Computers Opns Res. Vol. 17, No. 4, pp. 425-426, 1990 0305-0548/90 53.00 + 0.00

Printed in Great Britain. All rights reserved Copyright 0 1990 Pergamon Press plc

A NOTE ON A PROOF OF SPT OPTIMALITY FOR SINGLE- MACHINE SEQUENCING PROBLEMS VIA THE

TRANSPORTATION PROBLEM

T. C. E. CHENG* Department of Actuarial and Management Sciences, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

(Receioed June 1989; revised August 1989)

Scope and Purpose-We consider in this note the problem of sequencing a set of jobs on a single-machine to minimize the sum of job completion times. We solve the problem using the transportation model of LP. The solution shows that sequencing jobs in ascending order of processing times is optimal.

Abstract-In many single-machine sequencing problems the optimal job sequence is shown to be in the shortest processing time (SPT) order. This note offers a constructive proof of SPT optimality for these problems via solution of the transportation problem.

INTRODUCTION

Many single-machine sequencing problems are concerned Z(S), a function of the job sequence S, expressed as

Z(S) = i p,,t,, i=l

with minimizing an objective function

(1)

where [i] denotes the job in position i of S; Prrl and tt,] are the positional penalty and processing time of job [i], respectively; n is the total number of jobs simultaneously available for processing. Examples of such sequencing problems are n/l//X,, n/l//XFi and n/l//XLi, all of which have Pfil= (n - i + l), where Ci, Fi and L, denote the completion time, flow-time and lateness of job i, respectively [ 11.

It is a well-known result that Z(S) can be minimized by arranging Pti, in non-increasing order and tIi] in non-decreasing order, or vice versa [l]. This is what we refer to as the SPT optimality result, since the optimal sequence is invariably in the SPT order with respect to the positional penalties. In this note we prove the SPT optimality result via the transportation model of LP. This proof has some pedagogical value that demonstrates vividly the art of problem formulation in action.

TRANSPORTATION PROBLEM

Without loss of generality, we assume the jobs are initially arranged in non-decreasing order of . .

processing times, i.e. t, < t2 < . . . <t, and the positional penalties in non-increasing order, i.e. P, 2 P* >, . * f > P,. The minimization problem (1) can be formulated as a transportation problem as follows:

Minimize : Z(S) = i i tiPjXij* (2) i=l l=l

*T. C. E. Cheng is Professor of Operations Management in the Faculty of Management and Adjunct Professor of Industrial Engineering in the Faculty of Engineering, University of Manitoba, Canada. He obtained his PhD in operations research from the University of Cambridge, England. His research interests are in scheduling, simulation and inventory management. A member of the editorial advisory board of this journal, he has published over 50 papers in various scholarly journals such as European Journal oj Operational Research, IIE Transactions, International Journal of Production Research, Journal of the Operational Research Society and others. He is a registered professional engineer in the province of Manitoba.

425

426 T. C. E. CHENG

Subjectto ids Xij= 1, 1 <j<n (3)

,$r xil= 1, l<i<n (4)

Xij=OOrl, l<i<n, l<j<n (5)

where xii denotes the assignment of job i to position j of sequence S. Letting Cij = tip, denote the cost of assigning job i to position j in S, we first apply the northwest

corner rule [2] to set up an initial transportation tableau to yield a degenerate solution which requires n - 1 additional basic variables. We then introduce n - 1 artificial assignments to those cells next to each of the assigned cell in the initial tableau and proceed to solve the problem using the modified distribution (MODI) algorithm.

According to MODI, two sets of auxiliary cost variables, denoted by Ui, 1 G i < n, and o,, 1 < j < n,

are necessary for the optimality test. Associated with each assignment (i, j) is a pair of auxiliary variables ui and Oj related to the cost cij as follows:

cij =Ui+Uj. (6)

Next, an opportunity cost hij is calculated for each empty cell in the transportation tableau using the formula

hij = C,j- UI - Uj. (7)

Adding the artificial assignments and calculating all auxiliary and opportunity cost values, using the arbitrary initial condition u1 = 0, yields the following opportunity cost values:

h ii =cU- u, - u, 2 0 for all empty assignment (i, j).

Since the hii’s of all empty cells are non-negative, the solution (i.e. diagonal assignment) is optimal according to MODI. Therefore the optimal solution is to match the shortest job with the largest positional penalty, the second shortest job with the second largest positional penalty and so on, which indirectly proves the SPT optimality result.

Acknowledgements-This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OPGO036424. The comments of two anonymous referees are very much appreciated.

REFERENCES

1. K. R. Baker, fntroducrion to Sequencing and Sc/teduling. Wiley, New York (1974). 2. R. Bronson, Operorions Research. McGraw-Hill, New York (1983).