342 European Journal of Operational Research 70 (1993) 342-349 North-Holland

An approximation algorithm for the m-machine permutation flow shop scheduling problem with controllable processing times

E u g e n i u s z N o w i c k i

Institute of Engineering Cybernetics, 50-372 Wroctaw, Poland

Received June 1991; revised January 1992

Technical University of Wroctaw, ul. Janiszewskiego 11 / 17,

Abstract: The paper deals with the m-machine permutation flow shop scheduling problem in which job processing times, along with a processing order, are decision variables. It is assumed that the cost of processing a job on each machine is a linear function of its processing time and the overall schedule cost to be minimized is the total processing cost plus maximum completion time cost. A 4-approximation algorithm for the problem with m = 2 is provided; the best approximation algorithm until now has a

3 worst-case performance ratio equal to 3- An extension to the m-machine (m _> 2) permutation flow shop 1 problem yields an approximation algorithm with a worst-case bound equal to ½(p + ~p(m - 1) ) + ~ +

O(1/~/pm ), where p is the worst-case performance ratio of a procedure used, in the proposed algorithm, for solving the (pure) sequencing problem. Moreover, examples which achieve this bound for p = 1 are also presented.

Keywords: Flow shop scheduling; Approximation algorithms; Worst-case analysis

1. Introduction and problem formulation

Models of jobs (operations, activities) in which processing times are decision variables influencing the overall project cost are commonly used in the area of project planning and control [2]. Their motive in the field of sequencing and scheduling is of the same nature, that is, they are justified in situations where jobs can be accomplished in shorter or longer durations by increasing or decreasing additional resources. Some standard sequencing problems with controllable job processing times have been considered, e.g. by Vickson [15,16], Van Wassenhove and Baker [17], Ishi et al. [7], Grabowski and Janiak [4], Daniels [1], and Nowicki and Zdrzatka [11] (see also the review paper [12]).

In this paper we consider the m-machine permutation flow shop problem with controllable job processing times, which is a generalization of the pure sequencing problem denoted as F II C ~ (see [6] for notation).

Let J = {1, 2 . . . . , n} denote the set of jobs to be scheduled and M = {1, 2 . . . . . m}, the set of machines. Each job is performed first on machine 1, next on machine 2, and so on, until it completes execution on

Correspondence to: Dr. E. Nowicki, Institute of Engineering Cybernetics, Technical University of Wroelaw, ul. Janiszewskiego 11/17, 50-372 Wrociaw, Poland.

037%2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

E. Nowicki / An approximation algorithm for the m-machine permutation flow shop scheduling problem 343

machine m. It is assumed that each machine processes a job without interruption and the job processing orders on all the machines are identical. For each job j, j ~ J, there are defined: (i) processing time a i r - Xij on machine i, 0 <_x~j < uij, where x~r is the time by which the 'normal '

processing time air has been compressed (shortened), and u~j is the maximum compression; uij <_ air for i ~ M a;

(ii) cost per unit of compression cir on machine i; cir >_ 0 for i E M. Denote by x = (x11, x12 . . . . , Xl . . . . . . Xml, Xm2 . . . . , Xmn) a vector of compressions, and by X = {x:

0 _<x~r _< u~r, i ~ M , j E J} the set of all feasible compressions. Let 7r be a permutation of J a n d / / b e the set of all such permutations. Permutation ~r defines the job processing order, where ~r(k) denotes the job which is in position k in 7r. Denote by Cm~,(x, 7r) the maximum completion time for compression x ~ X and processing order ~r ~ / / . It is easy to verify that

ki

Cmax(X , "if) " ~ - max ~_, ~_~ (ai~(k) --Xi~(k)). (1) l = k 0 < k l < " " <km-l <kra=n iEM k=ki_ 1

The total scheduling cost for compression x ~ X and processing order ~ - ~ H is defined as K ( x , ~r) = WCmax(X, zr) + Y'.i ~ ME r ~ 1CiyXir, where w, w > 0, is the cost per unit of maximum completion time. Without loss of generality, we can assume w = 1. The problem is to find "n-* ~ / / and x* ~ X m i n i m i z i n g

K ( X , "iT) : Cmax(X , 71-) --[- E E ci jx i j (2 ) i ~M j~J

under the constraints

7r~I1 , x ~ X . (3)

In this paper, we shall need some additional notation. An instance I, of the problem (2)-(3) is a specification of the data n, m, ao., u~r, cgr, i E M , j ~ J . We define K * ( I ) = K ( x * , "rr*) to be the total scheduling cost if an optimal compression x * and an optimal processing order ~-* are used for instance I. K n ( I ) is the total scheduling cost if an approximation algorithm H is used for instance I. The worst-case performance ratio for an algorithm H is defined as ~7 H = min{t: K n ( I ) / K * ( I ) < t, for each I} [3]. Further on, to simplify notations, the argument I will be omitted in K u ( I ) and K * ( I ) .

For problem (2)-(3) with m = 2, which has been studied in [11], it has been shown that the decision version of the problem is NP-complete (even when the processing times on one machine are fixed and all

3 C~r are identical) and an approximation algorithm with a worst-case performance ratio equal to ~ has been found.

In this paper another approximation algorithm for problem (2)-(3) with m = 2 is provided. Its 4 worst-case performance ratio is equal to 3. Extending these results to the m-machine (m > 2) permuta-

1 tion flow shop problem, we provide an approximation algorithm with a worst-case bound equal to ~(p 1 + ~/p(m - 1) ) + ~ + O(1 /v /pm ), where p is the worst-case performance ratio of a procedure used, in

our algorithm, for solving the (pure) sequencing problem F II Cm~x (for some fixed x). Moreover, examples which achieve this bound for p = 1 are also presented.

2. Approximation algorithms

We observe (similarly as in [11]) that if cir > 1 for some i ~ M and j ~ J , then there exists an optimal schedule (x* , zr*) such that x* = 0. Hence, if cir > 1, then we may set cir = 0 setting simultaneously u~r = 0. This observation enables us to assume, without loss of generality, that cir < 1, i ~ M, i ~ J.

If all the processing times are fixed (i.e. x is fixed) then problem (2)-(3) reduces to the well-known sequencing problem F [I Cmax. On the other hand, when the permutation ~- is fixed, the problem can be

a It is assumed that a zero processing time on a machine corresponds to a job performed by that machine within infinitesimal time.

344 E. Nowicki / An approximation algorithm for the m-machine permutation flow shop scheduling problem

formulated as a linear program. This suggests the following heuristic approach H [11]. Firstly, for given z ~ X determine 7r'~/~H, minimizing Cmax(Z, zc) (this is the pure sequencing problem), and then determine x n ~ X , minimizing K(x , 1r H) subject to x ~ X . Obviously, the quality of such an algorithm strongly depends on the selection of z. In [11] for m = 2, ziy --- (1 - ci/)uij, i ~ M , j ~J , is proposed.

In this paper, we analyze an approximation algorithm such that p, g

Zi j ' ~ -g (c i j )U i j - -X i j , i ~ M , j ~ J , (4)

where

g ( y ) = max min - - - , 1 , 0 , y ~ [0, 1], (5) am a

or= 1 - p m / [ p + ~ p ( m - 1) ]2. (6)

and p is the worst-case performance ratio of a procedure used, in our algorithm, for solving the (pure) sequencing problem F l[ Cmax, P ~ [1, m]. Variable ot and the function g depend on m and p, which, however, have not been expressed explicitly. It is easy to verify that ot ~ [0, 1], g ( y ) ~ [0, 1] for each y ~ [0, 1], and x g ~ X . Finally we have the following approximation algorithm H(g) .

Algorithm H(g) Step 1. Find a processing order 7r rag) to the (pure) sequencing problem F 1] C m a x with processing times

aij - x g., i ~ M, j ~ J, using a procedure with the worst-case performance ratio p. Step 2. Determine x H(g) minimizing K(x , 7r H~g)) subject to x ~ X .

The processing order 7r H<g) for m = 2 can be found by Johnson's algorithm [8] in O(n log n) time (then p = 1). In this algorithm the jobs satisfying alj--xgj <_a2j--x~j should be arranged in order of nondecreasing a l j - xgj, followed by the remaining jobs in order of nonincreasing a 2 j - x~y. In the case m > 2, we can apply exact branch-and-bound algorithms (then p = 1), e.g. [5],[13] or polynomial approximation algorithms (e.g. the algorithm of Campbell et al. with p = [rn/2] [10], the algorithm of Nawaz et al. [9] or the algorithm of Shmoys et al. with p --- O(log2m) [14]).

The optimization problem in Step 2 can be solved using standard linear programming methods. This problem can also be solved by applying any procedure for finding a time-cost trade-off curve in activity network with linear cost-duration functions, see e.g. [2] (certain modification of the stop condition in the procedure is necessary).

3. Worst-case analysis of algorithm H(g)

In this section, we provided a worst-case analysis of algorithm H(g). Further considerations employ the auxiliary variable r such that

r = p + p ( m - p ) / ( 2 p + 2 ~ p ( m - 1) - 1),

1 1 where p ~ [1, m] and rn > 2. It can be verified that r = ~(p + ~p(rn - 1) ) + ~ + O(1/~/pm ). For p = 1 4 and rn = 2 we have r = 3.

Theorem. For each m >_ 2 and p ~ [1, m], K t ~ g ) / K * < r.

Proof. Firstly, we derive a lower bound on K*. Let m > 2, p ~ [1, m], x ~ X and ~- ~ H . It is clear that

Craax(X , ~-) ~ ( l / m ) • E ( a i / - x i / ) i~M j ~ J

E. Nowicki / An approximation algorithm for the m-machine permutation flow shop scheduling problem 345

and

Cmax(X, 71-) = O/Cmax(X , 71") + (1 -- o~)Cmax(X , 7r) ~ o~Cmax(X , n) + - - (1 - a )

m

Hence and by the definition of K(x, ~') in (2), we obtain

K ( x , 7r) =Cmax(X, 7r) + E ECiyXiy>--oth(x, ,i7-) + E E r i j ( x i j ) , i~M j~J iEM jEJ

where

E E ( a i j - - x i j ) . i~M j~J

(7)

h ( x , ~') ~ Cmax(X , 77") + E E (1 - g ( c i j ) ) x i j i~M j~J

and

1o( 1o) ri j(xi j ) ~ a i j+ otg(cij ) - - o r - - - +ci j xij.

m rn

Using the definition of Cmax(X , "/1") in (1) and the inequalities

(1 - g(Ci~r(k)))Xi~r(k) >__ 0,

we get

k ~ J - { k i _ l , k i _ l + l , . . . , k i } , i ~ M ,

h( x, 7r) = max l=ko_<kl_<... <_km_l <km=n

>_

k i

E E i~M k=ki_ 1

ki

max E E l=ko<-kl<-"" <-km-l<-km=n i~Mk=ki_ 1

(ai~(k)--Xi~(k)) + E E (1--g(ci~(k)))Xi~(k ) i~M k~J

( ai~(k) - g( ci~,(k))Xi~(k)) ]"

(8)

(9)

= max E (ai~(k)--xf~(k) = Cm~,( x~, ~)" l=ko<kl-<"" <km-l<k','=n i k=ki_ 1

It is easy to verify that the following equality holds for 0 < y < 1:

min{ag(y) - a +y , (1 - a ) / m } = min{y, ( 1 - a ) / m } .

Employing the definition of ro.(xij) and the last equality, we have

(( 1 o ) ) min rij(xij) = aij + min o l g ( c i j ) - - t X - - - - + C i j Uij, O

O<_xij'c. Uij m m ( 1o) 1 - a ( a i y _ u i ~ ) + m i n ag(ciy ) a + c i j , - - uij

m m

(1-°1 1 - a (ai j _ uij) + min cij , uij. m m

In view of the well-known property 'min max >_ max min' and from the definition of x g. in (4), we obtain

I )] minh(x, 7r) >_ max min E E (ai~r(k)--g(CiTr(k))Xilr(k) x~X l=ko_<kl<... <_km_l<km=n x ~ X [ i ~ M k=ki_l

346 E. Nowicki / An approximation algorithm for the m-machine permutation flow shop scheduling problem

Finally, from (7)-(9) and by the definition of p (i.e. min~nCmax(X g, 7r)> (l/p)Cmax(X g, ~rHtg))), we obtain the following lower bound on K*:

K* = min minK( x, Tr) ~ min min [ah( x, Tr) x~X ~ n x~X [ i ~M j~J rij( Xij) ]

o [ ,o { 'o}] ~--- --Cmax( xg, 7rH(g)) -[- E E - - - -m ( a i j - uij) q- min cij , Uij . (10)

P i~M j~J m

Now, the essential part of the proof will be carried out. By the definition, K n(e) =K(x 14(g), rr/4(g)) = rain, ~ xK(x, 7r/4(e)). Substituting *ij~/~(e) for x g. = g(cij)Uij , we obtain the inequality

KH(g) <-K( xg, 71"H(g)) = Cmax( xg, 71"H(g)) "¥ E E cijg(cij)Uij" (11) i~M jEJ

Applying the inequality Cma~(X, ~r) < ~i ~ M~-~j ~ j(ai j -- xij), we have

K 14(g) < man ~ ~ ( a i j - x i j + cijxij ) = E E [ ( a i j - u i j ) -F cijuij ] . (12) x~X i~M j~J i~M j~J

Let tr = ra/p. It is easy to verify that tr ~ [0, 1] and 1 - ~r = r(1 - a)/m. Taking the convex combina- tion of the right-hand sides of (11) and (12) with weights ra/p and r(1 - a)/m, respectively, we obtain

Cm (X , + g ( c i j ) + - - . (13) i~M j~J rn

Now, we show that the following inequality holds for y ~ [0, 1]:

g ( y ) + ~ y < m i n y, . (14) m rn

Indeed, employing the definition of g, we have

( p ) { ( 1 ( ) ( p l - a ) } 1-a } 1 - a l + p + a ( m - l - p ) - Y Y, Y ,--m-Y g ( y ) + ~ y = m a x rain + m m m

(;( ) ' ° ) 1 l + p + a ( m - l - p ) - Y Y, Y < max . (15) rn m

Next, using inequalities p > 1 and m > 2, we obtain

l + p + a ( m - l - p ) 1 + a ( m - 1) 1 - a l + a ( m - 1 ) 1 - a = + - - _ < + -

pro pm m m m 1--or

=2 + a < l - a + a = l . rn

Hence, for each y ~ [0, 1],

l ( l + p + a ( m - l - p ) ) 1 l + p + a ( m - l - p ) p m -Y y < p m y <y. (16)

Similarly, using the definition of a, we have

_ (m--1--p__)p__m ]2=[2mp~/p(m--1)_ +2mp2] 2 [ l + p + a ( m - - l - - p ) ] 2= m ( p + ~ p ( m _ l ) ) 2 ] [ (P+~/P(m-1)) 2

2mp ]2 = ( p + } / p ( m - l ) ) = 4 r n p ( l - a ) .

E. Nowicki / An approximation algorithm for the m-machine permutation flow shop scheduling problem 347

T a b l e 1

1 2 . . . n

a U n - 1 l + e . . . l + e

Ulj 0 1 ... 1 1

c U 0 ½ ... a2j 1 1 ... 1 u2y 0 1 ... 1

c2j 0 ~ ...

Hence , for each y ~ [0, 1],

- - p l ( l + P + a ( m - l - P ) ) [ l + p + a ( m - l - p ) ] 2 m - Y y - < 4m2p = 1-am (17)

Apply ing (16), (17) and the inequali ty (1 - a ) y / m < min{y, (1 - a) /m}, we obta in (14) f rom (15).

[] Finally, employing (14), (13) and (10), we get K H(g) <_ r K * , which comple tes the p roo f of theorem.

It follows f rom the p roof of the T h e o r e m tha t the opt imiza t ion Step 2 of a lgor i thm H(g) is not necessary. I t is sufficient to set xi~ (g) = x g if K ( x g, rr u(g)) _< g ( x h, "rrH(g)), and ~ij~H(g) _-x h = h(cij)ui j otherwise, where h ( y ) = 1, y c [0, 1]. Indeed , in this case we obta in the inequali ty

K n(g) = m i n{K(x g, rrn(g)), K ( x h, ~rn(g)) } _< t rK(x g, rr n(g)) + (1 - t r ) K ( x h, rr n(g))

<_ crK(x g, ~r n(g,) + (1-~) E E [ ( a o - u i j ) +cijuij], i ~ M j ~ J

which is identical with (13). Now, we show tha t the bound f rom the T h e o r e m is the best possible for m = 2 and p = 1 ( Johnson ' s

a lgor i thm in Step 1). For this pu rpose consider an instance specif ied by the da ta in Tab le 1. It is assumed tha t e is an arbi trar i ly small posit ive n u m b e r and n > 3. Tab le 2 p resen t s the appropr i a t e values of a i j - - x g u ' i ~ M , j ~ J (g(½)= ½).

Apply ing Johnson ' s a lgor i thm in Step 1 of Algor i thm H(g), we get 7r l-l(g) = (1, 2 . . . . . n). Next, in Step 2, we obta in x ~ (g) =X2naa (g) = O, ~t4(g) x ~ (~) O, j = 2, 3,. . , n - 1, x ~ (g) = e, Xzn~ (g) = 1. H e n c e ~ l j = E, . =

K n(g)=n - 1 + n - 1 + ½ ( ( n - 1)e + 1) .

O n the o the r hand, for the p e r m u t a t i o n ~.0 = (2, 3 . . . . . n, 1) and for x°~ = X°l = 0, x°j = 1, x°j -- 0, j = 2, 3 , . . . , n, we obta in

K* < K ( x °, zr °) = e ( n - 1) + n - 1 + 1 + l ( n - 1).

4 H e n c e KH(g)/K(x °, ~0) ~ ~ as e --> 0 and n ---> oo. This implies tha t Kn(g) /K * can be arbitrari ly close 4 ,17H(g) 4 to ~ and = -g = r.

T a b l e 2

1 2 ... n

a l i - x f j n - -1 l + e - - g ( ½ ) ... l + e - g ( ½ )

a2j -- x~j 1 1 - g(½) ... 1--g(½)

348

Table 3

E. Nowicki / An approximation algorithm for the m-machine permutation flow shop scheduling problem

i j

1 2 3 . . . k 2 k 2 + 1 k2+2

1 (1, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) (k, k, c) 2 (0, 0, 0) (1, 1, c) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) 3 (0, 0, 0) (0, 0, 0) (1, 1, c) (0, 0, 0) (0, 0, 0) (0, 0, 0)

k2 (0, 0, 0) (0, 0, 0) (0, 0, 0) (1, 1, c) (0, 0, 0) (0, 0, 0) k 2 +1 (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) (1, 1, c) (0, 0, 0)

Next, we show tha t the bound f rom the T h e o r e m is the best possible for p --- 1 (an exact b ranch-and- bound a lgor i thm in Step 1) and m = k 2 + 1, where k - - 2 , 3 . . . . . T o this end consider the instance specif ied by the da ta in Tab le 3. It is a s sumed tha t n = k 2 + 2, c = 1 / ( k + 1). Each triple in Tab le 3 deno tes (aii , uij, cij ~ for the respect ive i and j. Tab le 4 presen ts the appropr i a t e values of a i j - xg., i ~ M , j ~ J ( g ( c ) = ~).

Le t 7r' = (1, ¢rl, r r2 , . . . ,rrk, k 2 + 2), where ~r i = ( ik + 1, ik . . . . . (i - 1)k + 2), i = 1, 2 . . . . . k. It is easy to verify tha t Cm~x(X g, zr ') = min~ e nCmax( x s , rr). Then, we can set rr n(g) = rr ' . For this pe rmuta t ion we have xi~ (g) = O, i ~ M , j ~ J and K H(g) = 1 + k. Le t 7r ° and x ° be def ined as follows:

~ 0 = ( k 2 + 1, k 2 , . . . ,2 , 1, k 2 + 2) ,

x ° = 0 , i ~ M , j - - l , 2 . . . . , k 2 + 1 , x ° n = k , x ° n = 0 , i = 2 , 3 . . . . . m .

We obta in K * < K ( x °, ~ .0 )= 1 + c k = 1 + k / ( k + 1). H e n c e

K n ( g ) / K ( x °, rr °) = 1 + k 2 / ( 2 k + 1) = r and K H t g ) / K * = r.

Finally, we obta in ~7 H(g) = r for m -- k 2 + 1, where k = 2, 3 . . . . . No te that in the case k = 1 ( thus m = 2), a lgor i thm H ( g ) yields an op t imal solut ion ( Johnson ' s a lgor i thm genera tes the pe rmu ta t i on (2, 1, 3)).

I t is possible to const ruct examples for p = 1 and each m > 2 (without the restr ict ion m = k 2 + 1) for which K n < ~ ) / K * = r, as well. I t r emains an open quest ion whe the r the bound r is the best possible for p > 1 (a polynomial approx imat ion a lgor i thm in Step 1) and m > 2.

Finally, we observe tha t for examples f rom Tables 1 and 3 and p = 1 (an exact a lgor i thm in Step 1) we have 7r H(f) = zr n(g), for each funct ion f ~ F, where F is the family of all funct ions f rom [0,1] on [0,1]. H e n c e and f rom the T h e o r e m we conclude tha t a lgor i thm H ( g ) has the smallest worst-case pe r fo rmance rat io in the whole family of approx imat ion a lgor i thms { H ( f ) : f ~ F} for p = 1 and m = k 2 + 1, k = 1, 2, 3~ . . . .

Table 4

i j

1 2 3 . . . k 2 k 2 + 1 k 2 + 2

1 1 0 0 0 0 k ( 1 - g(c)) 2 0 1 - g(c) 0 0 0 0 3 0 0 1 - g(c) 0 0 0

".. i¢ 2 0 0 0 1 - g(c) 0 0 k 2 + 1 0 0 0 0 1 - g(c) 0

E. Nowicki / An approximation algorithm for the m-machine permutation flow shop scheduling problem 349

Acknowledgements

We wish to thank the referees for their helpful suggestions and comments regarding an earlier version of this paper.

References

[1] Daniels, R.L., "A multi-objective approach to resource allocation in single machine scheduling", European Journal of Operational Research 48 (1990) 226-241.

[2] Elmaghraby, S.E., Activity Networks, Wiley, New York, 1977. [3] Garey, M.R., and Johnson, D.S., Computers and Intractability. A guide to the Theory of NP-completeness, Freeman, San

Francisco, Ca, 1979. [4] Grabowski, J. and Janiak, A., "Job-shop scheduling with resource-time models of operations", European Journal of

Operational Research 28 (1986) 58-73. [5] Grabowski, J., Skubalska, E., and Smutnicki, C., "On flow shop scheduling with release and due dates to minimize maximum

lateness", Journal of the Operational Research Society 34 (1983) 615-620. [6] Graham, R.I., Lawler, E.L., Lenstra, J.K., and Rinnooy Kan, A.H.G., "Optimization and approximation in deterministic

sequencing and scheduling: A survey", Annals of Discrete Mathematics 5 (1979) 287-326. [7] Ishi, H., Martel, C,, Masuda, T., and Nishida, T., "A generalized uniform processor system", Operations Research 33 (1985)

346-362. [8] Johnson, S.M., "Optimal two- and three-stage production schedules with setup times included", Naval Research Logistics

Quarterly 1 (1954) 61-68. [9] Nawaz, M., Enscore, Jr., E.E., and Ham, I., "A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem",

OMEGA 11 (1983) 91-95. [10] Nowicki, E., and Smutnicki, C., "Worst-case analysis of an approximation algorithm for flow-shop scheduling", Operations

Research Letters 8 (1989) 171-177. [11] Nowicki, E., and Zdrzatka, S., "Two-machine flow shop scheduling problem with controllable job processing times", European

Journal of Operational Research 34 (1988) 208-220. [12] Nowicki, E., and Zdrza|ka, S., "A survey of results for sequencing problems with controllable processing times", Discrete

Applied Mathematics 26 (1990) 271-287. [13] Potts, C.N., "An adaptive branching rule for the permutation flow-shop problem", European Journal of Operational Research 5

(1980) 19-25. [14] Shmoys, D.B., Stein, C., and Wein, J., "Improved approximation algorithms for shop scheduling problems", in: Proceedings of

the 2nd ACM-SIAM Symposium on Discrete Algorithms, January, 1991. [15] Vickson, R.G., "Choosing the job sequence and processing times to minimize total processing plus flow cost on single

machine", Operations Research 28 (1980) 1155-1167. [16] Vickson, R.G., "Two single-machine sequencing problems involving controllable processing times", AIIE Transactions 12

(1980) 258-262. [17] Van Wassenhove, L.N., and Baker, K.R., "A bicriterion approach to time/cost tradeoffs in sequencing", European Journal of

Operational Research 11 (1982) 48-54.