Optimal design of multi-echelon supply chain networks under normally distributed demand

Abstract

This paper addresses the optimal design of a multiproduct, multi-echelon supply network under uncertainty of demand. The network consists of multiproduct production sites, warehouses and distribution centers and decisions about the selection of facilities and their capacity are taken. Furthermore, information about the flows of products transferred and the safety stock at each distribution center is derived. The lead time of an order to a customer is computed, using the probabilities of overstocking and understocking. All these decisions are incorporated into a single period mixed integer non-linear programming problem (MINLP) which minimizes cost. Linearization techniques for selected highly non-linear terms of the models are explored in order to reduce the computational effort for the solution of the model. Finally, a sensitivity analysis is performed by changing product demand parameters and assessing their effect on the supply chain structure.

Appendix

Proposition

The total, ordering, transportation, working inventory and replenishment cost is presented with the following term:

$$\sum_{i} \sum_{k} \biggl( \mathit{alp}_{ik} \cdot \sum_{l} \chi \cdot \mu_{il} \cdot X_{kl} \biggr) + \sum _{k} \sum_{l} \sqrt{2h \cdot ( F_{k} + g_{k} ) \cdot X_{kl} \cdot \sum _{i} ( \chi \cdot \mu_{il} )} $$

Proof

Due to the normally distributed demand, the handling cost of the inventory kept at a distribution center is represented by a (Q, r) policy, while orders can be expressed with the EOQ model. Transportation, inventory and working costs are given by the following mathematical type (Shen et al. 2003):

$$ F \cdot n + \biggl( g + \mathit{alp} \cdot \frac{D}{n} \biggr) \cdot n + h \cdot \frac{D}{2 \cdot n} $$

(A.1)

The first term refers to the standard total ordering cost per year (Fn), the second term refers to the annual transportation cost $( g + \mathit{alp} \cdot \frac{D}{n} ) \cdot n$ and the third term refers to the annual inventory cost ($h \cdot \frac{D}{2 \cdot n}$). If a cost function is considered with respect to n (number of replenishments), the following function:

$$ C(n) = F \cdot n + \biggl( g + \mathit{alp} \cdot \frac{D}{n} \biggr) \cdot n + h \cdot \frac{D}{2 \cdot n} $$

(A.2)

Setting the first derivative of the cost function equal to zero (C′(n)=0) in order to obtain a minimum for this function, the following equation occurs:

$$ F + g - \frac{D \cdot h}{2 \cdot n^{2}} = 0 $$

(A.3)

In order to obtain the critical point, the above equation is solved with respect to n. This yields the following:

$$ n^* = \sqrt{\frac{D \cdot h}{2 \cdot (F + g)}} $$

(A.4)

The second derivative of cost function is:

$$ C'' ( n ) = \frac{D \cdot h}{2 \cdot n^{3}} $$

(A.5)

In order to test whether n ^∗ is the minimum of cost function, it is substituted in the second derivative. The following derives from the substitution:

$$ C'' \bigl( n^* \bigr) = \frac{D \cdot h}{2 \cdot ( \sqrt{\frac{D \cdot h}{2 \cdot (F + g)}} )^{3}} > 0 $$

(A.6)

Based on second order derivative theorem for univariate functions, n ^∗ is the minimum of cost function C(n) whereas substituting n ^∗, the minimum cost is obtained by the following:

$$ \mathit{alp} \cdot D + \sqrt{2 \cdot h \cdot (F + g) \cdot D} $$

(A.7)

In term (A.7), corresponding to the minimum cost of C(n), demand is further analyzed by the product of annual mean demand of product i from customer l with the binary variable X _kl which refers to the connection of distribution center k with customer l. Thus, the total ordering, transportation, working inventory and replenishment cost after applying indices, the following desired term is formulated:

$$\sum_{i} \sum_{k} \biggl( \mathit{alp}_{ik} \cdot \sum_{l} \chi \cdot \mu_{il} \cdot X_{kl} \biggr) + \sum _{k} \sum_{l} \sqrt{2h \cdot ( F_{k} + g_{k} ) \cdot X_{kl} \cdot \sum _{i} ( \chi \cdot \mu_{il} )} $$

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