Supply Chain Optimization, Design, and Management by Ampazis Nicholas

Author:Ampazis, Nicholas...
Language: eng
Format: epub
Published: 2014-05-21T23:30:30.057000+00:00

Demand X is assumed to be a positive stochastic random variable with probability density function f(x) and cumulative distribution function F(x). The unit purchase cost paid to supplier i (i = 1, 2) is denoted by ci. The unit selling price is denoted by s and it is assumed that s > ci (i = 1, 2). The surplus stock that remains unsold at the end of the period can be sold to a secondary market at a unit salvage value r; it is assumed that r < ci (i = 1, 2). Moreover, k indicates the ‘loss of goodwill’ cost due to the shortage in products/parts; thus the total lost sales cost is the sum of the opportunity cost (s - ci) plus k.

Disruptions may occur only once for each of the two suppliers, during the selling period with probability pi (i = 1, 2). Moreover, when a disruption occurs a constant percentage of the order quantity Qi, denoted by yi (i = 1, 2), will be available in time to satisfy the demand during the selling period (the remaining quantity will not be delivered or it will be delivered after the end of the selling period). As a result, in case of a disruption the available quantity will be yiQi (see Figure 1), with yi capturing the ‘severity’ of the impact of the supply disruptions. Vlachos and Dekker (2003) discuss how yi can be modeled in an inventory system with commercial returns. The yi variables could model either the case of a limited production rate in a previous supply chain echelon (disruption in production) or the case of delayed delivery (disruption in transportation). In the first case, yi models the effective production capacity as a result of a production failure, while in the second yi corresponds to the part of the order quantity that may be utilized, since a specific part of the demand of the selling period might be lost in the case of late delivery.

Initially, when none of the supply channels face a disruption, that occurs with probability (1-p1)(1-p2), the expected single period profit G0(Q1,Q2) is obtained by the classical newsvendor problem analysis:

(2)

When a disruption occurs only to the first supplier (probability (1-p2)p1) only a fraction of Q1, initially ordered to the first supplier (y1Q1), can be now used to fill demand along with the order from the second supplier. The expected profit G1(Q1,Q2) is then expressed by the following equation:

(3)

Similarly, when a disruption occurs to the second supplier (probability (1-p1)p2), the expected profit G2(Q1,Q2) is expressed by:

(4)

Moreover, when disruptions occur at the same time to both suppliers (probability p1p2), the total available quantity to satisfy demand is y1Q1 + y2Q2, and the expected profit G12(Q1,Q2) is provided by:

(5)

Finally, the objective function of the optimization model (P) is the total expected profit G(Q1,Q2) obtained as the weighted sum of the expected profits given in (2), (3), (4) and (5) taking into account the probabilities of disruptions on none, on the first, on the second and on both supply channels respectively.

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