Stochastic joint replenishment problem with quantity discounts and minimum order constraints

Abstract

This paper analyzes a logistics system involving a supplier who produces and delivers multiple types of items and a buyer who receives and sells to end customers. The buyer controls the inventory of each item by ordering at a preset time interval, which is an integer multiple of a base cycle, to meet the stochastic demands of the end customers. The supplier makes contracts with the buyer that specify that the ordered amount is delivered at the start of each period at a unit price determined by a quantity discount schedule. The contract also specifies that a buyer’s order should exceed a minimum order quantity. To analyze the system, a mathematical model describing activities for replenishing a single type of item is developed from the buyer’s perspective. An efficient method to determine the base cycle length and safety factor that minimizes the buyer’s total cost is then proposed. The single item model is extended to a multiple items joint replenishment model, and algorithms for finding a cost-minimizing base cycle, order interval multipliers, and safety factors are proposed. The result of computational experiments shows that the algorithms can find near-optimal solutions to the problem.

Appendix: Convexity of \(\varvec{TC}\left( {\varvec{T},\varvec{ k}} \right)\)

Hessian matrix of the expected cost function is

$$\left[ {\begin{array}{*{20}c} {\frac{{8{\mathcal{A}}\left( {T + L} \right)\sqrt {T + L} - \sigma khT^{3} - \sigma BG_{u} \left( k \right)\left( {3T^{2} + 12TL + 8L^{2} } \right)}}{{4T^{3} \left( {T + L} \right)\sqrt {T + L} }}} & {\frac{{h\sigma T^{2} + B\sigma \left( {1 - F\left( k \right)} \right)\left( {T + 2L} \right)}}{{2T^{2} \sqrt {T + L} }}} \\ {\frac{{h\sigma T^{2} + B\sigma \left( {1 - F\left( k \right)} \right)\left( {T + 2L} \right)}}{{2T^{2} \sqrt {T + L} }}} & {\frac{B}{T}\sigma \sqrt {T + L} f\left( k \right)} \\ \end{array} } \right].$$

The Hessian matrix is not positive definite because the determinants of the first and second principal submatrices are not always positive. Thus, the cost function is not a convex function. Additionally, since the (2, 2) entry of the Hessian matrix is positive in the domain of the variables, the cost function is strictly convex on k for a given T.