Joint Pricing And Lot Sizing Problems: Models And Algorithms

Lot sizing problem deals with the case that under some constraints, the production (order) department of the firm decides when to produce (order), which item to be produced (ordered), how many products to be produced (ordered) in order to minimize the total cost in the planning horizon. By making rational decision on production progress and allocating properly a kind of resources, lot sizing problems contribute to reduce the overall costs and enhance the firm's competitiveness.In the classical lot sizing literature, decisions on pricing and lot size are made by decentralized policy in which the marketing department sets the price, the market responds with a specific demand, and the production department makes the lot sizing decision that minimizes the total cost while satisfying demand. This policy usually results in conflict between price and market share so that the firm's profit can not be maximized. Market share increases, but marginal profit may vastly decrease if the price is set on the low side, on the other hand, marginal profit increases, but market share may vastly decrease if the price is set on the high side.In contrast, the cooperative process determines the price and lot sizing at a time. The cooperative approach can rectify these drawbacks mentioned above. Moreover, the coordinated policy makes production activity more flexible and reduces production variability.By introducing price as decision variable into Wagner-Whitin model or its extensions, this paper studies coordination of pricing and lot sizing problems. The details are gives as follows.1. Coordination of pricing and lot sizing problem with backlogging is considered. (l)By introducing dynamic pricing into Zangwill model, the cooperative problem is formulated as non-linear mixed-integer program. After analyzing the characteristics of the optimal solution, an exact algorithm based on dynamic programming is developed, by which the optimal production planning and optimal price are solved in O(T~3) time. (2) By introducing fixed pricing into Zangwillmodel, the cooperative problem is formulated as non-linear mixed-integer program. An exact algorithm based on iteration is presented, by which the optimal production planning and optimal price are solved in O(T~6 log T) time.2. Coordination of dynamic pricing and lot sizing problem with inventory bounds is addressed. The cooperative problem is formulated as non-linear mixed-integer program. By the means of the concept of subplan, the problem can be solved in in O(T~4) time by first computing all possible subplans and then searching the optimal combination or concatenation of subplans using dynamic programming.3. Coordination of pricing and lot sizing problem with capacity constraints is considered. (1)Assuming that there is one-to-one correspondence between price and demand in each period, price is function of demand, and capacity is time-varying, the cooperative problem is formulated as mixed-integer program. After analyzing characteristics of the optimal solution, an exact algorithm based on dynamic programming is developed, by which the optimal production planning and optimal price are solved in O(T~6) time. (2)The problem of jointly determining prices and productionschedules is addressed for a set of items that are produced on the same production equipment. Under the assumptions that the production setup costs are negligible, that capacity is constant in the planning horizon, and that demand is seasonal but price dependent, the special structure of the problem is exploited to develop a solution procedure based on iteration which includes a linear programming problem solved by dual theory and a simple non-linear programming problem.4. Dynamic prices as the decision variables are introduced into multi-item capacitated lot sizing problem with backlogging (MCLSPB). The problem is formulated as a mixed-integer program. Lagrangean relaxation is applied to decompose the multi-item capacitated problem into a set of single-item uncapacitated sub-problems. A feasible solution can be found easily, unlike MCLSPB. Given some small numberε(less than 0.1), the gap (percent distance between upper and lower bound on the objective) is less thanεthrough several iterations in cooperative policy. In contrast, the gap is less than e through 50-70 iterations in decentralized policy.5. In the case of market segmentation, the problem of setting prices and choosing production quantities is studied for a single backlogging product over a finite horizon for a manufacturer facing price-sensitive demands. Under assumption that demand is linear function of price in each period for each item, the cooperative problem is formulated as a quadratic programming model. After characterizing properties of the optimal solution, a dynamic programming-based algorithm is proposed, by which the optimal pricing strategy and the optimal production planning are solved in O(T~3)time.6. In the case of market segmentation, the problem of setting prices and choosing production quantities is considered for a single product over a finite horizon for a constant capacity-constrained manufacturer facing price-sensitive demands. Under assumption that there is one-to-one correspondence between price vectors and demand vectors in each period for each item, the cooperative problem is formulated as a quadratic programming model. After characterizing properties of the optimal solution, a dynamic programming-based algorithm is developed, by which the optimal pricing strategy and the optimal production planning are solved in O(T~6) time.The extensive research results presented above not only enrich the content of production-inventory control theory but also provide the models and algorithms for integration of the production-inventory sub-system and marketing sub-system in the enterprise management software, such as MRPII, ERP et al, and provide more sufficient scientific evidence for decision makers making production decision and pricing decision.