JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 83, No. 1, pp. 199-205, OCTOBER 1994

TECHNICAL NOTE

Optimal Ordering Policy of a Sequencing Model'

B. ALIDAEE 2

Communicated by D. L. Brito

Abstract. In this note, we are concerned with the study of a sequenc- ing problem applicable to situations where the optimal choice among n! sequences is sought. A class of sequencing problems is proposed. Based on the adjacent pairwise interchange of two objects, necessary and sufficient conditions for an optimal ordering policy are given. Examples from the literature are considered and shown to be special cases of the proposed model. The results of this paper improve recent results given in Refs. 1 and 2.

Key Words. Optimal sequencing, pairwise interchange policy.

1. Introduction

In this note, we are concerned with the study of a sequencing problem applicable to situations where the optimal choice among n! sequences is sought. For example, consider the problem of inspecting multicharacteris- tics components (Refs. 1, 3, 4), when there is a component with n critical multicharacteristics that have to be inspected. These characteristics are unequal in their effect on the fitness for use, and some are of critical importance to human life (Ref. 1). Inspecting each characteristic requires costs which depend on the type of tests and equipment needed in carrying out the quality assurance procedures. It is assumed that, with probability Pi, the characteristic i will not pass the inspection. The inspections of

1The author would like to thank a referee for comments that improved the presentation of the paper. He also thanks Professor R. Combs of West Texas A&M University for comments.

2Assistant Professor, Mathematics and Physical Sciences Department , West Texas State University, Canyon, Texas.

199 0022-3239/94/1000-0199507.00/0 �9 1994 Plenum Publishing Corporation

200 JOTA: VOL. 83, NO. 1, OCTOBER 1994

characteristics are independent; they are done one after another until one is rejected or all the characteristics pass the inspection successfully.

In general, the sequencing model can be stated as follows. There are n objects with no precedence relations. Associated with each object i is a positive weight wi, a nonzero number pi, and a cost function f~: R N - - . R + , where R N are all the ordered subsets of the set of objects; for example, {1, 2, 3} is different from {2, 1, 3}. The objective is to find an ordering

= (Jl . . . . , j , ) of n objects so as to minimize the total cost, given by

TC(rt) = ~, f j , ( j , . . . . . j , ) . (1) i = 1

It is assumed that fj~, i = 1 , . . . , n, is independent of the ordering of the objects J l , . . . ,J~-l). For each function f:~, i = 1 . . . . . n, we assume that

{fji(J, . . . . . j,,, l,j,,+ l, . . . , j , ) - - f j i ( J ] . . . . . J a , J a + l , " . " , J i ) }

= ptwj~H(J, . . . . . Ji -1) , (2)

for any Ir = (Jl . . . . . J~-x), where B is a partial sequence. In Eq. (2), H is a positive set function defined on all subsets of n objects and H ( ~ ) = 1. Under the above assumption, the function T C can easily be minimized as shown in the following theorem.

if

where

Theorem 1.1. A sequence it minimizes the total cost T C if and only

h( j~) <_ h ( j z ) < " " < h( .~) , (3)

h ( j i ) =p:~/wj~, i = 1 . . . . . n.

Proof. Let n be a sequence of the n objects, and let n' be a sequence obtained from rc by changing the position of ith and (i + 1)th objects. Then, we have

rc( ) - rc( 3

= { f j , ,+ , (B , j , , j ,+ , ) - - f j i+ , (B , j ,+ , ) } - {fj,(B, j e+l , j , ) - f j , ( B , j i ) }

= l-l(S)pj, w:+, - H(S)p:,+,w:,

= H(B){p:,w:,+~ -p./,+ lw:,}. (4)

Since H ( B ) >__ O, and wi > O, i = 1 . . . . . n, then

T C ( n ) < TC(n ' ) , if and only if h (h) < h(h) - h(j i+ 1).

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Since h(l) =Pt/Wt > 0 depends only on the object/, I = 1 . . . . , n, then for three objects i,j, k we have: if h(i) < h(j) and h(j) < h(k), then h(i) < h(k), which proves the transitivity of h. From the transitivity of h, it follows that n is an optimal policy if and only if

h(j ,) <h(j2) < " " <h(j~). []

The following examples from the literature all satisfy the conditions of Theorem 1.1.

2. Examples

Example 2.1. See Refs. 1, 3, 4. We are given a component with n characteristics such that each characteristic has to be inspected separately. The inspections of characteristics are independent of each other. Character- istic i has an inspection cost C~ > 0 and a probability of rejection Ri, 0 < R~ < 1. The objective is to find the ordering inspection policy so as to minimize the total cost, given by

r - - I

TC(u) = C j , + ~ Cj~ I-I ( 1 - R j , ) , (5) r = 2 t ~ l

where r~ = (j~ . . . . . Jn) is an ordering policy and Jr is the rth characteristic to be inspected.

In Ref. 3, it was mentioned without any proof that the ordering policy which satisfies

C, IR, < c2tR~ <_... <_ c . IR. (6)

minimizes the cost function TC. Recently, in Refs. 1 and 4, proof of optimality of (6) has been presented. While the proof in Ref. 4 is similar to our proof in this paper, the proof given in Ref. 3 is based on several other results and uses mathematical induction. Their main result is stated in the following theorem.

Theorem 2.1. Given a component with n characteristics, assume that each characteristic i has a cost of inspection C,. > 0 and a probability of rejection Ri, 0 < Ri < 1. Then, the sequence that satisfies (6) minimizes TC.

The above theorem (also, the proof presented in Ref. 4) provides only a sufficient condition for the optimal inspection policy. It does not provide

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a necessary condition; however, the authors gave the following result (Corollary 4.3 in their paper).

Corollary 2.1. The sequence chosen in accordance with the rule

(1 - R , )C~ _<(1 - R 2 ) C 2 < _ . . . _ ( 1 - R . ) C .

does not minimize the total expected cost.

(7)

Using the definitions

wi : C i , Pi = - - R i , i = 1 . . . . . n,

)Ifj,(ji)) = Cj~, i = 1,

A ( J , . . . . . J, i 1 - - Rjt), i = 2 . . . . n,

~" l = l

i - - 1

H ( j l . . . . ,J i ) = I-I (1 - Rj,) , i = 1 , . . . , n, t = l

n ( ~ ) = 1,

it is clear that the conditions of Theorem 1.1 are satisfied and an ordering n is optimal if and only if

C1/R , <_ C21R2 < _ " " <- Cn/Rn.

Example 2.2. See Ref. 5. Assume that there are n acceptable candi- dates to fill a position. Suppose that the estimated benefit from candidate i, i --- 1 . . . . , n, for the next M years is E~ and the probability of acceptance of the job offer by him/her is 0 < R~ < 1; the probability of rejecting the job offer is 1 - R i . Also assume that this probability does not change during the entire selection process. When a candidate receives an offer, there is a fixed period of time during which he/she can accept or reject the offer; during such time, no other offers are being given to any other candidates; as soon as one candidate accepts the offer, the search terminates. The cost of offering the job to a candidate is a constant C. The net expected benefit from a candidate less the cost for offering him/her the job is RiE~ - C. The objective is to determine an optimal selection ordering (j~ . . . . ,Jn) among n ! permutations of n candidates so as to maximize the net expected benefit, defined by

TC(rt)(R:E:I-C ) + ~ (']:I I (I--Rjt))(RjrEjr--C ). r ~ 2 \ t = l

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Now, define

wi = (R iE , �9 - C) , Pl = - R t , i = 1 . . . . , n,

I f a~(ji) = (Rjigji -- C), i = 1, /

f j i ( j l . . . . . Ji ) R j i g i - - C 1 - Ryt), i = 2 . . . . . n, t = l

i - - I

H ( j , . . . . . j , ) = I-I ( 1 - R j , ) , i = l , . . . , n , t = l

= 1.

Clearly, the problem satisfies the condit ions o f Theorem 1.1 for maximiza- tion, and a sequence n is opt imal if and only if

h ( j l ) <_ h(j2) _ < " _< h( jn ) , I

where

h ( j i ) = R j i / (R j iE j i - C), i = 1 , . . . , n.

Example 2.3. See Refs. 2 and 6. Consider a set o f n independent, single-operation jobs to be processed by a single machine. Once the processing o f a job starts, it continues until the processing o f all jobs is finished. No preempt ion is assumed. Let

ti, F,. = tl + " + ti, f i ( t ) = ai exp(t)

be the processing time, flow time (complet ion time), and cost funct ion o f job i, i = 1 . . . . , n, respectively. The objective is to find the ordering o f the jobs so as to minimize the total cost, given by

it

TC( ) = f , , ( r : , ) . i = 1

Using the definitions

w~ = ai exp(t~), p; = exp(t~.) - 1, i = 1 . . . . . n,

f : ~ ( J l , - . . ,J;) = ~j~ exp(EjO,

H ( j , , . . . , h ) = exp(F:..i_~), i = 1 . . . . , n,

H ( ~ ) = l,

it can easily be concluded that the condit ions o f Theorem 1.1 are satisfied, and a sequence rc is opt imal if and only if

h ( j ~ ) -< h(j2) < " < h ( j , ) ,

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where

h(j ,) = (exp(tj,) - 1)/aj, exp(tj,), i = 1 . . . . , n.

Example 2.4. See Ref. 7. This problem is the same as problem stated in Example 2.3, except that, instead of an exponential cost function, we have a linear function,

ff(t) = ~it, i = 1 . . . . , n.

Define

wi = ~i, pi = ti, i = l . . . . , n,

f j i ( Jl . . . . . Jl ) ----" o~ji(Fj,),

H ( j l . . . . . J l ) = 1, i = 1 . . . . . n.

Again by Theorem 1.1, a sequence ~ is optimal if and only if

h ( j , ) < h(j2) < " < h ( j , ) ,

h( j~) = tj~/~ji, i = 1 . . . . . n.

3. Conclusions

We considered a sequencing problem applicable to situations where the optimal choice among n! sequences is sought. A class of sequencing problems that contains many sequencing problems from the literature was proposed. Based on the adjacent pairwise interchange of two objects, necessary and sufficient conditions for the optimal ordering policy have been presented. The results improve previous results given in special cases.

References

1. DUFFUAA, S. O., and RAOUF, A., An Optimal Sequence in Multicharacteristics Inspection, Journal of Optimization Theory and Applications, Vol. 67, No. 1, pp. 79-86, 1990.

2. LAWLER, E. L., and SIVAZLIAN, B. D., Minimization o f Time-Varying Costs in Single-Machine Scheduling, Operations Research, Vol. 26, No. 4, pp. 563-569, 1978.

3. RAOUF, A., JAIN, J. K., and SATHE, P. T., A Cost Minimization Model for Multicharacteristic Component Inspection, IIE Transactions, Vol. 15, No. 3, pp. 187-194, 1983.

4. LEE, H. L., On the Optimality o f a Simplified Multicharacteristic Component Inspection Model, IIE Transactions, Vol. 20, No. 4, pp. 392-398, 1988.

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5. KWAN, C. C., and YUAN, Y., .4 Sequential Selection Problem, Decision Sciences, Vol. 19, No. 4, pp. 762-770, 1988.

6. GLAZEBROOK, K. D., and GITTEN, J. C., On Single Machine Scheduling under Precedence Constraints and Linear or Discounted Costs. Technical Report, Math- ematical Institute, University of Oxford, Oxford, England, 1979.

7. SMITH, W. E., Various Optimizers for Single-Stage Production, Naval Research Logistics Quarterly, Vol. 3, No. 1, pp. 54-66, 1956.