Practical solution approaches to solve a hierarchical stochastic production planning problem in...

This paper deals with a Hierarchical Stochastic Production Planning (HSPP) problem of Flexible Automated Workshops (FAWs), each with a number of Flexible Manufacturing Systems (FMSs). The problem includes not only the standard (demand, capacity and material supply) uncertainties but also uncertainties

in processing times, necessity for rework and scrap. In contrast to most work that only considers either single period or infinite horizons, we also considers multiple time periods and multiple products. One objective of this paper is to determine a cost minimizing production plan for each FMS taking into consideration work-in-process inventory, work centers overload and underload, cumulative over- and under-production of finished products over a finite time horizon. The HSPP problem is formulated as a stochastic nonlinear programming model whose constraints are linear but whose objective function is piecewise linear. To facilitate the solution procedure, the model is first transformed into a deterministic nonli near programming model and then into a linear programming model. For medium- or small-scale problems, Karmarkar's algorithm is applied to obtain the solution. For large-scale problem, an interaction/prediction algorithm is used. The effectiveness of these approaches is benchmarked against the linear programming method in Matlab 5.0 in various HSPP settings.

1. Introduction

Stochastic Production Planning (SPP) in a Flexible Automated Workshop (FAW) in China is an important problem. Since the scope of Production Planning (PP) problems generally prohibits a monolithic modeling approach, a Hierarchical Production Planning (HPP) approach has been widely advocated in the PP literature (Davis and Thompson, 1993) which can further be classified into two categories: deterministic or stochastic.

The most common deterministic approaches employ the following concepts: (i) product disaggregation (Bitran et al., 1981; Graves, 1982; Davis and Thompson, 1993; Simpson and Erenguc, 1998; Simpson, 1999); (ii) temporal decomposition (Malakooti, 1989; Qiu and Burch, 1997); (iii) process decomposition (Villa, 1989; Yan, 1997; Yan and Jiang, 1998); and (iv) event-frequency decomposition (Kimemia and Gershwin, 1983). On the other hand, existent articles on stochastic production planning problems mostly focus on uncertainties in demand, capacity and material supply in either a single-period or infinite-horizon setting (Bassok and Akella, 1991; Ciarallo et al., 1994; Ishikura, 1994; Kasilingam, 1995; Hwang and Medini, 1998).

By way of exception, Bitran and Leong (1992) examine deterministic approximations to multi-period, multi-item production planning problems in environments with stochastic process yields and substitutable demands. Yan (2000a,b) proposes two new approaches to optimally decompose with (and without) respect to delay interaction of production plans for FAWs. By constructing linear quadratic models of SPP problems and using interaction/prediction, FAW's medium-term product demand plans are decomposed at a high speed into short-term plans that are to be executed by FMSs in the FAW. These approaches combine the principles of both temporal and process decomposition with the organizational structure of the FAW and are capable of solving very large HSPP problems.

A key drawback in Yan's approaches is that the overproduction penalty and the underproduction penalty in the objective function are the same, and so are the overload penalty and the underload penalty. In practice, while overproduction often results in higher Finished-Goods Inventory (FGI), underproduction would often lead to revenue loss and poor customer service. Hence penalty cost due to underproduction is usually higher. On the other hand, while underload usually results in low resources utilization, overtime wages resulting from overload could be several times as high as those for normal hours. Hence overload penalty is usually higher than underload penalty. Yan's SPP models are thus limited in finding practical solutions for the FAWs.

In this paper, an HSPP model with a nonlinear objective function and linear constraints is proposed to overcome the above problems. In addition to the common (demands, capacities and material supply) uncertainties considered in the PP literature--we also consider uncertainties due to processing times, rework and scrap. Moreover, the structure of the system under consideration is quite complex as each FAW could consist of a number of FMSs, each of which could be made up of several work centers. Since we also consider multi-period and multi-product, finding practical approach to solve the resulting stochastic nonlinear programming poses a key challenge.

Most stochastic programming problems in the literature are solved for single-period or an infinite-horizon (Bitran and Leong, 1992; Higle et al., 1994; Norkin et al., 1998). On the other hand, solution for multi-stage stochastic optimization problems typically requires tree-like decision-making structure (Mulvey and Ruszczynski, 1995) that is not suitable for the solution of HSPP problems. Even if the problem can be decomposed into short-term plans where closed form optimal solution exists, the solution could only be expressed in terms of the distribution functions associated with the stochastic variables that may be difficult to translate into executable plans by FMSs in terms of the type and quantity to produce. To circumvent this problem, we first approximate the stochastic nonlinear programming model using a deterministic nonlinear programming model so that the model can be decomposed into series of short-term plans expressed in terms of expectations of stochastic variables that are executable by the FMSs .

For solution procedure, since the gradient of the objective function of the deterministic programming model is piecewise constant, it is difficult to solve the model directly by such nonlinear programming methods as Newton's method and the gradient method. For convenience, the model is next transformed into a linear programming model by adding additional constraints where small-scale or prototype problems can be solved using simplex method on a personal computer.

For practical use in a general workshop (that typically contains thousands of constraints and variables), we propose that Karmarkar's algorithm be applied for the medium- or small-scale problems and an interaction/prediction approach based on Karmarkar's algorithm for the large-scale problems as it is well-known that Karmarkar's algorithm is faster than the simplex method (Adler et al., 1989) for large-scale problems.

2. Stochastic production planning model for FAW

The FAW under consideration in this paper is generalized from the flexible automated workshop of the CIMS in Chengdu Aircraft Industry Company, which is equipped with two FMSs and two flexible direct numerical control systems. To solve the general problem of HSPP, we suppose that an FAW consists of a shop included, the BRT says that government failure often is a bigger problem than market failure.

"The Roundtable's perspective is simply that the risk of government failure is bigger. It is essentially a view about what is the best institutional arrangement for a particular area of public policy. It may be education; who is the best provider, who can drive the costs down and get the maximum benefit for a given cost? In assessing the status quo we start from a default proposition that voluntary/market arrangements will lead to better quality outcomes. It is not a perfect paradigm."

The comments on the role of Government and who is looking out for the majority in the waged economy is open ended but the perception of the BRT as a self-serving, new right think-tank of establishment businessmen whose views represent only a small portion of the New Zealand economy persists.

Ashley Balls is a strategic planning consultant, generally to the legal profession and public sector. co-jones.pmf@clear.net.nz