Volume discounting coordinates a supply chain effectively when demand is sensitive to both price and sales effort

In this paper, we use a simple and parsimoniousmodel to investigate the performance of volume discounting schemes (hereafter ‘‘[VD]’’) in a supply chain where the market demand is sensitive to both retail price ‘‘p’’ and sales effort ‘‘e’’ — hereafter called a ‘‘(p,e)-channel.’’ The problem is analyzed as a manufacturer-leading Stackelberg game. We first present, for the deterministic-system-parameter situation, contract-designing procedures under two contract formats; namely, a ‘‘regular’’ version of [VD] (hereafter ‘‘[RVD]’’) and a ‘‘continuous’’ version of [VD] (hereafter ‘‘[CVD]’’). Our solutions show that [RVD] cannot perfectly coordinate this (p,e)-sensitive channel; moreover, very often [RVD] leads to a lower channel efficiency than the simple price-only contract. In contrast, we show that [CVD] leads to perfect channel coordination — a significant result since most contract formats have been shown in the literature to be unable to coordinate a (p,e)channel. Next, we consider the more realistic situations in which the manufacturer is uncertain about one of the system parameters — specifically, either the market size ‘‘a’’ or the effort cost ‘‘Z’’. Our results show that, if Manu is uncertain about a, [RVD] becomes useless but the manufacturer can still use [CVD] to benefit himself. When the manufacturer is uncertain about Z, [CVD] remains useful (as expected); however, surprisingly, [RVD] can outperform [CVD] when both the mean value and the uncertainty of Z are sufficient high. These results underline the necessity of evaluating a contract format under various forms of systemparameter uncertainties — often at the expense of analytical tractability. & 2012 Elsevier Ltd. All rights reserved.