SCHEDULING TO MINIMIZE THE WEIGHTED SUM OF COMPLETION TIMES WITH SECONDARY CRITERIA

Richard N. Burns

Austrcilion Nntiontrl 1 ‘niversity Ctrnberni, Australiti

ABSTRACT

A result of Smith previously puldished in this journal 131, on the use of secondary criteria in scheduling prohlems. is shown to be incorrect and a counter example is presented.

Heck and Roberts [LZ] suggested that their paper would be extended in the same way Smith’s algorithm was. .A new algorithm is given that converges to a local optimum for both problems.

1. BACKGROUND

The probleni of minimizing the sum of weighted completion times subject to a secondary con- straint has long been considered to be similar to minimizing the sum of completion times subject to a constraint. Smith [31 1956 presented an algorithm for the n-job, 1-machine case where he minimized the weighted completion times subject to the constraint that all jobs were completed by their due dates. Recently Heck and Roberts 121 presented an algorithm for minimizing the sum of completion times subject to not increasing the maximum tardiness calculated by the due date sequence. They claimed that their result could be extended to the problem of weighted completion times in a manner similar to Smith’s algorithm. Such is not the case since Smith’s algorithm did not find the optimum sequence.

weight a, and completion time c I . The problem under consideration is to find a sequence to

Using the notation of Conway, Maxwell and Miller [l], let job i have process time p , , due date d, ,

I ,

minimize C aici i = 1

i subject to C p j - d i c T for i = l , 2, . . ., n

j= 1

For T=O in (2) the problem is the one considered by Smith [3] and for T= maximum tardiness the problem is the one of Heck and Roberts [ 2 ] .

2. A COUNTER EXAMPLE TO SMITH’S ALGORITHM The method given by Smith to solve (1) subject to ( 2 ) with T = O is as follows: “If all jobs can be

125

126 R. N. BURNS

job 1

Pi 4 3 di 8 ai 1

pilai 4

completed by their due dates, an order which minimizes the weighted sum may be obtained. This order has its last job one with the largest value of pilai from those with due date as large as the total processing time of all jobs.”

Consider the following example:

2 3

2 9 10 4 3

314 213

Using Smith’s algorithm, job 2 would be placed last, since job 2 and 3 both have due dates greater than or equal to the total process time and 314 > 213. Codidering the remaining two jobs, again both

job 1 and 3 may be last and 4 > 213. Therefore the optimum sequence is 3, 1, 2 with z aici=48.

However, the sequence 2, 1, 3 also has all jobs completed by their due date and aici=46. Hence

Smith’s algorithm does not give the correct sequence. When considering job k to be placed last it is necessary to check that the resulting sequence satis-

fies (2). Hence a more precise statement of Smith’s algorithm would be as follows: ALGORITHM 1: (a) Order the jobs in the order of increasing due dates. It is assumed that all

jobs are on time by this schedule. (b) Place job k last where

3

i= 1

3

i= 1

&”>pi a k ai

for all i such that the resulting sequence will satisfy (2).

(c) Reduce n by 1 and return to (b) until all jobs have been sequenced by this method.

3. LOCAL OPTIMUM Two sequences S and S‘ are said to be adjacent if one can be formed from the other by a single

interchange of two jobs. A sequence is feasible if and only if it satisfies (2). 71

DEFINITION: A sequence is a local optimum for (1) subject to (2) if z aici is less than or equal i= 1

to the sum of weighted completion times of all feasible adjacent sequences. In order to check for a local optimum a method of comparing two adjacent sequences will now be

developed. Let S be a sequence with the jobs numbered 1,2 , . . . , n and let S’ be an adjacent sequence having

jobs k and L interchanged. Without loss of generality we can assume that job L appears after job k in the sequence S.

SCHEDULING TO MINIMIZE COMPLETION TIMES

For i < k - 1 and for i * L ,

127

F o r i = k , k + l , . . . , L - 1 ,

C l ! = C i + ( P L - p k ) .

Therefore

n n L

The following lemma follows directly from the definition of a local optimum and equation (3). LEMMA 1: A sequence S is a local optimum if and only if for all feasible sequences differing from

S by having jobs L and k interchanged with k < L the expression

(4)

(5)

For the special case where all the weights are one then equation (3) becomes

n n

Note that for the unweighted case the difference between two adjacent sequences is only a function of the two terms being interchanged and their distance apart. Lemma 1 would have an equivalent form for the unweighted case where equation (5) would be substituted for equation (4). The proof that Algorithm 1 converges to a local optimum (for the unweighted case) is straight forward and in fact was done correctly by Smith as he essentially developed equation (5).

For the weighted case the difference between adjacent sequences is a function of the jobs between the two interchanged jobs. In the papers by both Smith and Heck and Roberts a very special case of equation (3) was used. When L = k + 1 equation ( 3 ) becomes the following.

k - I k - 1

n n

128 R. N. BURNS

Equation (6) is the test used in step (b) of Algorithm 1. LEMMA 2: Algorithm 1 (Smith) converges to a constrained stationary sequence S having the

property that C aici for S is less than or equal to the sum of weighted completion times for the subset

of feasible adjacent sequences formed by interchanging only pairs of jobs that are adjacent in S.

n

i = 1

PROOF: Let the jobs in S be numbered in ascending order, 1 , 2 , . . . , n. Assume the Lemma is false. Then there exists S' a feasible sequence adjacent to S with

and S and S' differ only in the k and k + 1 st position. By (6)

P k P k + J ->-. a k a k + l

When Algorithm 1 was choosing the job for the k + 1 position job k was rejected and later placed in position k. Since S' is also feasible, job k must have been eligible to go in the k + 1 st position. By step (b) of Algorithm 1

which contradicts the result of the assumption. The counter example presented shows that considering only interchanges between jobs that are

adjacent in a sequence is not a sufficient condition for a sequence to satisfy Lemma 1. The following algorithm uses equation (2) to generate a local optimum.

ALGORITHM 2: 1. Schedule the jobs in increasing order of due date. (It is assumed that all jobs are completed by their due date. If such is not the case a solution to (1) subject to (2) with T equal to the maximum tardiness is straight forward.)

2. For k= n - 1, n - 2, . . . , 1, find the first value of k satisfying the following three conditions.

(7)

If such a k is found, interchange job k and n. Reset n to its initial value if necessary and return to the beginning of step 2. If no k is found satisfying (7), (8) and (9) reduce n by 1. For n 3 2 return to the be- ginning of step 2. For n= 1 the optimum sequence is the current one.

SCHEDULING TO MINIMIZE COMPLETION TIMES 129

THEOREM 1: Algorithm 2 converges to a local optimum. PROOF: Every time step two is successful the sum of the weighted completion times is reduced.

Since this can only happen a finite number of times the algorithm terminates. Conditions (7) and (8) determine which term in the sequence can be interchanged with job n and equation (9) is equivalent to having the right hand side of equation (3) greater than zero. Step 2, will only terminate after n has passed (7), (8) and (9) for all values of n from 1,2 , . . ., n. Hence by Lemma 1 the stationary point found by Algorithm 2 is a local optimum.

4. CONCLUSION Algorithm 2 on the counter example would first find the stationary point that Smith’s method

stopped at, and then moves directly to the global optimum. It is not always the case that the local optimum found by Algorithm 2 is the global optimum.

Consider the following example:

ai

3 Algorithm 2 would generate a local optimum sequence 1 , 3 , 2 with 2 aici=l%. The global optimum

is 2. 1 , 3 with C aici= 136. At present there is no simple way of finding a global optimum for permu-

tation sequences.

i = l 3

i= I

REFERENCES

[l] Conway, R. W., W. L. Maxwell, and L. W. Miller, Theory ofScheduling (Addison-Wesley, Reading,

[2] Heck, H. and S. Roberts, “A Note on the Extensions of a Result on Scheduling with Secondary

[3] Smith, W. E., “Various Optimizers For Single-Stage Production,” Nav. Res. Log. Quart., 3, 59-66

Mass., 1967).

Criteria,” Nav. Res. Log. Quart., 19, 403-405 (1972).

(1 956).