In this work we address the steady state optimization of a supply chain model that belongs to the class of vendor managed inventory, automatic pipeline, inventory and order based production control systems (VMI-APIOBPCS). The supply chain is optimized with the so-called normal vector method, which has specifically been developed for the economic optimization of uncertain dynamical systems with constraints on dynamics. We demonstrate that the normal vector method provides robust optimal points of operation for a number of scenarios. Since the method strictly distinguishes economic optimality, which is treated as the optimization objective, from dynamical requirements, which are incorporate by appropriate constraints, it provides a measure for the cost of stability and robustness as a desired side-effect.
Supply chain dynamics have been investigated with a variety of methods. Wang et al. (2009) review recent publications and group the existing methods into three classes, namely (i) those based on control theory, (ii) those based on behavioral science, and (iii) practitioner approaches. Approaches that belong to the first class (e.g., Disney and Towill, 2002a and Lin et al., 2004) use transfer functions found with Laplace- or z-transforms to analyze the stability of continuous time and discrete supply chain models, respectively. Investigations that fall into category (ii) are usually used for models which describe business games. For this class Lyapunov exponents (e.g., Hwarng and Xie, 2008 and Larsen et al., 1999) can be used to characterize different types of dynamic behavior. The third category groups those approaches that are based on extensive simulation studies (e.g., Nagatani and Helbing, 2004 and Sahin et al., 2008).
Supply chain dynamics play a crucial role in optimization. Unfortunately, economic optimality and optimal dynamical properties can often not be attained simultaneously. This is not surprising, since any non-trivial optimization pushes the optimized system to some or all of its operational boundaries, among them possibly its stability boundaries. There exist various approaches to considering dynamical properties when optimizing supply chain operation. Sarmiento et al. (2007) minimize the area under the curves of step responses in order to achieve stable operation. Similarly, Disney et al. (2000) and Disney and Towill (2002b) minimize the area under the step response of the actual inventory. In addition, these authors minimize a measure of the Bullwhip effect.
If optimization results in a mode of operation that lies on, or close to, any operational or stability boundaries, it is important to take model uncertainty into account. For an optimal point on, or close to, any such boundary, even a slight change in parameters may result in a violation of that boundary. Model uncertainty is particularly important, since parameters of supply chain models are quite imprecise. Following Disney et al. (2000) we consider uncertainties of up to 25% in the present paper.
Model uncertainty has been accounted for by several authors. Stochastic programming (e.g., Azaron et al., 2008), fuzzy programming (e.g., Mitra et al., 2009 and Lin et al., 2010), and probabilistic programming (e.g., You and Grossmann, 2008) are three popular approaches to handle uncertainties in supply chain optimization (see Mitra et al., 2008 for a summary of recent literature). Note that these approaches do not guarantee stability or other dynamical properties of the resulting optimal points, however.
In the present paper we apply the so-called normal vector method (Mönnigmann and Marquardt, 2002). The normal vector method was developed for solving optimization problems in which stability or related dynamical properties have to be guaranteed for systems with uncertain parameters. In contrast to the methods used by Disney et al. (2000), Disney and Towill (2002b) and Sarmiento et al. (2007), the normal vector method does not treat the dynamical properties as an optimization objective, but dynamical properties become constraints of a constrained optimization problem. Note that this implies a separation of economic optimality, which is considered in the cost function, from required dynamical properties, which are treated in the constraints. In particular, there is no need for combining economic optimality and dynamical requirements into a multi-objective cost function.
The normal vector method has originally been designed for the optimization of continuous-time systems that can be modeled by sets of parametrically uncertain ordinary differential equations (Gerhard et al., 2008, Mönnigmann and Marquardt, 2002, Mönnigmann and Marquardt, 2003, Mönnigmann and Marquardt, 2005 and Mönnigmann et al., 2007). It has recently been extended to discrete-time systems with uncertain parameters (Kastsian and Mönnigmann, 2008 and Kastsian and Mönnigmann, 2010). In fact, supply chain optimization served as an example in these publications. The separation of economic optimization from dynamical constraints for robustness is achieved for the first time in the present paper, however.
The paper is organized as follows. Section 2 introduces the model of the considered supply chain. This model belongs to a class of systems referred to as vendor managed inventory, automatic pipeline, inventory and order based production control system (VMI-APIOBPCS for short) (Disney and Towill, 2002a and Disney and Towill, 2002b). Section 3 describes the cost function. Section 4 presents the result of an optimization without any constraints on the dynamics, which serves as a reference in the remaining sections. This is followed by a discussion of the stability of the supply chain in Section 5. The normal vector method is introduced in Section 6 and applied to the supply chain in Section 7. A summary and conclusion are given in 8 and 9, respectively.
In this paper we have shown that the optimization method known as the normal vector method can successfully be applied to solving supply chain optimal design problems. In the normal vector method, economic optimality is treated in the objective functions, while desired dynamical properties, such as guaranteed stability, are addressed by augmenting the optimization problem with appropriate constraints. This is in contrast to previous approaches, where economic and dynamical properties have been combined into one multi-objective function. We claim that the strict separation of economic from dynamic properties simplifies to quantify the impact of desired dynamical properties on optimality of operation. Furthermore, the results demonstrate that normal vector method provides a systematic approach to treating uncertain parameters.
