A Model for Partial Product Complementarity and Strategic Production Decisions under Demand Uncertainty

Abstract

This paper considers a general industrial setting where multiple manufacturers each produce a different product and sell it to the markets. These products are partially complementary in the sense that there is a common demand stream that requests all these products as complementary sets and there are streams of individual demands each requesting only one of the products. All demands are uncertain and may follow any general, joint distributions. Facing demand uncertainties, the manufacturers each choose a production quantity for its product with an objective to maximize its own expected profit. We formulate the problem as a non‐cooperative game to study the strategic interactions of such firms and their implications to supply chain performance. We show that such a game may have numerous equilibria. Among all the possible equilibria, however, we prove that there always exists a unique one that maximizes each and every manufacturer's profit, and we derive an explicit solution for this Pareto‐optimal equilibrium point. We further study the optimal solution for a centralized system and compare it with the decentralized solution. Managerial insights are drawn as to how system parameters and control mechanisms affect firms' decisions and performance.

Capacity games for partially complementary products under multivariate random demands

“…There exist two switching points of a 0 such that if a 0 is small (a 0 ≤ 90 in the example), we have p * 0 < p * , and if a 0 is medium (a 0 ∈ [95, 225]), then we have p * 0 = p * , and if a 0 is large (a 0 ≥ 230), we have p * 0 > p * . As illustrated by the example in Table 2, our numerical results also indicate that if the number of firms (n) is small (n ≤ 2 in the example), each firm would charge a lower price for the dedicated demand D than for the common demand D 0 , i.e., p * < p * 0 ; if the number of firms is medium (n ∈ [3,5]), each firm would charge the same price for both demands, i.e., p * = p * 0 ; if the number of firms is large (n ≥ 6), each firm would charge a higher price for the dedicated demand than for the common demand, i.e., p * > p * 0 . Intuitively, if the number of firms (n) increases, the competition among the firms gets stronger in the common market, or the system becomes more decentralized, so each firm tends to lower its price p 0 , while the price charged to fill a unit of the dedicated demand stream, p, could either increase or decrease in n.…”

“…and q i ≤ q 0 −i , then (5) implies that both E i ( q ) and E i ( q) become independent of q 0 −i and q 0 −i , respectively. Therefore, we have E i ( q ) ≥ E i (q i | q S ) ≥ E i (q i | q S ) = E i ( q).…”

“…, n, each firm shall try to satisfy the common demand D 0 as much as possible and then its dedicated demand D i . This special case has been analyzed in [5]. For the sake of comparison with the other two cases, we summarize the results of [5] in this subsection.…”

“…In par with Proposition 1 of section 3.1, we restate Proposition 1 of [5] below: PROPOSITION 2: For any i = 1, . .…”

“…In par with Theorem 1 of section 3.1, we restate Theorem 2 of [5] below: THEOREM 2: Without loss of generality, assumeQ 1 = min i=1,...,n {Q i }. There always exists a unique Pareto-optimal Nash equilibrium, denoted by (Q …”