Stochastic Planning Problems Research Articles

Practical solution approaches to solve a hierarchical stochastic production planning problem in a flexible automated workshop in China

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-inprocess 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 nonlinear 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.

Hierarchical stochastic production planning with delay interaction and demand constraints

This paper explores the hierarchical stochastic production planning (HSPP) problem of flexible automated workshops, each with a number of flexible manufacturing systems (FMSs), the part transfer between which is delayed by a time period (i.e. delay interaction). The problem not only includes uncertainties in the demand, capacities, material supply, processing times, necessity for rework and scrap but also considers multiple products and multiple time periods. The objective is to devise a production plan which tells each FMS how many parts to produce and when to produce them so as to obtain the lowest cost under the condition of demands being just met. Herein, the HSPP problem is formulated by a stochastic non-linear programming model whose constraints (including demand constraints) are linear but whose objective function is piecewise linear. For the convenience of solving the stochastic non-linear programming model above, it is approximately transformed into a deterministic non-linear programming model and further into a linear programming model. Then, Karmarkar's algorithm and an interaction/prediction algorithm are used to solve the model, the former for the medium- or small-scale problems and the latter for the large-scale problems. With the help of the implementations of the above-mentioned algorithms and through many HSPP examples, Karmarkar's algorithm, the interaction/prediction algorithm and the linear programming method in Matlab 5.0 are compared, the result of which shows that the proposed approaches are very effective and suitable for not only ‘push’ production but also ‘pull’ production.

Hierarchical stochastic production planning for the highest business benefit

This paper addresses the hierarchical stochastic production planning (HSPP) problem of flexible automated workshops (FAWs), each with a number of flexible manufacturing systems (FMSs) the part-transfer between which is a delay of a time period. The problem not only includes uncertainties in the demand, capacities, material supply, processing times, necessity for rework, and scrap, but also considers multiple products and multiple time periods. The objective is to develop a production plan which tells each FMS how many parts to produce and when to produce them so as to obtain the highest business benefit. Herein, the HSPP problem is formulated by a stochastic nonlinear programming model whose constraints are linear but whose objective function is piecewise linear. For the convenience of solving the stochastic nonlinear programming model above, it is approximately transformed into a deterministic nonlinear programming model and further into a linear programming model. Because the scale of the model for a general workshop is too large to be solved by the simplex method on a personal computer within acceptable time, Karmarkar's algorithm and an interaction/prediction algorithm, respectively, are used to solve the model, the former for the medium or small scale problems and the latter for the large scale problems. By the implementation of the above-mentioned algorithms and through many HSPP examples, Karmarkar's algorithm, the interaction/prediction algorithm and the linear programming method in Matlab 5.0 are compared, the result of which shows that the proposed approaches are very effective and suitable for not only “push” production but also “pull” production.

Hierarchical stochastic production planning for flexible automation workshops

This paper presents the hierarchical stochastic production planning (HSPP) problem for flexible automation workshops (FAWs) in agile manufacturing environments, which is a multiple-period multiple-product problem with random material supply, demands, capacities, processing times, rework and waste products. To solve the HSPP problem, a mathematical model is built up first. Then, an algorithm for HSPP is deduced in detail by using a stochastic interaction/prediction approach. The corresponding software package named as stochastic interaction/prediction algorithm (SIPA) has been developed and is presented in this paper, through which examples of HSPP have been studied, and which show that the algorithm can optimally decompose medium-term random product demand plans of an FAW into short-term stochastic production plans to be executed by FMSs in the FAW. Finally, the application of the algorithm is presented in detail through one of those examples.

On solving stochastic production planning problems via scenario modelling

Linear Porgramming models for stochastic planning problems and a methodology for solving them are proposed. A production planning problem with uncertainty in demand is used as a test case, but the methodology presented here is applicable to other types of problems as well. In these models, uncertainty in demand is characterized via scenarios. Solutions are obtained for each scenario and then these individual scenario solutions are aggregated to yield an implementable non-anticipative policy. Such an approach makes it possible to model correlated and nonstationary demand as well as a variety of recourse decision types. For computational purposes, two alternative representations are proposed. A compact approach that is suitable for the Simplex method and a splitting variable approach that is suitable for the Interior Point Methods. A crash procedure that generates an advanced starting solution for the Simplex method is developed. Computational results are reported with both the representations. Although some of the models presented here are very large (over 25000 constraints and 75000 variables), our computational experience with these problems is quite encouraging.

Benefit-Cost Analysis of a Proposed Runway Extension

This paper describes a benefit-cost analysis of a proposed $1.4 million runway extension at Kelowna Airport, in Kelowna, British Columbia (Canada's westernmost province). Lengthening the runway as proposed would allow Pacific Western Airlines to operate their Boeing 737 aircraft out of Kelowna with heavier payloads, thereby reducing the occurance of situations where the combined passenger and cargo demand exceeds the available weight capacity of the aircraft. The problem is complicated by the fact that the maximum payload weight permitted on take-off varies considerably with the air temperature and wind speed and direction at the time of departure. On hot summer days, when the problem is most severe, it is sometimes necessary to remove all of the cargo from the aircraft, and, at times, to limit the number of passengers taking the flight to less than the number of seats. An alternative means of achieving the objectives of the runway extension was also considered: lowering the average demand per flight by simply increasing the number of flights out of Kelowna. The “Runway Extension” and “Additional Flights” options were evaluated in terms of their net economic benefits, as compared with the “Do Nothing” option. Net economic benefit is the present value of economic benefits less the present value of economic costs. The economic benefits depend upon the extent to which unsatisfied passenger and cargo demand would be reduced, and on the unit economic costs of unsatisfied passenger and cargo demand. Estimates of the reductions in unsatisfied demand were obtained by means of a probability model that is similar to ones used to solve single-period, stochastic inventory planning problems. The unit economic costs were not estimated; rather, the economic benefits were expressed as linear functions of the unit economic costs of unsatisfied passenger and cargo demand. The economic cost of the “Runway Extension” option was the cost of constructing and maintaining the runway extension; the cost of the “Additional Flights” option was the cost to Pacific Western Airlines of providing the additional flights out of Kelowna. The numbers of such additional flights were chosen to produce the same amount of unsatisfied passenger demand as that which had been estimated for the “Runway Extension” option. Those responsible for deciding whether the runway should be extended were asked to judge the values of the unit economic costs of unsatisfied passenger and cargo demand, thus determining which of the three options was preferred on economic grounds. Prior to the study. Transport Canada had formally recommended the proposed runway extension to the Treasury Board (the Cabinet Committee of the Canadian Government that controls government expenditures). The Treasury Board requested the Department to reconsider the project, and as a result, a benefit-cost analysis was undertaken. Because of the conclusions of the study. Transport Canada decided to postpone indefinitely the lengthening of the runway. The runway may still be lengthened sometime in the future to accommodate aircraft that require a longer runway than does the Boeing 737, but only if a subsequent analysis shows that the resulting benefits would exceed the costs. Having been successfully applied at Kelowna Airport, the methodology developed in this study has now been adopted by Transport Canada for evaluating proposed runway extensions at other Canadian airports.

Stochastic Horizons for the Aggregate Planning Problem

This paper develops a methodology for showing how forecast horizons for stochastic planning problems relate to the planning procedures and the information system within the organization. To illustrate the approach we have chosen a relatively straightforward production problem where the firm can meet a fluctuating demand pattern through a combination of overtime and inventory-related options. We show that the optimal production plan is monotonically non-decreasing in the demand sequence so that bounds can be placed on the optimal first period production plan for an N-period problem. These bounds together with information and computationally-related costs are used in specifying a methodology for determining forecast horizons. An illustrative example suggests that such horizons are likely to be small in many realistic cases. The concluding section indicates how this methodology can be utilized for specifying stochastic horizons for more general aggregate planning decisions in organizations.

Further Results on Planning Horizons in the Production Smoothing Problem

A generalized version of the Modigliani-Hohn production smoothing model is analyzed, and two types or classes of theorem are proved. (Theorems 1 and 2): The planning horizon (defined as a minimal interval of sufficient length to yield optimal current decisions) depends critically on the existence of “bottleneck” conditions; in the case of the original M-H model, a bottleneck arises out of the inventory nonnegativity constraint, and it is here shown that an additional interesting bottleneck condition arises out of an inventory storage capacity constraint. Whether a particular type of bottleneck defines a horizon will depend, interestingly, on the terminal inventory condition that is imposed for the original optimization process. (Theorems 3 and 4): The sensitivity of horizon length to two economic parameters, the discount factor and the marginal storage cost, will depend on whether the horizon-determining bottleneck is due to a binding inventory capacity constraint, or a binding nonnegativity constraint. The capacity constraint will yield “perverse” horizon sensitivities first noted by Charnes, Dreze and Miller [Charnes, Abraham, Drèze, Jaques, Miller, Merton. 1966. Decision and horizon rules for stochastic planning problems: a linear example. Econometrica 34 307–330.] in their work on the warehousing problem. In light of these results, the importance of horizons as a programming concept can be questioned.

Planning Horizons for Production Smoothing with Deterministic Demands

In Part I of this paper, we develop an algorithm for finding planning horizons for the deterministic production smoothing problem when all demand must be met from regular production, under rather general assumptions for the production, production smoothing, and holding cost functions. (In Part II, planning horizons will be developed when the model is extended to include backlogging and overtime.) The techniques developed here are essentially forward-looking and marginal-cost-balancing in nature, rather than total-cost-minimizing and backward-looking such as dynamic programming, or total-cost-minimizing and omni-looking in nature such as linear programming. The fact that the planning horizon theorems do not depend on having discount rates less than one illustrates that the approach developed here is fundamentally different from ordinary infinite horizon dynamic programming techniques. The algorithm is “user oriented” in the sense that only a small amount of forecasting work and computation ordinarily must be done to determine the horizon; the nature of the algorithm also makes the exact dependence of the horizon on the forecast clear for sensitivity analysis. Firms facing a seasonal demand pattern will often find that the horizon occurs within the first several periods after the peak period if there is a sufficient enough drop in demand afterwards. This result complements the findings of Modigliani and Hohn and provides insight into the nature of the optimal policy for stochastic planning problems.

Decision and Horizon Rules for Stochastic Planning Problems: A Linear Example

Analytic decision rules and horizon rules are developed for the warehousing problem. A stochastic warehousing problem is defined and solved, first by backwards induction, then by means of a forward working algorithm. The influence of holding costs and discount rates on forecasting horizons is illustrated.