Pareto-optimal wholesale price contracts

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. 

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 ... 

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...

Next we consider the BB contracts with wholesale price w Pareto-optimal order quantity. The associated probability functions of achieving the supplier s and the retailer s profit targets are, respectively ... 

Notice that wholesale price w in aBB contract plays m important role for the contract to be Pareto optimal. When c upper bound w in order for the BB contraet to be Pareto optimal. On the other hand, when wlower bound w - c-v) m order for the BB eontractto be Pareto optimal. When w = iD, the BB contract is Pareto optimal regardless of the buy back price b, as long as the general requirement of w - (c - v) < b < w holds. 

Obviously the Pareto-optimal WP contract (see Theorem 1) can coordinate the supply chain with satisficing objectives, which results in the probability pair [1,1-F((r + t) r - c))]. However, the Pareto-optimal BB contracts (see Table 3) cannot coordinate the supply chain with satisficing objectives. This is because all three resulted probability pairs are dominated by either [1,1-F((t + /( -< ))] or [ -F t + t) r - c)), 1]. As for Pareto-optimal QF contracts (see Table 4), the probability pair from qXQFI is dominatedby [ -F Jt + t)l r-cJ), 1]. Therefore, the Pareto-optimal QF contracts in set QFl cannot coordinate such a supply chain. However, the Pareto-optimal QF contract in set QF2, which is a WP contract with wholesale price w, coordinates the supply chain. Hence, for a QF contract to coordinate the supply chain under the satisficing objective, it has to degenerate into a WP contract. 

Observe that, in our example, under the coordinating contract, the retailer always makes more profit than without the coordinating contract. This is not true in general under the low unit-revenue and high variability the retailer may be better off without the contract for high wholesale prices. The wholesaler, however, might be better off without coordination for low wholesale prices. Thus, Pareto optimality of the coordinating contract depends on the problem parameters. 

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