Bruce Faaland

Scheduling tasks with due dates in a fabrication/assembly process

In fabrication and assembly processes, end-product due dates play a significant role in scheduling tasks to minimize earliness and lateness penalties. We develop a heuristic technique for this problem that solves a sequence of maximum flow problems to identify improved schedules. This method compares favorably with finite loading, another scheduling heuristic. Our computational results include the solution of a problem involving 26,100 tasks scheduled on 52 work centers.

Optimal pagination of B-trees with variable-length items

Two algorithms are developed for the optimal organization of B-trees (and variations) with variable-length items. The first algorithm solves a problem posed by McCreight, that of finding a pagination of n items that minimizes the sum of key lengths promoted to the next higher level of the tree. The algorithm requires O(n log n) time and O(n) space. The second algorithm constructs the minimum depth tree in O(n 3 log n) time from the n items. Both methods rely on dynamic programming arguments and can be interpreted as shortest-path problems. Practical approaches for implementing the algorithms are discussed.

Cost-Based Scheduling of Workers and Equipment in a Fabrication and Assembly Shop

Many manufacturing firms that use Material Requirements Planning MRP cannot deliver products on schedule and within budget. Faced with bewildering bottlenecks, erratic process flows, and unrealistic due dates, they are unable to develop accurate schedules for their raw material acquisitions, workforce, and equipment. Their MRP plans must be translated into a workable schedule, one which determines when individual tasks will be performed by workers at work centers. There is a clear need for such an enhancement to MRP, a means to operate on detailed task data, yet capable of producing a schedule that directly relates to the MRP plan, the master production schedule, and the resource plan. We describe a method for determining feasible and cost-effective schedules for both labor and machines in a job shop. The method first sequences tasks at resources and then minimizes overall earliness and lateness cost by solving a series of maximum flow problems. By using the model as an enhancement to a companys MRP system, we simulated the cost effects of redeploying its workforce. Although the model was not used for real-time scheduling, it served a strategic role in workforce expansion and deployment decisions.

Interior Path Methods for Heuristic Integer Programming Procedures

This paper considers heuristic procedures for general mixed integer linear programming with inequality constraints. It focuses on the question of how to most effectively initialize such procedures by constructing an “interior path” from which to search for good feasible solutions. These paths lead from an optimal solution for the corresponding linear programming problem (i.e., deleting integrality restrictions) into the interior of the feasible region for this problem. Previous methods for constructing linear paths of this kind are analyzed from a statistical viewpoint, which motivates a promising new method. These methods are then extended to piecewise linear paths in order to improve the direction of search in certain cases where constraints that are not binding on the optimal linear programming solution become particularly relevant. Computational experience is reported.

Technical Note—The Multiperiod Knapsack Problem

In the multiperiod knapsack problem the decision maker faces a horizon of m periods. Associated with each period are a number of types of items, each with a value and weight. Subject to the requirement that the cumulative capacity of the knapsack in each period i cannot be exceeded by items chosen in periods 1, …, i, the decision maker chooses the most valuable knapsack possible. A branch and bound algorithm exploits the special structure of the multiperiod knapsack problem by calculating bounds by the direct solution of linear programs with m constraints in 0(m) operations. Computational experience is reported on problems ranging in size up to 200 constraints and 1,000 general integer variables.

Economic lot scheduling with lost sales and setup times

We introduce a new modeling framework for the classic economic lot scheduling problem that permits lost sales if they lead to higher profits. The model also accounts for setup times at a facility, but assumes no explicit incremental setup cost in the objective. Despite assumptions of deterministic demands, production rates and setup times, the cost of carrying inventory may make lost sales during a cycle economically attractive. Statistical analysis on randomly generated problems ranging in size from 100 to 1000 products indicates that the computation time grows by the square of the number of products.

A Weighted Selection Algorithm for Certain Tree-Structured Linear Programs

We solve a simple class of linear programming problems with tree-structured constraint matrices. Our procedure separates the problem into a sequence of continuous knapsack problems, each of which requires linear time to solve, for total solution time no worse than proportional to the number of nonzero entries in the original constraint matrix. The model arises in portfolio selection, advertising, and inventory and production planning.

The Accelerated Bound-and-Scan Algorithm for Integer Programming

This paper presents a new implicit enumeration algorithm for solving the pure integer linear programming problem. The theory of equivalent integer programming problems is first used to reformulate the problem. A technique originally used with particular success in the bound-and-scan algorithm to deal with only a subset of the variables is extended to all of the variables in the restructured problem. In addition to the resulting basic enumeration scheme, the algorithm includes a scanning procedure and a method for identifying constraints that become redundant during the course of the algorithm. Computational experience on standard test problems is reported.

Technical Note-Solution of the Value-Independent Knapsack Problem by Partitioning

The value-independent knapsack problem is solved by considering a related problem in the theory of partitions of numbers. The solution technique is compared to standard algorithms for the knapsack problem.

On the number of solutions to a diophantine equation

Abstract Let A 1 , …, A r , x 1 , …, x r , and A be known positive integers. Let f(A) be the number of integer solutions (x1, …, xr) satisfying the Diophantine equation ∑ j=1 r A j x j = A and the conditions 0 ⩽ x i ⩽ x j , j = 1, …, r. This paper expresses f(A) recursively as a linear function of f(0), f(1), …, f(A−1).