Keywords

1 Introduction

Nowadays, the problems involved in the efficient management of a supply chain (SC) constitute a significant area of research. A glance at the wide range of publications on supply chain management (SCM) reveals that this subject has been widely studied from different perspectives and disciplines in the literature. Schneeweiss [1] provides a review of how decision-making problems in SCM are differently managed from the perspectives of applied mathematics, operations research, economics and artificial intelligence. From whichever point of view, the current cornerstone of SCM is to consider the supply chain globally rather than as a set of individual activities. This implies identifying the logistic activities, recognizing the organizations and/or decision makers involved in the SC and integrating them in an efficient manner. The challenge in SC integration is to coordinate the activities involved so that the performance can be improved by reducing costs, increasing service levels, providing a better use of resources and offering an effective response to the changes in the market place [2].

The aim of this paper is to identify the mathematical programming approaches proposed in the literature to model the relationship amongst SC agents and their interdependencies. We pay attention to papers which use deterministic mathematical programming models when dealing with an integrated problem in SCM. We have searched under “supply chain management” in the available databases Scopus, Sciencedirect, ProQuest and Google Scholar. The results show an increasing number of works on SCM in recent years. Based on these results we have selected papers which include either in the abstract or the keywords the terms program, mathematical or algorithm. The paper is organized as follows. Section 2 explores the literature on SCM and identifies the classification criteria proposed in previous reviews of integrated SC problems. Sections 3 to 6 go on to discuss recent reviews on the subjects as well as interesting papers not included in these reviews. Finally, Sect. 7 sets out our findings and conclusions as well as areas of interest for further research.

2 Classification Criteria

When analyzing SCM, Beamon [3] distinguishes between the production planning and inventory control process and the distribution and logistics process. Both kinds of process interact with each other to produce an integrated SC. Min and Zhou [4] state that a SC consists of two main business processes: material management (inbound logistics) and physical distribution (outbound logistics). These processes consist of three levels of decision hierarchy: competitive strategy, tactical plans and operational routines. Location-allocation decisions and demand planning are included in the first level, which can be considered as long-term planning. Tactical problems or medium-term planning are the inventory control and the production/distribution coordination. Finally, vehicle routing/scheduling is considered as part of the operational routines or short-term planning. Goetschalckx et al. [5] review mathematical programming models for the efficient design of global logistics systems that take into account strategic and tactical considerations. Melo et al. [6] review facility location models which also involve decisions about the design of the supply chain network. They include papers dealing with tactical and operational decisions such as capacity allocation, inventory, procurement, production, routing or transportation modes. Schmid et al. [7] focus on the classical vehicle routing problem but their search is extended to incorporate tactical and operational problems related to production, warehousing and inventory. Okongwu et al. [8] study the impact on the performance of the SC of the way in which firms plan tactical aspects such as procurement, production and distribution and argue in favor of their integration.

As a conclusion, the following logistic activities can be singled out in SCM: procurement, facility location, production, inventory management and transportation. Based on these and on the optimization models which have been proposed to deal with integrated SCM, we have classified the selected papers into four categories: location/routing, sourcing/production and supplier selection/inventory, location/inventory and inventory/routing, and production/distribution. In what follows, we include for each category a description of the problem and explore the relevant literature. The recent reviews discussed provide an important set of references. Only those papers either missing from these reviews or published more recently are explicitly mentioned here.

3 Location/Routing Problem (LRP)

This problem is defined for a network with, in general, a homogeneous set of vehicles with limited capacities which are shared by all depots. The goal is to determine which depots must be opened and decide the routes for each open depot and its customers. Early papers dealing with this problem considered uncapacitated depots. However, in more recent papers capacity constraints on depots and vehicles are addressed. The comprehensive review by Nagy and Salhi [9] published in 2007 proposes a classification scheme that takes into account four key aspects: the structure of the problem which involves entities and their organization; the type of input data which can be deterministic or stochastic; the planning period which classifies papers on location/routing in single or multi-periods; and the solution method that distinguishes between exact or heuristic procedures. Prodhon and Prins [10] provide a literature review on the subject until 2013 in which extensions of the classical location/routing problems are included, such as multi-echelon problems, applications with multiple objectives or uncertain data. Drexl and Schneider [11] review papers until 2014 and consider only problems where the selection of the facilities is not implicitly determined by the routing decisions. Both recent reviews provide an updated bibliography on the subject.

Amongst the papers not included in the previous reviews, Amiri [12] proposes a mixed integer program which is solved using Lagrangean relaxation combined with a heuristic procedure. Ross and Jaramayan [13] formulate a strategic model as a binary integer program to determine the location and routing solution. Then, an executional model, which is a linear program, provides the amount of product shipped between nodes. To solve the model they integrate simulated annealing and tabu search. Olivares-Benitez et al. [14] generate approximate efficient sets for the biobjective mixed integer program which minimizes the cost of the location/routing problem and the longest transportation time from the plants to the customers. Borges Lopes et al. [15] propose an evolutionary algorithm whose chromosome represents a complete solution i.e. the collection of routes and uses local search in the mutation phase.

In general, the models proposed in the literature are binary integer programs or mixed integer programs with a single objective function which is the sum of the opening costs and the routing costs. A few papers in this category consider multiobjective optimization models. The objectives take into account the location or the operational costs as well as the covered demand, the work-load balance or the longest transportation time. In problems which involve hazardous waste management, a social cost is introduced which takes into account social rejection or transportation risk. A few papers propose exact approaches for solving the problem mainly based on branch-and-cut algorithms. For large-size instances, metaheuristics have been proposed aiming to find good quality solutions in short computing times. Thus, simulated annealing, tabu search, evolutionary algorithms or adaptive large neighborhood search metaheuristics have been developed.

4 Sourcing/Production Problem (SPP) and Supplier Selection/Inventory Problem (SSIP)

Purchasing is one of the most strategically activities in SCM because it provides the opportunity to reduce costs across the entire SC. Since the cost of raw materials represents a substantial percentage of the total product cost, an essential task in the purchasing process is supplier selection. In addition, a relevant problem in SCM is to determine the appropriate level of inventory at each stage involved in a SC. Both supplier selection and inventory management are closely related to the production problem. The production of many different products with short life cycles and the need to provide customers a better service at a reduced cost lead to the integration of production and inventory related decisions. Usually, the manufacturer purchases the raw material and/or semi-finished items from several preferred suppliers. These raw parts are stored at the manufacturing facility or transformed into final products, as they are requested to satisfy demand from retailers.

Aissaoui et al. [16] present a literature review until 2007 on supplier selection. They focus especially on the problem of determining the best mixture of suppliers and allocating orders so as to satisfy different purchasing requirements. Ben-Daya et al. [17] propose a nonlinear mixed integer model for the sourcing/production problem which is approximately solved by applying derivative-free methods. The single objective function is computed as the sum of the costs incurred by the supplier and the manufacturer (production setup cost, raw material ordering and holding costs, and finished product holding cost) and the retailers (ordering cost and finished product inventory holding cost). Sawik [18] proposes a mixed integer programming problem for determining the latest delivery dates of product-specific parts. Two different approaches are compared. In the monolithic approach, the manufacturing, supply and assembly schedules are determined simultaneously. In the hierarchical approach, first an assignment of customer orders over the planning horizon and thereby the finished products assembly schedule is determined, and then the parts manufacturing and the supply schedules are decided. Concerning the supplier selection/inventory problem, in general only the minimization of the total system costs is considered when tackling both the supplier selection and the lot size decision. Keskin et al. [19] and Ventura et al. [20] formulate mixed integer nonlinear programming models. Cárdenas-Barrón et al. [21] propose a mixed integer linear model which is solved with a reduce and optimize approach. Glock [22] presents a continuous mathematical model and suggests heuristic solution procedures. Ustun and Demirtas [23] propose a multiobjective mixed integer linear programming model whose aim is to maximize the total value of purchasing, to minimize the total cost and the total defect rate and to balance the total cost between periods. The most preferred efficient solutions are determined by considering the decision maker’s preferences. Rezaei and Davoodi [24] propose two multiobjective mixed integer nonlinear models. The three objective functions are based on cost, quality and service level. They develop a multiobjective evolutionary algorithm to approximate the set of Pareto solutions. Huang et al. [25] model the problem as a three-level dynamic non-cooperative game. The authors propose both analytical and computational methods to compute the Nash equilibrium.

5 Location/Inventory Problem (LIP) and Inventory/Routing Problem (IRP)

Location/inventory problems involve integrated decisions on the location of facilities such as distribution centers or factories, and inventory management. The aim is to minimize transportation and facility fixed costs as well as inventory holding and handling costs. These problems are closely related to the inventory/routing problems which have received more attention in the literature. The inventory/routing problem can be seen as an extension of the vehicle routing problem where the aim is to satisfy the demand of customers while minimizing the total distribution cost. In the inventory/routing problem, the amount to supply to each customer (inventory allocation decision) as well as the time for delivery is determined in order to minimize the inventory cost. Moin and Salhi [26] present an overview of the area of inventory routing and classify papers according to the planning horizon: single-period, multi-period and infinite horizon models. Two more papers on this topic are worth mentioning. Tancrez et al. [27] propose a nonlinear continuous mathematical model with a single objective function obtained as the sum of periodic transportation, inventory holding, fixed operational costs and handling costs. Melo et al. [28] propose a mixed integer linear program whose single objective function accounts for the total net SC cost defined as the sum of supply, transportation and inventory holding costs and fixed costs for operating the facilities. In both papers a heuristic procedure is used to solve the problem.

The review by Andersson et al. [29] covers inventory management and routing from an operations research perspective until 2009. It takes into account the relation between science and practice by considering papers which consist of both case studies based on real applications and theoretical contributions based on idealized models. Coelho et al. [30] give a comprehensive review of inventory routing problems until 2013, by categorizing them with respect to their structural variants and the information available on customer demand. Yu et al. [31], Day et al. [32] and Coelho and Laporte [33] deal with mixed integer linear models. The papers by Berman and Wang [34] and Kuhn and Liske [35] propose nonlinear models. Song and Savelsbergh [36] give a different perspective and focus on how to measure the performance of the distribution strategies and also on which factors contribute to the complexity of the problem.

Most papers on this topic propose a mixed integer optimization model with a single objective function which reflects the overall costs. Branch-and-cut techniques are used to develop algorithms to exactly solve small-scale problems. When considering real-life instances, most algorithms are based on heuristics which take advantage of decomposing the problem into several sub-problems.

In this category we also include papers which focus on the strategic design of a SC network and deal with location, routing, production and inventory. The papers by Ambrosino and Scutellà [37], Melo et al. [38], Hinojosa et al. [39], Thanh et al. [40], Manzini and Bindi [41], Sadjady and Davoudpour [42], Guerrero et al. [43], formulate mixed integer models with a single objective function aiming to operate the SC network at minimum cost.

6 Production/Distribution Problem (PDP)

Production and distribution planning are tactical decisions. In these problems, the assignment of available capacity in production sites, the inventory allocation to maintain a level of service of a retail or distribution facility and the distribution of the stocks to retailers have to be determined. Mula et al. [44] review mathematical programming models for SC production and transport planning until 2009. The authors propose eight classification criteria: supply chain structure, decision level, modeling approach, shared information, purpose, limitations, novelty and practical application. A later review until 2014 by Díaz-Madroñero et al. [45] propose a classification framework based on production, inventory and routing aspects, objective function structure and solution approach. Fahimnia et al. [46] provide a comprehensive review until 2010 and a classification of production/distribution models based on their degree of complexity and the level of simplification of the real-life scenario used. Adulyasak et al. [47] review until 2014 problem formulations and solution techniques to solve a production routing problem jointly optimizing production, inventory, distribution and routing decisions.

Other papers in this category include Elhedhli and Goffin [48], Manzini [49], Raa et al. [50] and Low et al., [51] which consider a single objective function accounting for the total cost. Liu and Papageorgiou [52] develop a multiobjective mixed integer linear model to simultaneously minimize total cost, total flow time and total lost sales. The authors apply the epsilon-constraint method and the lexicographic minimax method to find an efficient solution. Chan et al. [53] also consider several criteria (operating cost, order fulfillment lead time, and equity of utilization ratios) which are weighted to obtain a single objective function by using the analytic hierarchy process. Dawande et al. [54] present a study of conflict and cooperation aspects arising in SC when there is a conflict between the goals of the manufacturer and distributor. They conclude that the cost saving resulting from cooperation is usually significant. When collaborative planning is possible, Selim et al. [55] develop a multiobjective linear programming model which is solved by using a fuzzy goal programming approach. However, this collaboration is not always possible. Calvete et al. [56] focus on a hierarchical production/distribution planning problem that takes into account the existence of two decision makers who control the production and the distribution processes, respectively, and do not cooperate because of different optimization strategies. A bilevel mixed integer programming model is formulated for solving the problem. Amorin et al. [57] propose a multiobjective mixed-integer model to solve the integrated production and distribution planning of perishable goods which takes into account the value of freshness of the goods in addition to costs.

Most papers discuss either linear programming or mixed integer programming models. In contrast, nonlinear programming and multiobjective programming modeling approaches are scarcely applied in practice. Concerning the algorithmic approaches, only a few papers propose exact approaches, these being mainly based on branching processes improved with the introduction of valid inequalities or Lagrangean relaxation. Metaheuristic methods have also been proposed such as genetic algorithms, tabu search or ant colony optimization.

7 Findings, Conclusions and Further Research

In this paper we have reviewed the existing literature on integrated problems in SCM which connect several SC partners and which are related to strategic, tactical and operational decision making. The aim is to provide the researcher with a summary of the mathematical approaches which have been proposed to deal with these problems. The focus is on reviewing the manner in which the mathematical approach models the interdependencies among supply chain parties. Table 1 summarizes the findings. The analysis reveals that the mathematical models formulated in the papers are binary/integer/mixed-integer optimization problems, either linear or nonlinear. Most of them involve a single objective function which reflects the overall cost of the system and implicitly assumes a centralized decision making process. This mathematical approach allows the single decision maker to simultaneously solve all the problems involved. Only a few papers formulate multiobjective programs which allow us to take into account several conflicting objectives or the existence of several decision makers who are ready to collaborate. Even fewer papers introduce bilevel programming or game theory concepts to model the decision making process in decentralized systems without collaboration among partners. However, collaboration in the SC is not an easy task. Kampstra et al. [58] analyze the reality of the SC and describe the gap between the interest in supply chain collaboration and the relatively few recorded cases of successful application. Huang et al. [59] analyze the impact of sharing information on SC dynamics and demonstrate that understanding the advantages and limitations of each modeling approach can help researchers to choose the correct approach for studying their problem. It is worth pointing out that even in monolithic models, different mathematical programming formulations can capture the different points of view of agents in the SC such as customers and suppliers. Kalcsics et al. [60] illustrate how the mathematical program proposed depends on the decision maker the problem focuses on. Bertazzi et al. [61] also emphasize the impact of the objective function on the solution. It can reflect different decision policies and/or different decision makers, i.e. diverse organizational aspects of a company with a different emphasis on what is important in making a decision. In a decentralized decision making environment in which the SC agents operate independently in an organizational hierarchy, non-monolithic models such as bilevel programs are more appropriate. Cao and Chen [62] and Calvete et al. [56, 63] provide examples which show the differences between monolithic and non-monolithic models.

Table 1. Supply chain management problems and references in the paper.
Full size table

In our opinion, the mathematical approaches described in the literature under review (single objective programs, multiobjective programs and bilevel programs) contain the foundations of the deterministic optimization models that allow the integration of several processes of the SC involving either collaborative or non-collaborative decision makers. Nevertheless, it would be necessary to extend their use, especially the more complex models, in the context of real-system applications. It is worth mentioning the existence of a substantial number of papers devoted to SC configurations based on the concept of interdependence among partners in the SC. This contrasts with the scarce number of papers which use multiobjective optimization or hierarchical optimization for modeling the complexity of the decision processes involved in SCM. A future line of research should fill this gap and analyze the impact of using different mathematical approaches on the modeling of the SC performance.