Joint pricing and production planning of multiple products

Abstract

Many industries are beginning to use innovative pricing techniques to improve inventory control, capacity utilisation, and ultimately the profit of the firm. In manufacturing, the coordination of pricing and production decisions offers significant opportunities to improve supply chain performance by better matching supply and demand. This integration of pricing, production and distribution decisions in retail or manufacturing environments is still in its early stages in many companies. Importantly it has the potential to radically improve supply chain efficiencies in much the same way as revenue management has changed the management of the airline, hotel and car rental industries. These developments raise the need and interest of having models that integrate production decisions, inventory control and pricing strategies.In this thesis, we focus on joint pricing and production planning, where prices and production values are determined in coordination over a multiperiod horizon with non-perishable inventory. We specifically look at multiproduct systems with either constant or dynamic pricing. The fundamental problem is: when the capacity limitations and other parameters like production, holding, and backordering costs are given, what the optimal values are for production quantities, and inventory and backorder levels for each item as well as a price at which the firm commits to sell the products over the total planning horizon. Our aim is to develop models and solution strategies that are practical to implement for real sized problems.We initially formulate the problem of time-varying pricing and production planning of multiple products over a multiperiod horizon as a nonlinear programming problem. When backorders are not allowed, we show that if the demand/price function is linear, as a special case of the without backorders model, the problem becomes a Quadratic Programming problem which has only linear constraints. Existing solution methods for Quadratic Programming problem are discussed. We then present the case of allowed backorders. This assumption makes the problem more difficult to handle, because the constraint set changes to a non-convex set. We modify the nonlinear constraints to obtain an alternative formulation with a convex set of constraints. By this modification the problem becomes a Mixed Integer Nonlinear Programming problem over a linear set of constraints. The integer variables are all binary variables. The limitation of obtaining the optimal solution of the developed models is discussed. We describe our strategy to overcome the computational difficulties to solve the models.We tackle the main nonlinear problem with backorders through solving an easier case when prices are constant. This resulting model involves a nonlinear objective function and some nonlinear constraints. Our strategy to reduce the level of difficulty is to utilise a method that solves the relaxed problem which considers only linear constraints. However, our method keeps track of the feasibility with respect to the nonlinear constraints in the original problem. The developed model which is a combination of Linear Programming (LP) and Nonlinear Programming (NLP) is solved iteratively. The solution strategy for the constant pricing case constructs a tree search in breadth-first manner. The detailed algorithm is presented. This algorithm is practical to implement, as we demonstrate through a small but practical size numerical example.The algorithm for the constant pricing case is extended to the more general problem. More specifically, we reformulate the timevariant problem in which there are multi blocks of constant pricing problems. The developed model is a combination of Linear Programming (LP) and linearly constrained Nonlinear Programming (NLP) which is solved iteratively. Iterations consist of two main stages: finding the value of LP’s objective function for a known basis, solving a very smaller size NLP problem. The detailed algorithm is presented and a practical size numerical example is used to implement the algorithm. The significance of this algorithm is that it can be applied to large scale problems which are not easily solved with the existing commercial packages.