On some sequencing problems

The (mxn) sequencing problem may be characterized as follows: There are m machines which can produce a piece consisting of n parts. Each part has a determined order in which it is processed through the machines. It is assumed that each machine cannot deal with more than one part at a time and that the processing required for each part can be accomplished only on one machine. That is, the machines are all specialized so that alternate machines for the same processing on a part is not possible. The problem is to find the best production plan consisting in sequencing the different parts so as to make the whole amount of time from the beginning of work till the piece is completed the shortest possible. Such a plan is called an optimum one. In the first 4 sections of this paper, the problem (2xn) is solved for the (2xn) case in which the order in which parts come on the machine is not constrained by further assumptions. The remainder of the paper then takes up: 1) the (3xn) problem of Bellman-Johnson (viz. the technological processing order through the machine is the same for all parts) for several new special cases; 2) the 2xn problem of sequencing when delay times must also be considered; and, 3) some properties of an approximating method for solving (mxn) problems, including a delineation of cases when the approximating method will yield optimal solutions.     

The truck assignment problem

Let us consider the following problem: A group of cement factories produces several types of cement, but each factory produces only one type. There is also a group of purchasers and each purchaser may need several types of cement. The amounts supplied and the demands are assumed to be known for each cement factory and each purchaser. Each cement factory has several trucks, but there is only one dispatcher for ail of the trucks from all of the factories. It is assumed that the entire lcad of each truck is for one purchaser only. A truck begins its workday by leaving its base depot loaded and ends its day by returning to it empty. During the day it may be required to transport cement from any of the cement factories. The distances from various factories to individual purchasers are known. The problem to be solved is that of finding a truck schedule such that cement in the needed quantities is delivered daily to the individual purchasers and in such a manner that the total truck-kilometers traveled will be as small as possible. This paper presents the method of solution, though the assumption of an 8-hour workday may not be met. On the other hand, there are methods developed for effecting a variety of cyclic routings which can be used to lend considerable flexibility to the schedules.     

Extreme solutions of the two machine flow-shop problem

The paper provides a new theoretical framework to identify extreme solutions of the two machine flow-shop problem. Some remarkable properties of these solutions have been developed. As a result the problem of generating minimal solutions can be decomposed into a number of smaller subproblems.     

Special cases of the flow-shop problem

The paper examines all known special cases of the mXn flow-shop problem. It provides solution procedures to three new special cases along with the optimality proofs. The theory of the new special cases is based on the critical path concept.     

Solvable cases of the flow-shop problem without interruptions in job processing

This article examines solvable cases of the flow-shop problem without interruptions in job processing where the completion time is being minimized. All those cases utilize Gilmore-Gomory's algorithm. A lower completion time bound is also provided.     

A note on the flow-shop problem without interruptions in job processing

This paper deals with a flow-shop problem where the n jobs are being processed uninterrupted by m machines. A comprehensive theory based on “an earliest starting time of a job” concept produced the most efficient solution method for a variety of optimization criteria. The paper also rectifies several known results in this area.     

Mathematical aspects of the 3 × n job-shop sequencing problem

The paper discusses mathematical properties of the well-known Bellman-Johnson 3 × n sequencing problem. Optimal rules for some special cases are developed. For the case min B_i ≥ maxA_j we find an optimal sequence of the 2 × n problem for machines B and C and move one item to the front of the sequence to minimize (7); when min B_i ≥ max C_j we solve a 2 × n problem for machines A and B and move one item to the end of the optimal sequence so as to minimize (9). There is also given a sufficient optimality condition for a solution obtained by Johnson's approximate method. This explains why this method so often produces an optimal solution.     

Johnson's approximate method for the 3 × n job shop problem

The effectiveness of Johnson's Approximate Method (JAM) for the 3 × n job shop scheduling problems was examined on 1,500 test cases with n ranging from 6 to 50 and with the processing times A_i, B_i, C_i (for item i on machines A, B, C) being uniformly and normally distributed. JAM proved to be quite effective for the case B_i ? max (A_i, C_i) and optimal for B_i, ? min (A_i, C_i).     

Elimination methods in the m × n sequencing problem

This paper considers elimination methods in solving the sequencing problem where no passing is permitted. An elimination method consists of reducing (according to some criterion) the initial set of n solutions to a smaller set. A crucial question arises as to whether this reduced set contains an optimal solution. The answer is affirmative if this elimination criterion implies condition (3).