Download pdf - An approximation algorithm for a single-machine scheduling problem with release times, delivery times and controllable processing times

74 European Journal of Operational Research 72 (1994) 74-81 North-Holland

Theory and Methodology

An approximation algorithm for a single-machine scheduling problem w th release times, delivery times and controllable processing times

Eugeniusz Nowicki Technical University of Wroctaw, Institute of Engineering Cybernetics, ul. Janiszewskiego 11 / 17, 50-372 Wroctaw, Poland

Received April 1991; revised January 1992

Abstract: The paper deals with a single-machine scheduling problem with release times, delivery times and controllable job processing times. It is assumed that the cost of performing a job is a linear function of its processing time, and the total schedule cost to be minimized is the total processing cost plus maximum completion time cost. In consequence, the processing order of jobs and their processing times are decision variables. A (p + 1/3)-approximation algorithm for the problem is provided, where p is the worst-case performance bound of a procedure for solving the pure sequencing problem.

Keywords: Single-machine scheduling; Approximation algorithms; Worst-case analysis

1. Introduction and problem formulation

In the real-life applications of scheduling research, apart from the machines, processing a job requires additional resources, such as facilities, manpower, funds, and so, which implies that jobs can often be accomplished in shorter or longer durations by increasing or decreasing the additional resources. Generally, this situation is usually very difficult to analyze. One of the simplest and most basic forms of resource allocation is represented by the t ime/cos t trade-off, [3]. There are situations where compressing a job is possible, but it entails extra costs, and such an action would be rational only if these additional costs are compensated by the gains from job completion at an earlier time. In the t ime/cos t model, the time required to perform a job can be reduced by the application of additional nonrenewable resources (measured by their cost). In consequence, job processing time can be considered as a decision variable.

Studies on the standard sequencing problems with controllable job processing times have been initiated by Vickson [12]. Vickson has analyzed the single-machine sequencing problem denoted as 1//)Zw~Tj (see [6] for notation). He has assumed that the cost of performing each job is a linear function of its processing time and a schedule cost is equal to the total processing cost plus a cost associated with

Correspondence to: Eugeniusz Nowicki, Technical University of Wrockaw, Institute of Engineering Cybernetics, ul. Janiszewskiego 11/17, 50-372 Wroctaw, Poland.

0377-2217/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0377-2217(92)00163-V

E. Novicki / A single-machine scheduling problem 75

the job completion times (the total delay cost). The problem is to find a job sequence and processing times minimizing the schedule cost. Other sequencing problems with controllable job processing times have been considered e.g. in [2,4,8,9,13]. Recently, in [10] a review of current results has been presented.

In this paper we consider the well-known sequencing problem denoted as 1/r j ,q j /Cma × with controllable job processing times. The problem is formulated as follows:

Each of n jobs from the job set J = {1, 2 . . . . . n} has to be processed on one machine. Jobs preemption is not allowed. For each job j, we define:

(i) ready time rj, rj > 0, (ii) processing time aj - x j, 0 _< xj < u j, where xj is the time by which the normal processing time a i is

shortened (compressed) and uj is the maximum compression, aj > u j, (iii) cost per unit of compression c j, cj > 0, (iv) delivery time q j, qj >_ 0.

At most one job can be processed at a time, but all jobs may be simultaneously delivered. Let ~r be a permutation of the job set J, and /7, the set of all permutations; ~- denotes a processing order of jobs. Denote by Cmax(X, ~-) the maximum completion time for compressions x = ( x ~ , x 2 . . . . . x , , ) ~ X , and processing order ~- e H, where X = {x ~ ~" : 0 _< xj < u j, j ~ J}. It is easy to verify that

C .... (x , 7r) = max r~(il)+ ~ (%( i ) -x ,~ ( i ) ) +q=(i:) . l <-il <i2<-n i=it

The total compression cost is equal to Ej ~ scjxj and does not depend on ~'. The total scheduling cost for compression x ~ X and processing order ~- ~ / 7 is defined as K ( x , ~r) = cCmax(X, ~ ) + E j~sc j x j , where c, c > 0, is the cost per unit of maximum completion time. The problem is to find ~-* ~ / 7 and x* ~ X minimizing

K ( x , "IT) =cCmax(X , 77") + E C j X j (1) jEJ

under the constraints

~ H , x ~ X . (2)

Without loss of generality, we can assume c = 1. The pure sequencing problem 1/rj,qj/Cmax, which extension is discussed in this paper, is equivalent

to the problem 1 / r j /Lma x with release times, due dates instead of delivery times and maximum lateness criterion. Another equivalent statement with non-bottleneck machines instead of release times and delivery times has been proposed by Graham et al. [6]. It can be noted that, due to the forward-back- ward symmetry, the delivery-time model has been studied more frequently than others. The problem 1/r j ,q j /Cma x has received considerable attention in past twenty years and has been employed among others in scheduling jobs on critical machine, in approximation algorithms for job-shop problem and as lower bound for the flow-shop and job-shop problems. Like the problem l / r j ,q j /Cmax, the problem (1)-(2) has an important applications as a simplified model of more complex scheduling situations in sophisticated machine and resource environment.

The sequencing problem 1/rj,qj/Cma x with controllable job processing times has been formulated as a single-objective optimization problem (analogous to the formulation in [12] for the problem 1 / /ZwjTj ) . Other formulations of sequencing problems with controllable job have been presented in [10]. For example, in a bicriterion formulation of these problems the set of all efficient points in (Cmax(x, ~r), Y:i~ j c j x j ) space is found. Note, that the algorithm which solves the problem (1)-(2) can be also applied for generating efficient points (for each c > 0 we obtain a single efficient point). Final selection among efficient points is made by the user, who can take into account e.g. the limitation of resource quantity.

The problem (1)-(2) has been studied in [10] and [11]. Due to its strong NP-hardness only approximation algorithms have been considered for real applications (n > 100). In [10] a 2-approximation algorithm has been shown. Zdrzaika, in [11] has proposed a (p + ½)-approximation algorithm, where p is the

76 E. Novicki / A single-machine scheduling problem

worst-case performance bound of a procedure used for solving the pure sequencing problem l/r~, q//Cma x. In this paper we present an approximation algorithm with the worst-case performance bound

1 equal to p + 7.

2. Approximation algorithm

First, we observe (as in [10,11]) that if c / > 1 for some j E J, then there exists an optimal schedule (x* , 7r*) such that x* = 0 (the function K(x, rr) is nondecreasing in the variable x/). Hence, if c/> 1 for some j, then we may set c / = 0, simultaneously setting u/= 0. This observation enables us to assume in what follows, without loss of generailty, that c/< 1, j ~J.

When all the processing times are fixed (i.e. x is fixed), the optimization problem (1)-(2) reduces to the pure sequencing problem l /r/ , qJCma x (see [1,5,6]); the problem is strongly NP-hard, but in practice it seems to be relatively easy to solve. On the other hand, when the permutation ~- is fixed, the optimization problem (1)-(2) reduces to a linear programming problem. This suggests the following heuristic approach to the problem (1)-(2). First, for the given x ' ~ X determine ~r r' ~ H , minimizing Cmax(X', 7r), and then determine x H e X , minimizing K(x, ~.H) subject to x EX. Obviously, the heuristic quality strongly depends on the selection of x' . As was mentiond in [11], the above heuristic algorithm has been run twice: first with x] = (1 - c/)%, and then with xj = uj. We propose a single run of the algorithm with

x j = m i n { 1 , } ( 1 - c / ) } u / , j ~ J . (3)

Thus we have the following approximation algorithm.

Algorithm H Step 1. Find a processing order T/"H for the pure sequencing problem 1/r/, q j / / C ma x with processing

times a / - x ] , j ~ J, using an algorithm with worst-case bound p. H H Step 2. Determine x minimizing K(x, Tr ) subject to x ~ X.

In Step 1 we can apply one of existing exact (then p = 1) branch-and-bound algorithms, e.g. [1,5], polynomial approximation schemes [7], or the polynomial approximation algorithm with worst-case

4 performance bound p = ~ [7] (the algorithm of Hall and Shmoys). The minimum x H in Step 2 can be found using the simplex method. This optimization problem can

also be solved by applying any procedure for finding a time-cost trade-off curve in an activity network with linear cost-duration functions, see e.g. [3] (certain modifications of the stopping condition in the procedure is necessary).

Now we give certain intuition explaining the selection of x' . It is quite natural to select x ' 'close' to (unknown) x *. Let us consider the selection xj = z(c)uj, where z is a function from [0,1] on [0,1] (then x ' ~ X ) . There exists an optimal schedule (x* , 7r*) such that x* = u~ if c / = 0 and x* = 0 if c / = 1. Hence we obtain that z(0) = 1 and z(1) = 0. Moreover, it is reasonable to take z as a nonincreasing function. These facts and the form of lower bound introduced in the next section suggest jointly the choice of the function z in the form (3) (i.e. as z(%) = rain{l, 4(1 - %)}).

3. Lower bounds

In this section we derive various lower bounds for K(x*, 7r*) -~ K *. For a ~ [0,1] we define

LB( a) ~- a min Cmax( X', ~') ,rr E II

+ E r a i n { ( 1 - a ) a j , ( 1 - o Q ( a j - u j ) + rain{c/, c , ( 1 - 4 a ) + ½a}u/}. j ~ J


Lemma. LB(a) <_ K * for any a ~ [0,1].

Proof. Let x ~ X, ~- ~ / 7 and a ~ [0,1]. The proof consists of three parts. First, we will introduce a lower bound on K(x , 7r) using two auxiliary functions f ( x , ~) and g(x, a). Next, certain lower bounds on rain, ~ x f ( x , 7r) and min~ ~ x g(x , a) will be derived. These bounds allow us to prove the inequality LB(a) _< K* in the third part.

First, we show that

K ( x , 7r) = Coax(X, zr) + ~_,cjxj>__af(x, 7r) + g ( x , a) , (4) j E J

where

f ( x , ~ ) ~ Coax(X, ~') + 2 max{0, 3 c i - ~ } x j j ~ J

and

g ( x , ~) ~ E [(1 - o 0 ( a j - x a ) + min{cj, cj(1 - ~a) + ½a}xj]. j c J

Applying the machine-based bound Coax(X, 7r)> Ej e g(a s - x i ) , we obtain

Cmax(X , 77") + E C j X j ~ O l C m a x ( X , 77") + ( 1 --O~) E ( a j - x j ) + ~_,cjxj. (5) j ~ J j ~ J j ~ J

1 Using the equality max{0, y} + min{0, -y} = 0 for y = (3cj - ~)axj, we have

4 cjxj = cjxj + a max{0, ~cj4 _ .~}xj~ + rain{0, ~ - ~cj}ax j

= a m a x { 0 , 4 c j - 1 } x , + m i n { c j , c j ( 1 - ~ a ) + ½a}xj, j ~ J . (6)

Finally, applying (6) the definition of the f ( x , 7r) and g(x, a), we obtain (4) from (5). Next, we derive a lower bound on m i n , ~ x f ( x , rr) and we calculate the value of m i n ~ x g ( X , a). The

following inequality holds

m i n f ( x , ~') >__ Coax(X', 7r). (7) x~)(

Indeed, employing the definition of Coax(X, rr) and inequalities

1 max{0, 4 - ~ } x = o ) > 0 , j ~ J - { i I i 1 + 1 , i2} ~C ~( j ) _ , . . . . ,

we have

f ( x , =

where

>_

i2 1 max r~(i,)+ E (a~(j)--X~r(j))+qr(i2) + E max{0, 43crr(j)-5)x~r(i)

1 <--i t <-i2<-n j=i~ J ~ J

1 max r~-(il) + ~ ( a ~ o , - x .o ) + max{ 0,4xc~o) - x}x.o)) + q~.(i2)

1 <_i~ <_i2<_n j.=i t

max h( x, 7r, i 1, i2), l<<_il<-i2<-n

h( x, rr, il, i2) ~ r~r(iO + i 2

[a~o ) - rain{l, 4(1 - c~o)))x~o)] + q,(i:). j = i I

(s)

78 E. Novicki / A single-machine scheduling problem

Applying the definition of the set X, inequalities

m i n { 1 , 3 ( 1 - c = ( j ) ) } > 0 , j~{ i l , i , +1 . . . . . i2}

and (3), we obtain

i2 rain h(x, 7, il, i2) =r=(il)+ ~ min x E X J =il x~r(J)~ [O'u~(D]

i2

= r~(i,)+ 2 [a~(j)- min{1, 4 ( 1 - c~(j))}u~(j)] + q~r(i2 ) j= i 1

i2

= rrr(ip + E (art( j ) -- X~( j ) ) + qlr(iz ). j= i l

Finally, from (8), the well-known property

rain max h(x, 7, i1, i2) > max min h(x, 7, il, i2) x ~ X l<_il<_i2<_n l<_il<_i2<_n x ~ X

and (9), we have

rain f (x , 7) > rain max h(x, 7, i1, i2) > x E X x ~ X l~il<_i2<_n

i2

= m a x rlr( i , )+ E ( a ~ r ( j ) - - x ~ ( j ) ) + q l r ( i z ) = C m a x ( X ' , 77-), l <-il <i2<n j=i l

[a~(j)- rain{l, 4(1 - c~d) ) } x=(j) ] + q~(i2)

max rain h( x, 7r, il, i2) l<i l<i2<n x ~ X

(9)

which completes the proof of (7). The expression g(x, a) is linear and separable in each of the variables xj. Because of this linearity and separability, the minimum over the set X is obtained by setting each of the individual variables to one of its extreme values. Hence, we obtain

ming(x ,a )= Emin{ (1 -a )a / , (1 -a ) (a j -u j )+min{c j , cj(1-4a)+½a}uj}. (10) x E X j ~ j

Now, the essential part of the proof will be carried out. From (4), (7), 7 ' and (10), we have

K * = min minK(x , 7 ) > rain min[af(x, 7) +g(x, a)] rr~H x ~ X ~rEH x ~ X

> ~nmin t[aminx~x f (x , 7) + x~xmin g(x , a)] > =~umin tl~Cm~x(X" 7) + x<xmin g(x, =)] = a m i n Cmax(X', 7 ) + min g(x, a) = LB(a) ,

zr~II x ~ X

which completes the proof of the Lemma. []

Further considerations employ the permutations 7'(x ' )~II such as Cmax(X' , 7 ' ( x ' ) ) = m i n = ~ t / Cr, ax(X', 7). To simplify notation, the argument x' will be omitted in 7'(x'). Using the definition of the x ' and 7 ' , we have

LB(a) - -aCm,~(x' , 7 ' ) + Y'~ [(1 - a )a j + min{O, ozxj-(1- cg)uj}]. j ~J

It follows from definition of p (worst-case performance bound) that

1 Cmax( X t, I"T') >_ ~-Cmax(X', "ITH).


From there, and by applying the lemma with a = 1 we have

K* >_ LB( I ) = Cmax(X', ~-') + E min{0, xj' - (1 -cy)uy} = Cm~x(X', ~"1 j~J

1 >__ - - C m a x ( X t, 7 F H ) .

p

3 3 r Similarly, for a = ~ and from inequalities aj >_ u i, ~xj - (1 - cj)uj <_ O, j ~ J, we obtain

K . > L B ( ~ ) 3 3 ' - (1 - %)uj] __ 7 C m a x ( X ' , gl") -'}'- E [ l aj -}- gXj j e J

3 t >__ ~-Cmax(X , ' w ' ) ~'- E min{ {, c,}uj

j e J

1 3 - - C m a x ( X ' , 3"1 ".1) -t- }7~min{¼, cj}uj.

P j c J

Lower bounds (11) and (12)will be employed in the worst-case analysis of Algorithm H. Let consider another lower bound for K* (see lemma)

(11)

(12)

LB ~ max L B ( a ) . e~ ~ [(),1]

Next we will show how to calculate LB. First note, that the function LB(a) is piecewise linear and concave. Let % ~ max{ 3, 1 - %}, j E J and a , + 1 ~ 1. It follows from the definition of the xj' and aj, that

c L x ~ ! = m a x { ¼ , 1 - % } m i n { 1 , 4 ( 1 - % ) } u j = ( 1 - c j ) u y , j ~ J .

Hence and using the inequality

3 __ X ! 1 3 , (1 - cy)u)] = E [a j - (1 - c~)uj] : LB(O) j~J j ~J j~J

we obtain the simple formula for the lower bound: LB = max I ~j_<n+l LB(aj). This lower bound will be employed in the experimental analysis of Algorithm H.

4. Worst-case analysis

For some problem instance, let K H be the value of the objective function (1) when Algorithm H is applied, i.e. K H ~ K(x H, Ira).

1 Theorem. For every problem instance KH/K * <_ p + 7.

Proof. From Step 2 of Algorithm H we obtain

K ~t=K(x H,Tr H) = m i n K ( x , ~ - H ) <K(x ' ,~r H)=Cma×(x' ,~H)+ ~cjx~. (13) xEX jEJ

1 It can be verified that cj(1 - cy) < 7. Hence, we have

~ c j x ~ = ~ m i n { % , 4(1-%)cj}uj<_ Y'~min{%,½}uj_< ~ m i n { 4 c j , ~}uj. (14) i~J J~J j c J j c J

80

Table 1

E. Novicki / A single-machine scheduling problem

j 1 2 3 4

rj 1+3e 0 0 0 ay e l + e l + e l + e uj 0 1 l l

1

qj 2 + 2e 0 0 0

From (13), (14), (12) and (11) we obtain

1 4 1 v t g H _~< Crnax ( x ' , "/7- H) q- ~ E min{cj, ~}uj < Cmax( X , 77" H) -t- 4K * - -Cmax( x , 77- H)

j e J P

< 4 K * + 1 - o K * = (O + -~)K ,

which completes the p roof of the Theorem. []

1 We now show that the bound p + ~ is the best possible when p = 1. To this end consider the instance specified by the data in Table 1. It is assumed that e is an arbitrarily small positive number . F rom (3), we

! ! v t 2 have x t = 0, x e = x 3 = x 4 ~. Applying Algor i thm H, we get rr n = (2, 3, 4, 1) in Step 1 and next, in Step 1 . 2 . The optimal permuta t ion is 2, we obtain x ~ = 0 , x ~ = l , x ~ = l , x4 n = 0 and K H = 3 + 6 e + 7

4 (2, 1, 3, 4), for which we have x• = x ~ =x~ ' =x~ ' = 0 and K * = 3 + 6e. Hence K H / K * ~ ~ as e ~ 0. It follows f rom the p roof of the T h e o r e m that the opt imizat ion Step 2 of Algor i thm H is not necessary

1 t to guaran tee the bound p + 7. It is enough to set x~ : = xj, j ~ J (see (13)). However , the per formance of Step 2 usually decreases the goal funct ion value and does not need much computa t ion time.

Algor i thm H and Algor i thm Z (taken f rom [11]) were coded in Pascal and run on an IBM PC AT. In Step 1 of both algori thms the p rocedure of Hall and Shmoys [7] was implemented. Test instances were genera ted in the way described by Carlier [1] (for the pure sequencing problem 1/rs, q J C ~ x ) . For each n e {50, 100, 200, 300, 400, 500} and k e {17, 18, 19, 20, 21} samples of 20 instances were obta ined (values rj, at, ui, qj, cj were chosen with uni form distribution between 1 and rmax, amax, Umax, qmax, Cmax, respectively). We set ama x = 5 0 , Uma x = 2 5 , rma x = qmax = nk, Cma x = 100 and c = 100 (see (1) for the definit ion o f c). Thus, 600 instances were tested. It should be stressed that instances with k ~ {18, 19, 20} were repor ted by Carlier as the hardest one. For each instance we calculated following values: LB, K n and r/A = ( (K A _ L B ) / L B ) * 100%, A ~ {Z, H} (it is quite obvious that ~TA >_ ( ( K A _ K * ) / K *)* 100%, A e {Z, H}). The computa t iona l results are presented in Table 2.

The results form Table 2 show the dominance of Algor i thm H. Namely, on 600 tested instances we obta ined average value of r/H and ~7 z equal respectively to 1.25 and 6.68%. We found K H < K z in 547

Table 2 Computational results

Number Number Average Average Number of instances for which Average comput. of of of of K u < K z K H = K z K n > K z time of Alg. H jobs instances r/H ~7 z (in sec)

50 100 1.21 6.53 90 5 5 2 100 100 1.12 7.61 89 6 5 7 200 100 1.26 5.98 92 6 2 14 300 100 1.34 6.73 89 4 7 24 400 100 1.27 6.42 93 2 5 42 500 100 1.31 6.81 94 3 3 63


cases, K H = K z in 26 cases and K H > K z in 27 cases only. Moreove r , the c o m p u t a t i o n t ime of

A l g o r i t h m Z is by its s t ruc tu re twice longer than tha t of A l g o r i t h m H (A lgo r i t hm Z is run twice for two

var ious x). U p to now the re is not known in the l i t e r a tu re any exact a lgor i thm for the p r o b l e m (1)- (2) . Though in

theory such an a lgo r i thm could be cons t ruc ted b a s e d on b r a n c h - a n d - b o u n d techniques , never the less the c o m p u t a t i o n t ime of this a lgor i thm can be eva lua t ed with r e spec t to the c ompu ta t i ona l t ime of an exact a lgor i thm for the pu re sequenc ing p r o b l e m 1 / r j , q j /Cma x and t he r e fo re it could a p p e a r ineffect ive. Fo r example , for e s t ima t ion pu rposes we have e m p l o y e d the bes t known exact a lgo r i thm (Car l i e r [1]) for the p r o b l e m 1 / r j , q j /Cma x in ca lcu la t ion of LB. F o r n = 500 the average c o m p u t a t i o n a l t ime of Car l i e r ' s a lgor i thm was 668 sec (12 ins tances f rom 100 t e s t ed have not been solved in t ime limit 1800 sec). It seems tha t an exact a lgor i thm for the p r o b l e m (1) - (2 ) would work s ignif icant ly longer .

Tak ing all these a rgumen t s into account , it seems tha t a pp rox ima t ion A l g o r i t h m H can be recom- m e n d e d for p rac t ica l appl ica t ions .

Acknowledgements

W e wish to t h a n k the r e fe rees for the i r he lpfu l sugges t ions and c o m m e n t s r ega rd ing an ea r l i e r vers ion of this pape r .

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