Linus Schrage

OR Practice-A Scenario Approach to Capacity Planning

Production capacity has always been one of the most important strategic variables for the major automobile companies. Decisions by individual companies concerning the overall level of capacity, the type of facility e.g., the level of flexibility, and the location of that capacity e.g., in the United States or abroad are discussed in great detail in the popular business press. In this paper, we describe a model developed for General Motors to aid in making decisions about capacity for four of their auto lines. The model incorporates elements of scenario planning, integer programming, and risk analysis. All the input and output is done using Lotus 1-2-3. Although the presentation is motivated by the particular application in the auto industry, the model represents a general purpose approach that is applicable to a wide variety of decisions under risk. An example in this paper uses actual data, appropriately transformed to ensure confidentiality.

A More Portable Fortran Random Number Generator

There are a number of situations in which it is desirable to have a random number generator that is machine independent. In general, it is useful if a program written in a high-level language produces results which are the same from machine to machine as long as the input to the program is the same. For example, the pseudorandom number generator used by the Control Data Corporation in its GPSS simulation program is the same as in IBMs GPSS, even though the generator is known to have defective statistical behavior. Apparently, compatibility is more valuable than statistical goodness. The program described here is an implementation of the generator described by Lewis et al. [6] and indirectly attributed to D.H. Lehmer. The code for the program appears in Figure 1. The generator produces a sequence of positive integers, IX, by the recursion:

Dynamic Programming Solution of Sequencing Problems with Precedence Constraints

Consider a set of tasks that are partially ordered by precedence constraints. A subset of tasks is called feasible if, for every task in the subset, all predecessors are also in the subset. The major results are 1 a method for enumerating all feasible subsets and 2 a method for assigning to each feasible subset an easily computed label that can be used as a physical address for storing information about the subset. These two results permit a very compact computer implementation of a dynamic programming algorithm for solving one-machine sequencing problems with precedence constraints. This algorithm appears to be much more efficient than previous ones for certain one-machine sequencing problems.

The Deterministic Dynamic Product Cycling Problem

Certain manufacturing situations involve a small number of items produced sequentially on the same facility with high changeover cost. Production schedules are characterized by long runs, and individual items are produced infrequently. At the same time, seasonal demand patterns require that the items be maintained in inventory in the right mix. This paper presents alternative formulations of this problem. It uses a Lagrangean relaxation approach to decouple the problem and provide lower bounds used in a branch and bound algorithm. Some experimental computations on small problems are reported.

Retail Inventory Management When Records Are Inaccurate

Inventory record inaccuracy is a significant problem for retailers using automated inventory management systems. In this paper, we consider an intelligent inventory management tool that accounts for record inaccuracy using a Bayesian belief of the physical inventory level. We assume that excess demands are lost and unobserved, in which case sales data reveal information about physical inventory levels. We show that a probability distribution on physical inventory levels is a sufficient summary of past sales and replenishment observations, and that this probability distribution can be efficiently updated in a Bayesian fashion as observations are accumulated. We also demonstrate the use of this distribution as the basis for practical replenishment and inventory audit policies and illustrate how the needed parameters can be estimated using data from a large national retailer. Our replenishment policies avoid the problem of “freezing,” in which a physical inventory position persists at zero while the corresponding record is positive. In addition, simulation studies show that our replenishment policies recoup much of the cost of inventory record inaccuracy, and that our audit policy significantly outperforms the popular “zero balance walk” audit policy.

The Queue M/G/1 with the Shortest Remaining Processing Time Discipline

A priority queuing model in which the processing times of jobs are known upon arrival and preemption without loss of time or processing already accomplished is studied. Priority is assigned to jobs according to the length of processing remaining with highest priority going to the job with least processing left. A preemption will occur whenever the processing time of a newly arriving job is less than the remaining processing time of the job then in service. The Laplace-Stieltjes transforms of the waiting time and time-in-system distributions are obtained and comparisons with other queuing disciplines are made.

Finding an Optimal Sequence by Dynamic Programming: An Extension to Precedence-Related Tasks

We discuss the dynamic programming approach to finding an optimal sequence of a set of tasks when the tasks are related by precedence restrictions. We describe how to use this approach in problems where no explicit precedence relations exist. Computer implementation considerations played an important role in its development. Computational results indicate that, when the curse of dimensionality can be dispelled, dynamic programming can be a useful procedure for large sequencing problems.

Formulation and structure of more complex/realistic routing and scheduling problems

We classify the features that seem to be encountered in real vehicle routing problems. Given these features, we try to indicate which features cause the greatest difficulty and which modeling approaches allow us to represent the greatest range of practical considerations or features.

Solving Resource-Constrained Network Problems by Implicit Enumeration-Nonpreemptive Case

This paper considers a scheduling problem that has both precedence constraints of a general form as in PERT-CPM problems and resource constraints of the form in the general job-shop scheduling problem, and gives an efficient enumerative procedure for generating all active schedules for this problem. Based on this enumerative scheme, the paper describes a branch-and-bound method for implicitly enumerating all schedules and determining the optimum; finally, it gives computational experience for the problem where the objective is minimizing the project makespan.

Stochastic Shop Scheduling: A Survey

In this paper a survey is made of some of the recent results in stochastic shop scheduling. The models dealt with include: (i) Open shops. (ii) Flow shops with infinite intermediate storage (permutation flow shops). (iii) Flow shops with zero intermediate storage and blocking. (iv) Job shops. Two objective functions are considered: Minimization of the expected completion time of the last job, the so-called makespan, and minimization of the sum of the expected completion times of all jobs, the so-called flow time. The decision-maker is not allowed to preempt. The shop models with two machines and exponentially distributed processing times usually turn out to have a very nice structure. Shop models with more than two machines are considerably harder.