Pareto-optimal order quantity

In the next three subsections, we design the Pareto-optimal contracts— that is, the Pareto-optimal contract parameters—for each contractual form. Moreover, we identify the associated Pareto-optimal order quantity and the maximal probabilities of achieving the target profits for the supplier and the retailer. 

It can be seen from Figure 2 that, given a WP contract with parameter set 9=w, the unique feasible Pareto-optimal order quantity is given by ... 

The answer to the above example is summarized in Table 1. Under a WP contract with a wholesale price w, q (w) and P/(w) denote, respectively, the associated Pareto-optimal order quantity and the retailer s maximal probability of achieving her profit target. The Pareto-optimal wholesale price is given by vP = 22. Note that under a feasible WP contract with a wholesale price w, the supplier s profit is deterministic and F (w) = 1 always holds. 

In Table 2, q is the corresponding Pareto-optimal order quantity for the satisficing objectives, and qj is the optimal order quantity for the objective of expected profit maximization. Notice that the performance of the supply chain... 

Table 1. The associated Pareto-optimal order quantity and the retailer s maximal probability of achieving her target profit under a WP contract with wholesale price w...

The three sets of BB contracts summarized in Table 3 are Pareto optimal. The second column of the table identifies the conditions for the contract parameter set 6 = w,b to be Pareto optimal. The associated Pareto-optimal order quantity and maximalprobabilities of achieving the target profits for the supplier and retailer are given by q 0), Pfff), and P 6), respectively. 

Next we consider the BB contracts with wholesale price w probability functions of achieving the supplier s and the retailer s profit targets are, respectively ... 

Forthe purpose of illustration, we considerthe same decentralized supply chain as the one in the previous sections. The terms oftrade between the two agents are specified by a simple WP contract. For simplicity, we assume the demand follows a uniform distribution on [0, z]. If the supply chain is centralized and both agents adopt the EPM objective, it can be verified that the Pareto-optimal order quantity is given by F ((r-c)/(r-v))= r-c)zl r-v) and the maximal expected profit is given by ... 

Therefore, the Pareto-optimal order quantity forthe supply chain is =max (q q). The associated probability for the supplier to achieve his target is 1, and the associated expected profit of the retailer is given as ... 

Therefore, both q 6) and q 9) are Pareto-optimal order quantities. It can be verified that the order quantity q 9) will eventually lead to the Pareto-optimal set QF2 in Table 4. On the other hand, if the retailer orders q 9), the probabilities for the supplier and the retailer are given by ... 

Since both PJqq 9),9) and PJ(q 9),9) increase with respect to w, we choose w (u,d)= w. The associated Pareto-optimal order quantity q (9) then becomes ... 

A WP contract is Pareto optimal if the wholesale price w = w. The associatedPareto-optimal order quantity and maximal probabilities of achieving the targets are given by ... 

IfZ) > q or D probability pair with the allocation rule . iq> D> q the order quantity q leads to the probability pair [1,1] for the supplier and retailer. Hence, q Pareto dominates q under the case oi q> D > q . In summary, the order quantity q is Pareto optimal. 

The relevant quantities for the Pareto optimization are the maximum displacement Xo of the friction device (this should be small in order to ensure safety of the supports) and the maximum structural displacement Xs (measured relative to the device, this should be small to ensure seismic isolation). As the quantities are random, characteristic values (i.e., mean value plus three times the respective standard deviation) are chosen for the objectives. 

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