Designing a stochastic distribution network model under risk

Abstract

In this paper, we present a stochastic supply chain network model under risk, in which three tiers of decision makers of suppliers, distribution centers (DCs), and customers seek to determine their optimal plans. Unlike other studies in the extant literature, we use financial risk measure to model and for the first time, a new methodology based on finance literature is used to optimize the conditional value-at-risk (CVaR) measure for the problem. In fact the aim of this paper is to show the benefits of considering CVaR to control the risks in the supply chain networks. Also in order to obtain the economics of scale, a piecewise linear transportation cost function is applied between suppliers and DCs. We formulate the problem as a convex mixed integer program. A two-phase (constructive and improvement) heuristic method is developed to solve the problem. In the constructive phase, an initial solution is built randomly, and in improvement phase, this solution is improved iteratively by using a hybrid algorithm combining Tabu search and simulated annealing methods. Numerical experiments demonstrate the practicability of considering CVaR in the model which is promising in the supply chain network area, and also the significant efficiency of the heuristic method in terms of CPU times.

Appendix

Definition of coherent risk measure: $\rho :\delta \to \Re$ (δ is a set of stochastic variables) is a coherent risk measure if it has the following four properties.

• a.
  $$ \matrix{{*{20}{c}} {X \geqslant Y \Rightarrow \rho (X) \geqslant \rho (Y)} \hfill & {\forall X,Y \in \delta } \hfill \\ } $$
• b.
  $$ \matrix{{*{20}{c}} {\rho \left( {X + c} \right) = \rho (X) + c} \hfill & {\forall X \in \delta, \,c \in \Re } \hfill \\ } $$
• c.
  $$ \matrix{{*{20}{c}} {\rho \left( {\lambda \,X} \right) = \lambda \rho (X)} \hfill & {\forall X \in \delta, \,\lambda > 0} \hfill \\ } $$
• d.
  $$ \matrix{{*{20}{c}} {\rho \left( {\lambda X + (1 - \lambda )Y} \right) \leqslant \lambda \rho (X) + (1 - \lambda )\rho (Y)} \hfill & {\forall X,Y \in \delta, \lambda > 0} \hfill \\ } $$

The convexity axiom (d) is a key property from both the methodological and computational perspectives. In a mathematical programming context, it means that ρ is a convex function of the decision vector. This, in turn, entails that the minimization of risk over a convex set of decision variables constitutes a convex programming problem, amenable to efficient solution procedures.