Simulating Complex Supply Chain Relationships Using Copulas

Introduction

Supply chains involve complex interactions between different aspects of the research question and it is extremely difficult (if not impossible) to derive closed-form analytical solutions, leaving simulation as the only available alternative. The use of simulation in functional areas of business is well documented and includes accounting, finance, marketing, and other areas (Pegden et al., 1995). The analysis of supply chain systems is no exception. Recently, Waller and Fawcett (2012) observed that “…as simulation methods and technology are advanced, metaheuristics exist to find near optimal solutions, and powerful optimization is now in the hands of nearly any manager, let us not just model these simple systems, let us model the realistic, complex systems—in for a penny, in for a pound.” The statement “When all else fails, there is simulation!” (Evers and Wan, 2012) is by no means an exaggeration.

The Council of Supply Chain Management Professionals defines supply chain management as encompassing “the planning and management of all activities involved in sourcing and procurement, conversion, and all logistics management activities. Importantly, it also includes coordination and collaboration with channel partners, which can be suppliers, intermediaries, third-party service providers, and customers. In essence, supply chain management integrates supply and demand management within and across companies.” When modeling this complex system, simulation may be the best alternative as the analytical tool (Waller and Fawcett, 2012). Swaminathan et al. (1998) also strongly argue for the use of simulation in supply chains.

This is because complex interactions between different entities and the multitiered structure of supply chains make it difficult to utilize closed-form analytical solutions. Benchmarking solutions provide insights into current trends but are not prescriptive. This leaves simulation as the only viable platform for detailed analysis for alternative solutions. Evers and Wan (2012) offer an excellent analogy in support of simulation. Griffis et al (2012), Rogers et al (2012), and Bartolacci et al (2012) also provide convincing evidence of the power of simulation as a methodological tool.

Constructing and modeling supply chains in practice, however, presents many challenges. Supply chains often involve “one or more families of related products”, involving different entities or organizations that have different objectives, that are “highly interdependent when it comes to improving the performance of the supply chain in terms of objectives such as on-time delivery, quality assurance, and cost minimization.” (Swaminathan et al., 1998). Hence, in order to effectively analyze supply chains using simulation, it is necessary to model the interdependence between different aspects of the model. Such interdependence may manifest itself in the demand for products, in the delivery of raw materials, in set up times, etc. Often, distribution of demand, delivery times, and set up times are not normally distributed, and must be described using non-normal distributions such as the Gamma, Weibull, Beta, and other similar skewed distributions (Burgin, 1975; Wagner et al, 2009). Furthermore, it is possible that the demand for a given product has a certain marginal distribution (say Gamma), while that of another has a completely different distribution (say Weibull), but the two demand distributions are related. Thus, in order to analyze the performance and management of supply chains and similar complex systems, the ability to simulate random variates whose marginal distribution and dependence are specified is a necessary pre-requisite.