A stochastic linear programming approach to hierarchical production planning
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A stochastic linear programming approach to hierarchical production planning D Kira, M Kusy and I Rakita Concordia University, Montreal, Canada This paper examines production planning decisions. The process is formulated as a hierarchical production planning (HPP) model under uncertain demand. A review of HPP articles indicates that while current models do consider uncertainty as a part of their solution methods, a deficiency persists since these models fail to incorporate the uncertain demand explicitly in the formulation of the problem. A stochastic linear programming model (SLP) is proposed to better reflect reality and to provide a superior solution. The model remains computationally tractable despite the precise incorporation of uncertainty and the imposition of penalties when constraints are violated. A problem is introduced which illustrates the superiority of the proposed model over those currently being applied. Keywords: hierarchical; stochastic linear programming; simple recourse; production planning
Introduction The ever-increasing complexity in production management has generated a demand on a global scale for more effective techniques to optimize the allocation of scarce resources and to efficiently contend with exogenous random events such as supply shortages and the receipt of unexpected orders. Decisions made at higher levels in an organization will impact on planning and scheduling on the shop floor. Hierarchical production planning (HPP), is a multilevel model for decision-making that includes, at each level, an optimization problem whose solution provides constraints for successively lower levels in a process which terminates when a detailed schedule is produced for a small horizon. The HPP approach has been applied to a number of practical production planning problems some of which appear in Jaikumar1, Armstrong and Hax2, Fisher and Jaikumar 3, Keong Leong et al 4 and de Kok5. Dempster et al 6 conclude that there are two main reasons for using HPP. To reduce complexity This occurs since HPP involves breaking up a large problem into a series of smaller subproblems where the extent of the interaction between levels is minimal. This claim is easily understood when it is recalled that a scheduling problem is classified as NP-hard. This means that the optimal solution depends on a magnitude of computations that is proportional to eNT, where N refers to the number of items in the global model and T is the time
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horizon. Since higher levels involve grouping of items into families, the number of computations in the global model dramatically exceeds those of the HPP model for moderately large values of N and T.
To reduce uncertainty A global optimization problem would necessitate the specification of data well in advance of when the plan would have to be operationalized and would clearly be subject to a substantial degree of uncertainty. H
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