the demand is dependent on the clinical outcomes. Levis and Papageorgiou (2004) presented a MILP model for long-term, multi-site capacity planning of a pharmaceutical firm under uncertainty while considering the trading structure of the company. Amaro and Barbosa-Póvoa (2008) presented a modeling approach for the sequential planning and scheduling of supply chain structures with reverse flows and applied it for a real pharmaceutical case study. Gatica et al. (2003) presented a MILP model to formulate a multi-stage, multi-period stochastic optimization model for a pharmaceutical inventory system under risk and potential returns. Papageorgiou et al. (2001) presented a MILP model for concurrent optimization of product development and introduction strategies as well as investment strategy and capacity planning in a pharmaceutical system.

Fig. 1, which is the revised version of a pharmaceutical supply chain structure presented by Laínez et al. (2012), illustrates a general pharmaceutical network, including the primary and secondary manufacturing sites; main distribution centers (DCs); local DCs; and demand points and/or retailers (i.e., hospitals and pharmacies).

The primary manufacturing sites are in charge of producing active ingredients, which normally involve either several chemical synthesis and separation stages to build up the complex molecules involved, or fermentation and product recovery and purification. The secondary manufacturers, which often outnumber the primary manufacturing centers, are in charge of producing final pharmaceutical products. Main DCs usually have copious storage capacity and are in charge of delivering the received products to the local DCs. Finally, local DCs that play a significant role in the whole chain are responsible for fulfilling customer demands directly (i.e. hospitals and retailers such as pharmacies). In the UK, about 80% of total demands are directly fed by local DCs (Shah, 2004). By assuming that most of ingredients are imported from overseas (as it is the case for Iranian drug manufacturers), therefore, without loss of generality, the concerned pharmaceutical network in this paper includes all aforementioned echelons except the primary manufacturers.

The dynamic and imprecise nature of quantity and quality of manufactured products in the pharmaceutical industry network results in a high degree of uncertainty in required data when designing such supply chains. Uncertainty in data is generally represented by two main forms (Mousazadeh et al., 2014). The first type is characterized by randomness that stems from the random nature of parameters. Stochastic programming methods are the most applied approaches to cope with this sort of uncertainty when there are distributional information about random data (see Santoso et al., 2005, Baptista et al., 2012). Nevertheless, in the absence of distributional information about random data (i.e., lack of historical objective data), which is the case in most of real-life situations, robust programming methods are widely applied. Among several works which have been done in this field, Pishvaee et al. (2011) presented a robust optimization approach to closed-loop supply chain network, considering minimization of total cost as the sole objective function. Mirzapour Al-e-hashem et al. (2011) proposed a multi-objective aggregate production planning problem under uncertainty for which the robust optimization approach was applied to cope with uncertain data. Ben-Tal et al. (2011) addressed the robust optimization approach for humanitarian relief logistics network design under demand uncertainty, and considering dynamic traffic assignment. Jabbarzadeh et al. (2014) developed a dynamic network design model for the supply of blood in disasters in which the uncertain nature of the input parameters was handled using the robust optimization approach.

The second type of uncertainty is epistemic that deals with ill-known and imprecise parameters arising from the lack of knowledge regarding the exact values of these parameters for which, possibilistic programming approaches are usually applied. In our model, there is an epistemic uncertainty on exact values of some critical parameters (such as demands, unit manufacturing and transportation costs) due to unavailability as well as the dynamic and imprecise nature of required objective data which have to be estimated mostly by relying on the subjective opinions/experiences of field experts. Accordingly, these parameters are assumed to be imprecise (i.e. possibilistic) in nature. In this way, it is assumed that a suitable possibility distribution based upon the subjective opinions of decision makers (DMs) and possibly some available objective data has been estimated for each imprecise parameter in the form of a trapezoidal fuzzy number. In this field, Pishvaee et al. (2012a) presented a credibility-based fuzzy programming approach for a bi-objective green logistics network under uncertainty. Zahiri et al. (2014a) addressed a robust possibilistic programming approach to location-allocation of organ transplant centers in a multi-period horizon. In another work, Zahiri et al. (2014b) proposed a novel multi-objective network design of the organ transplant chain under perishability and uncertainty. The epistemic uncertainty of the input data was handled using an interactive fuzzy approach. Salehi Sadghiani et al. (2015) developed a retail supply chain network design model under operational and disruption risks in which a mixture of random and imprecise (i.e. possibilistic) input data was handled by a mixed possibilistic-robust programming approach.

Collectively, most of the reviewed papers (see Sousa et al., 2005, Susarla and Karimi, 2012, Grunow et al., 2003, Hansen and Grunow, 2015) deal with only strategic decisions over a single period horizon while ours accounts for both strategic and tactical decisions over a multi period horizon allowing both long-term and mid-term decisions to be jointly addressed. Furthermore, only small proportion of the current works (see Levis and Papageorgiou, 2004, Hansen and Grunow, 2015) have accounted for one type of uncertainty while our work accounts for epistemic uncertainty in the critical input data within a robust programming framework leading to a robust possibilistic approach. In this way, the model enables the DMs to obtain robust solutions while coping with impreciseness of critical data. Lastly, majority of relevant papers aim at optimizing just a single objective mostly in the form of minimization of the total costs or maximization of the NPV, while our model makes possible a trade-off analysis between the two important objectives (i.e., minimization of the total costs and unsatisfied demands) in a bi-objective modeling framework. It should be noted that minimization of unsatisfied demands has not already been considered as an optimization objective in the related literature.

Consequently, the main contributions of this paper that differentiates it from other relevant papers could be summarized as follows:

  • Developing a bi-objective MILP model for a multi-product, multi-period pharmaceutical supply chain network design problem in which both strategic decisions (i.e. numbers, locations and production technologies of secondary manufacturers; numbers, locations and capacities of main DCs as well as numbers and locations of local DCs) and tactical decisions (i.e. product flows between facilities of different echelons in each period) are taken simultaneously into account.

  • Accounting for epistemic uncertainty of critical data within a robust framework through applying a robust possibilistic programming approach, which ensures yielding robust solutions while accounting for possible changes of imprecise data in the pre-defined ranges.

  • Providing a real case study to validate the proposed model and analyze the efficiency of utilized solution method.

  • Applying two different multiple objective decision making (MODM) approaches including the ɛ-constraint method for depicting the conflicting nature of objectives and providing the trade-off analysis between concerned objectives while achieving the respective Pareto frontier as well as the TH approach (Torabi and Hassini, 2008) to achieve a final preferred compromise solution for the problem in a guided interaction with the DM.

It is worth noting that the proposed model addresses an integration of strategic and tactical level decisions in a pharmaceutical supply chain. Nevertheless, in an effort to achieve a balance between practicality and tractability, we do not incorporate the detailed operational decisions in our model such as those related to production sequencing decisions while accounting for machine changeovers in the product items’ level. Nevertheless, the results of such integrated model could be used as the main inputs for making detailed decisions in the lot-sizing and scheduling level especially when applying a hierarchical planning framework. Interested readers are referred to Torabi et al. (2010) to see more details about the hierarchical planning (HP).

It is also noted that as the model is presented in an aggregate level to reflect its mixed strategic-tactical nature, we will be facing with some product families instead of end product items. Each product family consists of those product items with similar technological and marketing characteristics (e.g. same production process and demand seasonality) while different families may have quite different aspects, specifically in terms of production process/technology. Hence, without losing generality, we assume that each product family has its own manufacturing line (as it is often the case in the pharmaceutical industry) and therefore accounting for shared manufacturing resources which are usually common in practice, is not the case for our model. However, it is assumed that different product families are interrelated by sharing the main and local DCs when addressing the concerned supply chain planning problem. Furthermore, other production resources such as quality control and packaging resources that might be shared between different product families, are not usually bottleneck stages in the manufacturing centers, so they will not be addressed in the proposed decision model (of course without loss of generality).

The rest of the paper is organized as follows. Problem definition and its mathematical formulation followed by a linearization of developed model are presented in Section 2. In Section 3, a robust possibilistic programming approach is utilized to generate the crisp counterpart of the original model. Description of a real case study, validation of the proposed model, interpretation of achieved results from two MODM approaches and sensitivity analysis on the model's main parameters are provided in Section 4. Finally, the paper concludes with some remarks and recommendations for further research in Section 5.

Section snippets

Problem description and formulation

According to Fig. 2, the concerned supply chain network is a multi-product, multi-period, four-echelon pharmaceutical supply chain that includes several secondary drug manufacturing centers, main DCs, local DCs and customer zones (i.e., hospitals/clinics and pharmacies) with possible lateral transshipment flows between the local DCs.

It is noted that because of integration of strategic and tactical decisions, the model is studied under a multi-period planning horizon allowing finding the optimal

The robust possibilistic approach

Due to the dynamic and fluctuating nature of some critical parameters (including the demands, unit manufacturing costs, unit transportation and transshipment costs and safety stock levels) over the planning horizon and also unavailability and even non-obtainability of required historical data in the design phase, they have to be estimated mostly by relying on experts’ subjective opinions and experiences. Therefore, we formulate such imprecise parameters as possibilistic data in the form of

Implementation and evaluation

In this section, performance and usefulness of the proposed model are tested via an empirical case study. Based on the data collected from Iran's National Organization of Food & Drug, regarding the total national sales, amoxicillin 500 mg Cap was the most prescribed and consumed drug among around 5500 different types of drugs from 2004 till 2013 in Islamic Republic of Iran (around 900 million capsules per year). In fact, amoxicillin can treat a broad range of infections such as: chest

Conclusions

In this paper, a bi-objective mixed-integer programming model (MILP) was developed for a pharmaceutical supply chain network design problem. Since the problem deals with great degree of uncertainty in input parameters, a robust possibilistic programming approach was applied to cope with the inherent epistemic uncertainty in the model's main input data. Also, to validate the proposed model, it was tested on a real case study, i.e. the supply chain network design of amoxicillin 500 mg Cap in

Acknowledgements

The authors would like to thank Mohammad Hossein Mirsalehian, production expert at Pharma Chemie, TPICO (Tamin Pharmaceutical Investment Company) and Mr. Ehtezazi, the sale manager at SOHA Helal Distribution Company (Red Crescent Society) for their precious efforts in validating the structure of the proposed model as well as gathering required data for the case study.

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