Contj~ut. & Gjw Rrs., Vol. 2, pi. WXI. P~QUW Pm, 1975. Printed in Great Britain

A SEARCH ALGORITHM FOR THE GENERALIZED FLOWSHOP SCHEDULING PROBLEM

JATINDER N. D. GUPTA*

management luformation Systems Department, U.S. Postal Service. Washington. DC 20260, U.S.A.

Scope and purpose--Scheduling problems consist of processing a given number of jobs on a specified number of machines to optimize a known objective function. Though the terminology from manufacturing is used to describe a “job” and “machine.” the scheduling problems are quite general and equally apply to office Row of work, scheduling of computer operations, etc. The operation of multi-process computers. for example. can be modeled as a flowshop scheduling problem. In such a case, the processes will be viewed as ~aehi~es. and the computer programs will be considered as jobs.

Prior to the development of a scheduling technique, it is essential that an objective function be explicitly defined. When a manager faces a scheduling problem, he generally wishes to produce the maximum in the minimum possible time. This apparent objective may be misleading as there are interacting costs that should be considered. Therefore, a clear identi~cation of these costs, and the manner in which these costs are calculated becomes significant.

A mathematical formulation that reflects these interacting costs and possible structural characteristics of the problem can be of great help to the scheduling practitioner since this can lead to the development of suitable analysis and solution techniques.

This work is a pioneering effort in this field as it describes, for the first time, a composite cost measure for the scheduling problems and a possible technique for solving a specific flowshop scheduling problem.

Abstract-This paper considers the problem of scheduling a given number of jobs to be processed on a given number of machines in a flowshop. By a realistic analysis of the assumptions and current formulation of the problem, the generalized flowshop scheduling problem is defined and a search algorithm, based on the concepts of Lexicographic search, is proposed as a possible solution to the n-job, M-machine generalized Rowshop scheduling problem when the objective is to minimize total opportunity cost. The proposed a~orithm uniquely specifies a linear texicogmphic order relation and systematically generates schedules until the set of all feasible schedules is implicitly examined and the optimal schedule is identified. Computational results show that the proposed algorithm is comparatively more efficient than a complete enumeration procedure.

1. INTRODUCTION

Shops of multiple machines in which the work flow is unidirectional are called ffowshops. The unidirectional flow of work implies that the technological ordering of all jabs on all machines is identical and pre-determined. The general scheduling problem in a flowshop is one of finding the schedule (sequence) of a given number of jobs on a given number of machjnes to optimize a given measure of performance. This problem was first formulated by Johnson [9] as an n-job. 2-machine scheduling problem when the objective function is to minimize the total through put time (called the make-span) in which all jobs complete their processing on all machines. Subsequent developments in scheduling theory have been extensions of Johnson’s formulation, in that the number of machines is increased to the general case M(Z3). Recently, there has been considerable interest in finding suitable mathematical techniques to solve the flowshop scheduling problem and substantial progress has been made in the development of efficient algorithms for obtaining optimal or near optimal solution to the problem of scheduling n-job on M-machines in a flowshop to minimize make-span. The recent reviews by Day and Hottenstien~3], Elmaghraby[4], and Gupta[8] describe these developments. However, in spite of this progress in scheduling theory, there seems to be little practical application of the flowshop scheduling techniques. There seems to be a gap between the development of the theory and the practical situations where the problems are being faced. One factor contributing to this gap seems to be the measure of pe~ormance used in developing the flowshop scheduIing techniques. Whereas most research workers are concerned with developing efficient procedures to minimize

*Jatinder N. D. Gupta is currently General Manager, Statistical Systems Requirements Division, Management Information Systems Department, U.S. Postal Service, Washington, D.C. He holds a B.E. (Mech.) from University of Delhi, M. Tech (II? & OR) from Indian Institute of Technology, Kharagpur, India, and Ph.D. from Texas Tech University. Dr. Gupta has taught at Texas Tech, University of Ala~ma in Huntsville, Alabama A&M University, and Sangamon State University. He is the author of more than 3Oresearch papers and is the co-author of an introductory text: Operarions Research in Decision Making.

83

84 J. N. D. GUPTA

the total throughput-time (called the make-span), practicing scheduling experts debate the applicability of the make-span as the criterion of schedule selection[6, lo]. The real-world scheduling problems involve conflicting cost components and multiple measures of performance become important. The use of make-span as the representative of these cost components and multiple measures of performance raises several questions. Recent studies, performed to study the effect of schedule criterion on the selection of the schedule, indicate that make-span schedules may involve far larger opportunity losses than those selected according to, say, the penalty cost criterion [6,7, lo]. Thus, there is a reluctance to use make-span as the criterion of optimality or in other words, the extended version of Johnson’s problem finds little use in the study of general scheduling problems. In addition, the formulation of the flowshop scheduling problem as an extension of Johnson’s problem requires several restrictive assumptions like the simultaneous availability of jobs and machines, sequence independent setup times, etc. These restrictions, coupled with the use of the schedule selection criterion make the possibility of applying theoretical results to scheduling practice improbable (if not impossible).

This paper presents a realistic representation of the flowshop scheduling problem in its deterministic case where the above restrictions are relaxed and a generalized flowshop scheduling problem is described. The proposed formulation of the problem uses minimization of total opportunity cost as the measure for schedule selection. The proposed optimality criterion is general and is flexible enough to derive any other desired objective function[7]. When considered in the present proposed generalized formulation, the existing flowshop scheduling techniques cannot solve the problem. However, by suitable modifications, the lexicographic search technique [5] can solve the problem. These modifications are discussed and a search algorithm is described as a possible solution technique for the generalized flowshop scheduling problem.

2.PROBLEM DEFINITION AND FORMULATION

A typical definition of the generalized flowshop scheduling problem reads as follows: “Given n jobs to be processed on M machines in the same technological order, the process

time of job a, available at time A,, on machine m, available to take up jobs for processing at time B,, being t,,, (a = 1,2, . . . , n ; m = 1,2, . . . , M), and if job b precedes jobs a, the setup time on machine m for job a is shorn ; find the common order in which these n jobs should be processed on these M machines to minimize total opportunity cost.”

The mathematical formulation of the above defined problem requires an explicit determination of the total opportunity cost. The proposed formulation and the search algorithm make the following assumptions:

(a) Assumptions concerning jobs (1) A single job cannot be processed simultaneously by more than one machine. (2) The process time of each job is known and is deterministic, represented by

ti,(i=1,2 ,..., n; m=1,2 ,..., M). (3) Setup, transportation and material handling time of job i is known if job j precedes job i.

This is represented by sjim. (4) Jobs are processed as soon as possible. (5) Job i is available for processing at time Ai, i = 1,2,. . . , n. (6) All jobs follow the same operations sequence on all machines.

(b) Assumptions concerning machines (1) At most one job can be processed on a specific machine at a given time. (2) There is only one machine of each type in the shop. (3) Machine m is available to take up the job at time B,, m = 1,2,. . . , Al. (4) A machine is not kept idle if it can take up a job.

(c) Others (1) In-process inventory is allowed. (2) Same order of sequence is followed on all machines, i.e. no passing is allowed. (3) The opportunity cost components are linear functions of time. Since the flow of work in a flowshop is unidirectional and the numbering of machines is

A search algorithm for the generalized flowshop scheduling problem 85

arbitrary, the nomenclature of machines may be so adopted that jobs are processed on machine 1 first, machine 2 second, . . . , and machine M last. With such an indexing of machines and the above stated assumptions, mathematical formulation of the problem becomes relatively less complex. In order to formulate the flowshop scheduling problem mathematically, define a partial schedule as any subset of jobs whose order (sequence) of processing on machines is completely specified. Thus, if we are considering a 6-job flowshop problem and we specify that job 3 will be processed first, job 2 second, and job 5 third, then (3,2,5) will be called a partial schedule and will be represented by (T. Now, we can obtain another partial schedule by augmenting, say job 4 to partial schedule (3,2,5) to get (3,2,5,4). Mathematically, if we augment job 6 to partial schedule g, we will represent such augmentation as (~b. With this definition of partial schedule and the previously stated assumptions for the flowshop scheduling problem, the completion time of a partial schedule crab (formed by augmenting job a to the partial schedule uh), T(uba, m) may be computed from the following recursive relations:

where: T(aba, m) = max [T(&I, m - l), s40m + T(& m)] + tam (1)

T(uba, 0) = A,, T(@, m) = B,, for all II and m. (2)

Let:

X 0” :

yam :

Idle time at machine m before job a starts processing and machine m has finished processing previous jobs. Time for which job a waits before starting processing at machine m, after its completion on machine m - 1.

d, : Time by which job a is late. Urn: Unit cost of machine m.

warn : Unit waiting cost of job a at machine m. h,: Unit setup cost of job a at machine m. Da: Due date for job a.

Then, from the above definitions:

xom = max [ T(uba, m - 1) - T(ub, m), O] (3)

yam = max [T(ub, m) - T(ubu, m - I), 01 (4)

d. = max [ T(uba, M) - D,, 01. (5)

Assuming the opportunity cost to be linear functions of time, the total opportunity cost of schedule uba, g(ubu) is given by the expression:

g(flbu) = g(ub) + m$, (xlafn. urn + yam . warn + Sham . hm ) + pd. (6)

Then the general flowshop scheduling problem, described above, is one of minimizing g(ubu) where u ranges over all possible permutations of (n - 2) jobs not containing jobs a and b and jobs a and b range over all possible jobs, not equal to each other. This is a typical quantitative combinatorial problem and permits of a finite number of feasible solutions, in fact, the total number of feasible schedules is n ! Theoretically, therefore, it is possible to solve this problem by complete enumeration or integer programming techniques. However, computational require- ments of these procedures forbid the solution of even moderately small problems. Thus, suitable search techniques may provide a possible solution procedure for the problem.

3. LEXICOGRAPHIC SEARCH

The lexicographic search approach proposed here is an extension of an earlier work[5] and utilizes the concept of formation of words. The search for an optimal schedule can be considered analogous to the search of a desired word in a dictionary. However, because of the feasibility

86 J. N. D. GUPTA

restriction of the problem, the range of the desired words is restricted and the alphabets of the dictionary are thought to be numbers. For a given problem, the size of the alphabet equals the number of jobs, n. In order to describe the search procedure effectively, the general terminology of the dictionary is adopted. Thus, a job is referred to as a letter of the alphabet, a schedule is called a word and the number of jobs contained in a schedule (partial or complete) is termed the length of the word. In order to make this paper independently readable, the general concepts of lexicographic search are briefly discussed here.

Let W, and Wb be two vectors representing distinct and mutually exclusive subsets of n items, the concatenation of which results in a single permutation vector W = {W. : Wb}. The lengths of the vectors W, and W, are defined as the number of items contained in the subsets. If a and b represent the lengths of W, and W,, then, by definition, a t b = n. For example, if n = 6, and W, = [532] and W, = [ 1461, then Q = [532 1461. Considering the scheduling problem on hand, if n be the total number of jobs, W can represent a schedule if the numbers do not repeat. Further, W, corresponds to a partial schedule containing a jobs. For purposes of further analysis and in accordance with above discussion, W is called a word. A word of length i, will be represented as: W = (w,, w2,. . . , a, Xi+,, Xi+2,. . . ,x,) where o, ; (p = 1,2,. . . , i) represent the specified letters of the word and x, ; (q = i + 1, . . . , n) represent the unspecified letters. If i = n, the total number of items under consideration, all letters of the word W are specified. Such a word is called a complete word. If i = 0, all the letters of the word are unspecified and the word is called a null word. A word can represent a schedule (or a subset of all feasible schedules, if incomplete) only if its letters do not repeat and its length is greater than or equal to one. Such words will be termed as sensible words.

Each word has certain penalty associated with it which, in the case of general scheduling problem under consideration, is the total opportunity cost. The aim of the lexicographic search approach is to find a complete sensible word which has the least penalty (total opportunity cost). To achieve this end a lexicographic listing of words is formed which, if complete also reflects a hierarchy of the penalty associates with the words on the list. In order to reduce the examination of the complete list, the procedure used in the formation of the words is as follows:

(a) The unspecified letters of an incomplete word occur only at the right end. (b) A word is said to be a leader of another if the specified letters of the first are identical with

the letters in the corresponding positions in the latter, which has at least one more specified letter. Hence, the set of complete words denoted by a leader will include the sets denoted by its follower words as its subsets. Thus, for example, (4,5, 1, x4, . . . , x,) is a leader of (4,5,1,2,3,. . . , x,) but (4,&l, 3, xs, . . . ,x,) is not.

(c) Let two words W = (a,. . . , uk,xk+l,. . . ,x,) and W’ = (o:, w;, . . . , o;, XJ+I, . . . ,x:)

have the longest common leader of length i. If i C min (k, j) then W precedes W’ if OJ~+~ < w’,+,. In short, in lexicographic search, if two words of length greater than i have a longest common leader of length i, ordering of the (i t 1)-th letters in the word determines the precedence relation between W and W’.

If f(W) represents the total opportunity cost of a complete word W, then a decrease in this cost (penalty) can be obtained only by interchanging the sequence position of the letters of the word to give a new word W’. Then this interchange is acceptable only if f( W’) <f(W). Also, this interchange (or exchange) must satisfy other constraints, i.e. the word formed by the interchange must be sensible.

4.THE SEARCH ALGORITHM

For the general scheduling problem, as stated earlier, the letters correspond to jobs and f(W) is the total opportunity cost of the word (schedule) W. If W’ is a word of length k(k < n), then the opportunity cost of word w’ may be used to reject a subset of words which are necessarily worse than a known complete word of value 8. Thus, if f( W’) < 8, then, it is possible that some complete word can be formed with W’ as its leader which has the total opportunity cost less than 6. To achieve this formulation, a lexicographic search procedure is used which will examine all the incomplete words such that f( W’) < S.

4.1 Algorithmic logic Consider two words W = (aI, 02,. . . , ok, Xk+l, . . . , h) and W’ = (@l,a2~ * . * 3 Ok, Ok++13

A search algorithm for the generalized flowshop scheduling problem 81

.x$2,. . . , xi,). Word W’ differs from W by the (k + l)-th letter. If the opportunity cost of the incomplete word W be g( W, k), then from the relations (1) through (6), the recursive relation to tind g( W’, k + 1) is:

k = 0, 1,2,. . . , n - 1 (7)

where

. I

l = Wk+tr j=ok,k=1,2 ,..., n-l, g(WO)=O, j =O, ifk=O

and all other are termed as defined before. The search is started with a null word W and a sufficiently large value 6 * f( W’) for all W’.

Starting with a word W, larger and larger words are formed in a lexicographic order (i.e. the lowest numbered letters, if available, are specified first) till it is confirmed that the word on hand is better than the incomplete word W’. Thus, if W is a null word, the corresponding W = (1, x;, . . . ( xk). The opportunity cost of W’, g( W’, l), can be determined by using equation (7). If g( W’, 1) < 6, the word W is lengthened to w’, i.e. W = (1. x2,. . . ,x,). The next W’ = (1,2, x;, . . . , xi,). The procedure of calculating g( W’, k -t 1) and checking is continued until a complete word W =(1,2,3,. . . , n) is obtained. Then 6 is replaced by g( W, n).

Forfurthersearch, W=(1,2 ,..., n-2,x,-,,x,)and W’=(l,2 ,..., n-2,wL_,,x,)where wl-, = n. The procedure described in the preceding paragraph is followed and if at stage k, g( w’ , k + 1) > 8, then all unspecified letters greater than w :+1 are tried at sequence position (k i- 1)

in a lexicographic order. However, if none of the unspecified letters of W satisfy the relation g( w’, k + 1) < 8, then the kth letter of W is unspecified and the search continued as described above until the last letter (i.e., n) requires unspecif~ng at sequence position 1.

The procedure outlined above is curtailed enumeration. For the flowshop scheduli~ problem with no passing allowed, there are n ! possible schedules. Since the search procedure described here closely resembles enumeration, the maximum number of words tried can be n ! However, because of the algorithmic procedure and the dominance check, many of these words will be of length less than n. In fact, the author’s expereince on a large number of random problems indicates that many words are of length one.

The following step by step procedure will generate the optimal schedule: Step 1. Initialize: Set k = 0, w: = 1, g( W, 0) = 0 and 8 = 03. Enter step 2. Step 2. Compute g( W’, k + 1) by using equation (7) where W = (w,, w2,. . . , wk, xki,, . , . , x,)

is the word formed at stage k and w’ = (o,, w2,. . . , ok, to;+,, xL+~, . . . , XL) is the next possible word under consideration.

Is g(W’,k+l)<S? (a) Yes. Enter step 3. (b) No. Enter step 4. Step 3. Augment wir+, at the (k + 1)th sequence position in W, i.e. wk+I = o;+,. Set k = k t 1. Is k=n? (a) Yes. Enter step 6. (b) No. Let oh+, be the lowest unspecified letter in W. Return to step 2. Step 4. Is wh+, = n? (a) Yes. Enter step 7. (b) No. Set w:,, as the lowest unspecified letter in W, greater than w:&,. lf no such letter is

available, go to step 5, otherwise return to step 2. Step 5. Unspecify ukt the kth letter of W. Let k = k - 1. Return to step 4. Step 6. Set S = g( W, k). Store S and W. Let k = n -2, o ;+, = w:+,, unspecify &+?. Return to

step 4. Step 7. Is k 12 O? (a) No. Return to step 5. (b) Yes. Stop the search. The last complete word is the optimal solution.

88 J. N. D. GKJPTA

5. A NUMERICAL ILLUSTRATION

The working of the above algorithm is explained by solving a 3 job, 3 machine flowshop scheduling problem. For all a and m, the unit setup cost, h,, = 1, and expected return on capital r = 2. Tables l-4 give the problem data. (Note that C, is the raw material cost for job a.)

Table 1. P-time (t,,,,), A,, Di, pi, and C, for the problem

r., at m=

a 1 2 3 A, D, p. C.

I 13 7 II 0 41 I I 2 4 6 12 I 26 3 2 3 12 15 9 3 58 2 3

Table 2. Machine availabil- ity (B,,) and unit idle-costs

(UGl)

1 2 3 B” U:

0 IO 12 0 3 6

Table 3. Setup times (+,,,) for the example problem Table 4. Value added by machines (uim)

j Machine 1 Machine 2 Machine 3 i 012301230123 m

i I 2 3 11~120~311~21 2 2 1 22 3 2 3 0332x2 1 0 2 1 3322zo11=544p 2 I 0 2

3 2 I 1

the unit waiting cost of job i and machine m, w,,,, is given by[6].

m-l

wj, = r Ci + C vi,,, , m = 1,2,. . . , M In’=, >

Substituting the values of r, Cl, and Vi, into equation (8), wi, can be derived. Table 5 gives the values of wi, thus calculated.

The steps of the algorithm may be performed in a tabular form as shown in Table 6. As an illustration, consider row 4 of the table: W = (1, x2, x,). IV’ = (1,3, x,). k = 1, g( W, k) = 64 and step 2 is to be executed. In step 2, g( w’, k + 1) is to be calculated.

From equation (l), it is observed that T(k + 1,l) = 28, T(k + 1,2) = 43, T(k + 1,3) = 52. Using equations (3) and (4) x3] = 0, x32 = 5, x33 = 7, d, = 0. y31 = y32 = ys3 = 0.

Substituting the above values into equation (7), g( W’, 2) = 64 + 85 + 67 t 2 t 1 t 2 = 153 since g( W’, 2) > S (= 124), and w: = 3, step 7 is executed through which step 5 is executed as k = k - 1 = 0. W = @. Row 6 shows the further search.

The optimal schedule thus obtained is (2,3,1) with a cost of 68. Such a schedule is also feasible, since all permutations of II jobs are feasible and jobs can be processed in any of these n ! permutations.

Table 5. Unit waiting cost matrix (w,,)

m i 1 2 3

I 0 2 1 2 1 0 2 3 2 1 I

6. COMPUTATIONAL RESULTS

Apart from numerical comparison, there seems to be no other method to test the efficiency of any combinatorial algorithm. The algorithm was programmed on an IBM 360150 computer to solve 180 problems constructed with random times. The number of machines varied from four to six and jobs from four to six. The results of this study were compared with those obtained by

A search ~gorithm for the generalized fiowshop scheduljng problem

Table 6. The search table

Tlk+I,m) Row I( W G&,l~ 1 2 3 gfW’,k+il 5% Decision

IO ip 1 14 21 32 64 < k=k+l=l 2 1 (1,x,,.%) 2 19 30 46 114 < k=ktl=2 3 2 (1,2,x,) 3 33 48 59 124 < &=124,k=3,W=(l.2,3) 4 I (l,x,,x3) 3 28 43 153 > k=k-l=O SO@ 26 18

:; 19 < k=k+l=l

6 1 (2,xz,x,f 1 20 28 43 39 < k=k+l=2 7 2 (Zl,X,e,) 3 34 49 58 98 < &=98,k=3,W=(Z,i,3) 8 I (23x2, x,) 3 20 36 45 40 cl k=ktl=2 9 2 (2.3,~~) I 35 40 57 68 < 6 = 68, k = 3, w = (2.3.1)

100 sp 3 15 30 39 108 > Stop the search

complete enumeration. Table 7 shows the comparison of time required for computation and the average number of sequences generated by the proposed search algorithm and complete enumeration.

The results of the Table 7 indicate that the computation time of the proposed search ~gorithm is less than that of complete enume~tion. In addition, the percentage savings in combustion time increases with the increase in the size of the problem. This empirical verification thus shows that the proposed search algorithm is compa~tively more efficient than complete enumeration.

Table 7. Comp~son of search algori~m and enumemtion

Lexi-search Enumeration Computation Average Computation

time no. of time No. of n m <sed sequences (Set) sequences

4 : o-117 4.3 0.135 24 0,146 4.9 0.174 24

6 0.172 4.15 0.187 24

s” 0.527 8-95 0.681 120

5 0.619 I.4 0.780 120 6 0.724 7.2 0.891 120

“; 2.629 3585 3.300 720

6 3.168 11.45 4.644 720 6 3.768 12+25 HI64 720

7. DISCUSSION AND CONCLUSIONS

The generalized flowshop scheduling problem presented above encompasses several realistic situations and provides flexibility in the use of the optimality criterion. All existing formulations of the flowshop scheduling problem can be obtained from the proposed formulation of the problem. Thus, for example, if Ai = 3, =Oforall~and~,~~=Oforl~~~~~-1):and f&m = pj = him = 0 fo r a 11 i and m, then the proposed total o~po~unity cost criterion is equivalent to minimizing the span. Further, because him = 0, the problem takes the form of a static make-span ffowshop scheduling problem with sequence independent times as considered by Dudek and Teuton[3] and subsequently solved by Brown and Lomnickifl], Smith and Dudek [I I], and Cupta IS].

The flexibility of the problem formulation and the general nature of the proposed search algorithm make the proposed procedure applicable to solve the generalized flowshop scheduling problem for any objective function. Further, in addition to generating optimal schedules, the proposed search algorithm yields additional info~ation on partial schedules as search progresses. The near optimal schedules can be recognized at an early stage of search and the computation may be curtailed if only near optimal solutions, which are good enough in many practical situations, are desired. Thus, ifit is decided that it is not worthwhile going in for a schedule better than the current one unless the former has a total oppo~nnity cost less than the latter, say by a units, then at step 6 of the algorithm, the trial value S can be replaced by (8 - a) and the search continued as usual. This reduces the computational requirements considerably. If the total op~~~ity cost of the optimal schedule is between S and CS - a), the alg~~~rn will not recognize

90 J. N. D. GUPTA

the optimal solution and hence will yield a sub-optimal solution, the risk which was agreed upon already. However, if the total opportunity cost of the optimal schedule is less than (6 - a), the algorithm will establish this fact and iterate accordingly.

Finally, the efficiency of the proposed algorithm greatly depends on the convergence capability of the generated schedules to the optimal schedule. For the make-span criterion, it was possible to use certain bounding procedures [5] and obtain the initial trial solution by a heuristic rule. Developments in (1) heuristic rules to give quick and near optimal schedules for the generalized flowshop scheduling problem and (2) computationally feasible lower and upper bounds for the total opportunity cost seem to provide interesting research projects and may help find a practical solution to the problem.

REFERENCES

1. A. P. G. Brown and Z. A. Lomnicki, Some applications of the branch and bound algorithm to machine scheduling problems, Ops Res. Q. 17, 173-186 (1966).

2. J. E. Day and M. P. Hottenstein, Review of sequencing research, Nao. Res. Logist. Q. 17, II-39 (1970). 3. R. A. Dudek and 0. F. Teuton, Jr., Development of M-stage decision rule for scheduling n jobs through M-machines,

Ops Res. 12, 471-497 (1964). 4. S. E. Elmaghraby, The machine sequencing problem: review and extensions, Nav. Res. Logist Q. 15,205-232 (1968). 5. J. N. D. Gupta, A general algorithm for the n X M flowshop scheduling problem, Int. J. Prodn Res. 7.241-247 (1969). 6. J. N. D. Gupta, Economic Aspects of Scheduling Theory, Ph.D. Dissertation, Texas Tech University, Lubbock, Texas

(1%9). 7. J. N. D. Gupta, Economic aspects of production scheduling systems, J. Ops Res. Sot. Japan 13, 167-193 (1971). 8. J. N. D. Gupta, M-stage scheduling problem-a critical appraisal, Int. J. Prodn Res. 13, 267-281 (1971). 9. S. M. Johnson, Optimal two and three stage production schedules with set-up times included, Nav. Res. Logist. Q. I,

61-68. 10. M. L. Smith, A Critical Analysis of Flowshop Sequencing, Ph.D. Dissertation, Texas Technological College, Lubbock,

Texas (1968). 11. R. D. Smith and R. A. Dudek, A general algorithm for solution of the n-job, M-machine sequencing problem of the

flowshop, Ops Res. 15, 71-82 (1967) Errata, Ops Res. 17, 756.