Convex optimisation for aggregate production planning

Abstract

This paper presents a new solution approach to the problem of aggregate production planning (APP). As identified by many researchers, the APP cost function is convex and piecewise. Thus, the convex optimisation approach can be applied to the APP problem. Solving the APP problem using convex optimisation is attractive since it leads to an improved solution over the classical solution methods and it can be applied to a wider range of functions. The classical Linear Decision Rule model of APP is solved using convex optimisation and the resulting solution is compared to three solution approaches which have been historically used to solve this model. The results suggest that convex optimisation may be an effective approach for solving certain types of planning models. IntroductionIn general, production is a process of the transformation of inputs (raw materials, information, etc.) into finished products (goods or services). To effectively manage a large number of components of a manufacturing system, a considerable number of decisions should be made on several organisational levels. The taxonomy proposed by Anthony (1965) classifies managerial decisions into three categories: strategic planning, tactical planning and operations control. This taxonomy has been used by many researchers (see e.g. Shim et al. 2002;Kempf, Keskinocak, and Uzsoy 2011).Strategic planning decisions have long-term implications and are affected by both internal and external information. Tactical planning is a middle-level activity which connects strategic planning and operations control. For tactical planning, the basic problem to be resolved is the allocation of resources such as capacity, work force availability, storage and resource distribution over a medium range planning horizon. Operations control decisions deal with short-term operational and scheduling problems which require complete disaggregation of the information generated at higher levels of the taxonomy. At the tactical level, aggregate production planning (APP) models are concerned with determining the production rate and work force level that should be set over a given horizon in order to meet the total demand and minimise the total expected cost (Kempf, Keskinocak, and Uzsoy 2011).All APP techniques face a problem of trade-off between the accuracy of the model in capturing the relevant features of the production-planning environment and the resulting model complexity (Nam and Logendran 1992). The accuracy means that the model precisely describes the problem. The complexity implies that an optimal solution algorithm of solving the model exists and small changes in the model will not require a new algorithm. Some of APP models are not accurate, because they assume linear or quadratic cost function which does not match the data (Groff and Muth 1972; Kempf, Keskinocak, and Uzsoy 2011). Other APP models use algorithms that do not fit the complexity requirement or do not guarantee an optimal solution. The above-mentioned problems are one of the main reasons why APP techniques are not...

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