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This article was downloaded by: [University of Waterloo]On: 10 October 2014, At: 16:59Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
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A scheduling problem with machine flexibilityALAN P. MUHLEMANN a & A. GEOFFREY LOCKETT ba Department of Management Sciences , U.M.I.S.T. , Manchester, Englandb Manchester Business School , Manchester, EnglandPublished online: 24 Oct 2007.
To cite this article: ALAN P. MUHLEMANN & A. GEOFFREY LOCKETT (1975) A scheduling problem with machine flexibility,International Journal of Production Research, 13:1, 57-73, DOI: 10.1080/00207547508942974
To link to this article: http://dx.doi.org/10.1080/00207547508942974
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INT. J. PROD. RES., 1975, VOL. 13, NO. I, 57-73
A scheduling problem with machine flexibility
ALAN P. MUHLEMANNt and A. GEOFFREY LOCKETTt
A problem of scheduling the production of pharmaceuticals in a limited number ofvessels is presented. It has the characteristics of sequence dependent changeovertimes. multi-stage production, parallel usage of vessels and flexibility in the choiceof vessels. An optimizing procedure is discussed, but dismissed as computationallyinfeasible. Consequentlyt a heuristic is developed, and its performance evaluated.TypicaJ results are also reported.
IntroductionThis problem concerns the manufacture of a variety of pharmaceutical
products in a limited number of vessels. Quantities of these products aremanufactured for stock every three months, and a schedule is drawn up ofthe number of batches to be produced for each different product. The problemis then to allocate products to vessels in order to produce the requirements inthe shortest ·time. The objective 'minimize makespan ' is used becausevessel utilization is at a premium (there being another use to which the vesselsare put, on completion of the schedule), and stockholding costs are insignificant by comparison. Therefore, the problem is one of scheduling, the stockcontrol aspects being ignored in this paper.
Some of the products require one stage of manufacture, others require two.The first stage simply involves parallel usage of one or more vessels for apredetermined length of time. Those products having only one stage arefinished after this. For those with a second stage, there then follows a periodof parallel usage of another set of vessels. However, in this case, if one ormore of the vessels required in stage two is not available, the product mustwait in the first set. The vessels require cleaning out between differentproducts and some products have up to four alternative methods of production.There are seventeen different types of product and seven vessels, full detailsbeing given in Table I.
The structure of the problem is best illustrated with an example. Supposethe following are to be produced: one batch of product 5, one of product 15,two 'of product 10,two of product I and one of product 17. It can be seenfrom Table I, that for products 5 and 10 alternative methods of manufactureare available. Suppose that version (i) is selected for product 5; and forthe first batch of product 10, version (i) is selected, and for the second batch,version (ii) is selected. If the products are manufactured in the sequencegiven above, then the result can be represented diagrammatically in the formof a Gantt chart showing the state of each vessel at any point in time. Thisis given below in Fig. I, the overall time taken being 105 hours.
Received 21 June 1973.t Department of Management Sciences, U.M.I.S.T., Manchester, England.t Manchester Business School, Manchester, England.Published by Taylor & Francis Ltd., 10-14 Macklin Street, London WC2B 5NF.
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58 A. P. Muhlemann and A. G. Lockett
VesselsNo. Produot
version 1 2 3 4 5 6 7
1 1 372 2 36 363 3 544 4 (i) 112--------+ 125 (ii) 12..---1126 5 (i) 24 247 (ii) 24 248 6 (i) 64 649 (ii ) 64 64
10 7 (i) 54 5411 (ii) 54 5412 (iii) 54 5413 (iv) 54 5414 8 (i) 48 4815 (ii) 48 4816 (iii) 48 4817 (iv) 48 48
~
18 9 60 , 60 60r--"-----.
19 10 (i) 20 , 24 24r--"-----.
20 (ii) 20 , 24 2421 11 (i) 4022 (ii) 4023 12 60 6024 13 (i) 58 5825 (ii) 58 5826 (iii) 58 5827 14 36 3628 15 13+--2929 16 34..---130 ,530 17 24, i
Table 1. Time required (hours) in the vessels to produoe one batch of each produotfor each possible alternative method of produotion (version). Arrows indicatepreoedenoe relations, brackets parallel usage. Brackets are omitted whenthere is no ambiguity (i.e. only one stage). The olean out time is four hours.
A formulationManne (1960) shows how mathematical programming can be used to solve
a scheduling problem. Conway, Maxwell and Miller (1967) develop thisfurther and show how the simple job-shop scheduling problem can be formulated as a mixed integer programming problem. This formulation can beextended and modified to cater for the alternative methods of production,sequence dependent changeovers, and parallel usage of vessels, which ocourfor the problem desoribed in the previous section. Complete details of suohan extension can be found in MuWemann (1972), where the exact form of thevariables and constraints, together with an example of the formulation whenapplied to a small example are given.
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A schedUling problem with machine flexibility 59
vessel
1
3
5
6
7
Key:
5i I
1 I 1 ~ 17 I
I 10i I I!Jl 10ii I
5i I~
15 !;;I 10i !al71W&'/A
I 10i ~ 10ii I
I lOii I'
~ Oeaning
~ Waiting
o Processing
50 100time (hours)
Figure 1. Illustratory Gantt chart.
Although it is possible to formulate the problem to use mathematicalprogramming as a solution procedure, this formulation necessitates the introduction of not only integer variables but also quadratic eonstraints. Thisrules out the use of this approach for solving problems of a realistic size.Twelve jobs generate two hundred variables and even more constraints.Von Lanzcnauer (1970) confirms that, with the presently available integerprogramming codes, it is impossible to solve problems of a fairly realisticsize. With this in mind a heuristic approach to the problem has been taken.
A heuristic approachSchwartz (1964) describes an application of heuristics to a simplification
of the problem described here, and Lockett and Muhlemann (1972) report adegree of success in applying heuristics to a one machine problem with sequencedependent changeover times. The problem under discussion is far more complex and necessitated the development of a new heuristic which is describedbelow.
The delay matrix
The heuristic developed here is based on the concept of a delay matrix.Suppose the set of all products (including alternatives of the same product)are numbered 1 to p. (So using the data in Table 1, 1 refers to product 1, 2refers to product 2, 3 refers to product 3, 4 refers to product 4 version (i),5 refers to product 4 version (ii), and so on; with p equal to 30.) It is thenpossible to define a p x p matrix A where Ail is the delay caused to the product
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60 A. P. Muhlemann and A. G. Lockett
Second (i) 1 2 3 4 5 6 7 8 9 10--- 1- 2 3 4 5 6 7First (i) (i) (ii) (i) (ii) (i) (ii) (i)
1 1 37 0 0 0 0 0 41 0 0 412 2 0 36 0 40 0 0 0 40 0 03 3 0 0 54 0 0 0 0 58 58 04 4 (i) 0 116 128 112 16 0 0 128 128 05 (ii) 0 0 128 16 112 0 0 128 128 1166 5 (i) 0 0 0 0 0 24 28 0 0 07 (ii) 28 0 0 0 0 28 24 0 0 288 6 (i) 0 68 68 68 0 0 0 64 68 .09 (ii) 0 0 68 0 68 0 0 68 64 68
10 7 (i) 58 0 0 0 58 0 58 0 58 5411 (ii) 0 0 58 0 0 58 0 58 58 012 (iii) 0 0 0 0 58 58 0 0 58 5813 (iv) 58 0 58 0 0 0 58 - 58 58 5814 8 (i) 0 52 0 52 0 52 0 52 0 015 (ii) 0 0 0 0 52 52 52 0 52 52HI (iii) 0 52 0 52 0 52 52 52 0 017 (iv) 0 0 0 0 52 52 0 0 52 5218 9 124 64 64 64 0 0 124 64 64 12419 10 (i) 0 48 28 28 0 0 0 28 28 020 (ii) 0 48 28 0 28 0 0 28 28 2821 11 (i) 44 0 0 0 0 0 44 0 0 4422 (ii) 0 0 0 0 0 44 44 0 0 023 12 64 0 0 0 0 64 64 0 0 6424 13 (il 0 62 0 62 0 0 0 62 0 025 (ii) 0 62 0 0 62 0 0 0 62 6226 (iii) 62 62 0 0 0 0 62 0 0 6227 14 0 40 0 0 40 0 0 0 40 4028 15 0 33 0 33 0 46 46 33 0 029 16 134 139 0 139 0 168 134 139 0 13430 17 32 8 0 8 0 0 32 8 0 32
Table 2. The basic delay matrix for the problem described by Table 1.
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A 8cheduling problem with machine flexibility 61
Second (j) 11 12 13 14 15 16 17 18 19 20---- 7 8 9 10First (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) (i) (ii)
1 1 0 0 41 0 0 0 0 0 0 02 2 0 0 0 40 0 40 0 40 40 163 3 58 0 58 0 0 0 0 58 58 584 4 (i) 128 0 128 .116 0 116 0 128 128 1285 (ii) 128 116 128 0 116 0 1.16 128 128 1286 5 (i) 28 28 0 . 28 28 28 28 0 0 07 (ii) 0 0 28 0 28 28 0 0 0 08 6 (i) 68 0 68 68 0 68 0 68 68 689 (ii) 68 68 68 0 68 0 68 68 68 68
10 7 (i) 0 58 58 0 58 0 58 0 0 011 (ii) 54 58 58 58 0 0 58 58 58 5812 (iii) 58 54 0 58 58 0 58 0 0 5813 (iv) 58 0 54 0 0 0 0 58 58 5814 8 (i) 52 52 0 48 0 52 52 52 52 015 (ii) 0 52 0 0 48 52 52 0 0 5216 (iii) 0 0 0 52 52 48 0 52 52 017 (iv) 52 52 0 52 52 0 48 0 0 5218 9 64 0 124 64 0 64 0 60 64 6419 10 (i) 28 0 28 28 0 28 0 28 24 2820 (ii) 28 28 28 0 28 0 28 28 28· 2421 11 (i) 0 0 44 0 0 0 0 0 0 022 (ii) 0 0 0 0 44 44 0 0 0 023 12 0 0 64 0 64 64 0 4 0 O'24 13 (i) 0 0 0 62 0 62 0 62 62 3825 (ii) 0 62 0 0 62 0 62 0 38 6226 (iii) 0 0 62 0 0 0 0 2 38 3827 14 0 40 0 0 40 0 40 0 16 4028 15 0 0 0 33 46 46 0 33 33 029 16 168 168 134 168 0 139 168 139 139 030 17 0 0 32 8 0 8 0 8 8 0
Table 2 (Oontinued).
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62 A. P. Muhlemann and A. G. Lockett
Second(j) 21 22 23 24 25 26 27 28 29 30---- 11 12 13 14 15 16 17First (i) (i) (ii) (i) (ii) (iii)
1 1 41 0 41 0 0 41 0 0 41 372 2 0 0 0 40 40 40 40 40 0 403 3 0 0 0 0 0 0 0 0 0 04 4 (i) 0 0 0 116 0 ·0 0 116 116 1165 (ii) 0 0 0 0 116 0 116 0 0 06 5 (i) 0 28 28 0 0 0 0 0 0 07 (ii) 28 28 28 0 0 28 . 0 0 28 248 6 (i) 0 0 0 68 0 0 0 68 0 689 (ii) 0 0 0 0 68 0 68 0 0 0
10 7 (i) 58 0 58 0 58 58 58 0 58 5411 (ii) 0 0 0 0 0 0 0 0 58 012 (iii) 0 0 0 0 58 0 58 0 0 013 (iv) 58 0 58 0 0 58 0 0 58 5414 8 (i) 0 0 0 52 0 0 0 52 0 5215 (ii) 0 52 52 0 52 0 52 23 0 016 (iii) 0 52 52 52 0 0 0 52 0 5217 (iv) 0 0 0 0 52 0 52 0 0 018 9 124 0 124 64 0 124 0 64 124 12419 10 (i) 0 0 0 48 48 48 48 23 0 2820 (ii) 0 0 0 48 48 48 48 0 0 021 11 (i) 40 0 44 0 0 44 0 0 44 4022 (ii) 0 40 44 0 0 0 0 15 0 023 12 64 64 60 0 0 64 0 35 64 6024 13 (i) 0 0 0 58 62 62 62 62 0 6225 (ii) 0 0 0 62 58 62 62 0 0 026 (iii) 62 0 62 62 62 58 62 0 62 5827 14 0 0 0 40 40 40 36 0 0 028 15 0 46 46 33 0 0 0 29 0 3329 16 134 0 134 139 0 134 0 139 130 13930 17 32 0 32 8 0 32 0 8 . 32 24
Table 2 (Continued).
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A scheduling problem with machine flexibility 63
version corresponding to j if it is processed after the product version corresponding to i, in an empty system, It is the time lag between the commencement of processing product version corresponding to i and the earliest startof processing version corresponding to j. Three examples of the calculationof elements in the basic delay matrix are given below in Fig. 2. The entirebasic delay matrix for the problem described in Table 1 is given in Table 2.
So given a set of quarterly requirements, it is possible to generate thedelay matrix for this set by picking the appropriate elements from the basic
Figure 2. Three examples of the calculation of elements in the basic delay matrix !:J..(Note!:J. j i is the delay caused to the product version corresponding to j if itis processed immediately after the product version corresponding to i'in acompletely empty system. i and j are the numbers in the first column ofTable I, hence product 1 is numbered 1, product 10 version (i) is 19 andproduct 14 is 27). Case (a) product 1 before product 1 (delay !:J.1 1 =37 hours).Case (b) product 10 version (i) before product 14 (delay !:J.19 27=48 hours).Case (e) product 14 before product 10 version (i) (delay !:J.<n 10= 16 hours).
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64 A. P. Muhlemann and A. G. Lockett
delay matrix 6.. Suppose there are m batches of a product in this set, ingenerating the delay matrix for the set this is considered as m single batches,and so if the product has n alternative methods of manufacture, this resultsin mn rows and columns in the matrix, corresponding to this product. Thisis illustrated in the numerical example at the end of this section.
The heuri8tic HAt first sight a possible heuristic could be based on the solution to the
travelling salesman problem (this is discussed in detail in Little et al. 1963),defined by the delay matrix for a particular set of jobs (a quarter's requirements). This is complicated slightly by the presence of alternative methodsof manufacture which means that we are in fact dealing with a ' generalized'travelling salesman problem (see Srivastava et al. 1969). However, it ispossible to base a heuristic on this (full details are given in Muhlemann 1972),but this suffers from the following handicap :
Suppose the product versions corresponding to i, i' and i" are scheduledin the order i, i', in. If i does not delay i' and i' does not delay in, then thesetimes in the delay matrix are zero. However, it is quite possible that i delaysin, but simply solving the travelling salesman problem does not take thisinto account.
In order to overcome this, a heuristic has been developed. In generalterms, we build up the sequence by processing next the product versionwhich incurs the least total delay when considered with all the productversions already in the sequence (rather than simply the product versionimmediately before it in the sequence). It is described in detail below,followed by a summary of the steps for a numerical example.
Steps in the calculation
Let I = {it> ... , i,} be the set of quarterly requirements to be scheduled,and suppose D is the corresponding r x r delay matrix for the products including alternatives. So suppose we have 3 batches of product 1, 4 of product2, 3 of product 3 and' so on, 1= p, 1, 1, 2, 2, 2, 2, 3, 3, 3, ... }. It is bothnecessary to build up a sequence and select alternatives where necessary.
Let the sequence generated be labelled j 1 .•. , i. let S be the set of indicesof the matrix still to be considered. The procedure is as follows:
Set S={l, 2, ... , r}
and using matrix notation, DL,L, is the element in row L 1 and column L 2 ofthe matrix D.
Step 1 Let L 1 be the index of the row with the smallest element (if thereare ties pick that row with the least number of smallest elements).L1corresponds to a version of a product in I, label this jl. RemoveL 1 from S, along with any versions associated with that product.
Siep 2 Let L 2 be the element in S such that the delay caused to £2 by £1is minimized over all elements remaining in S. Mathematically L2is such that:
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A scheduling problem with machine flexibility 65
The product and version corresponding to L 2 is labelled i«: L 2and its associated versions are removed from S.
Step 3 Let La be the element in S such that the sum of the delay caused1;0 L 2 by Lv caused to La by L 1 and caused to La by L 2 is minimizedover all elements in S. Mathematically, La is such that:
DL1L, + DL,L, + DL1L, = min {DL1L, + DL'L + DL1L : LES},
f2(L) == DL1L, + DL,L + DL1L
and this is equivalent to
min {fl(L)+DL,L: LES}
The product and version corresponding to La is labelled ja; Laand its associated versions are removed from S.
General step
Lp is such that
Pi:.1 f DL<lLR=min{Pf
Q=1 R=Q+l Q=1
which is equivalent tomin {fp-2(L) + D1_
p_ 1L: LES}
The product and version corresponding to Lp is labelled jp; Lp and itsassociated versions are removed from S.
Hence the sequence j may be built up. Although at first sight the calculations appear rather laborious, use of the recurrence relation for fp(L)significantly reduces the effort involved.
An illustrative example
An example of the application of the heuristic to a small example ispresented below. The set of requirements is given in Table 3.
Number ofType batches Job No.
I 1 12 I 23 I 34 2 4,55 2 6,76 2 8,97 I 108 I II9 2 12, 13
12 1 1414 I 1515 1 16
Table 3. Example set of data.
1.l.P.R. E
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Job No.1 2 3 4 5 6 7 8 9
Job Job ~ ,--.-A-----.. ~ .--'"-0 ~ .-"----.No. type Job type
1 2 3 4 4 5 5 6 6(i) (ii) (i) (ii) (i) (ii) . (i) (ii) (i) (ii) (i) (ii)
1 1 - 0 0 0 0 0 0 0 41 0 41 0 0 0 02 2 0 - 0 40 0 40 0 0 0 0 0 40 0 40 03 3 0 0 - 0 0 0 0 0 0 0 0 58 58 58 584 4 (i) 0 116 128 - - 112 16 0 0 0 0 128 128 128 128
(ii) 0 0 128 - - 16 112 0 0 0 0 128 128 128 1285 4 (i) 0 116 128 112 16 - - 0 0 0 0 128 128 128 128
(ii) 0 0 128 16 112 - - 0 0 0 0 128 128 128 1286 5 (i) 0 0 0 0 0 0 0 - - 24 28 0 0 0 0
(ii) 28 0 0 0 0 0 0 - - 28 24 0 0 0 07 5 (i) 0 0 0 0 0 0 0 24 28 - - o· 0 0 0
(ii) 28 0 0 0 0 0 0 28 24 - - 0 0 0 08 6 (i) 0 68 68 68 0 68 0 0 0 0 0 - - 64 68
(ii) 0 0 68 0 68 0 68 0 0 0 0 - - 68 649 6 (i) 0 68 68 68 0 68 0 0 0 0 0 64 68 - -.
(ii) 0 0 68 0 68 0 68 0 0 0 0 68 64 - -10 7 (i) 58 0 0 0 58 0 58 0 58 0 58 0 58 0 58
(ii) 0 0 58 0 0 0 0 58 0 58 0 58 58 58 58(iii) 0 0 0 0 58 0 58 58 0 58 0 0 58 0 58(iv) 58 0 58 0 0 0 0 0 58 0 58 58 58 58 58
11 8 (i) 0 52 0 52 0 52 0 52 0 52 0 52 0 52 0(ii) 0 0 0 0 52 0 52 52 52 52 52 0 52 0 52(iii) 0 52 0 52 0 52 0 52 52 52 52 52 0 52 0(iv) 0 0 0 0 52 0 52 52 0 52 0 0 52 0 52
12 9 124 64 64 64 0 64 0 0 124 0 124 64 64 64 6413 9 124 64 64 64 0 64 0 0 124 0 124 64 64 64 6414 12 64 0 0 0 0 0 0 64 64 64 64 0 0 0 015 14 0 40 0 0 40 0 40 0 0 0 0 0 40 0 4016 15 0 33 0 33 0 33 0 46 46 46 46 33 0 33 0
Table 4. Delay matrix for the set of jobs given in Ta.ble 3.
0>0>
~
~
f~~
~
i
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......
Job No.10 11 12 13 14 15 16
Job Job ,No. type Job type
7 8 9 9 12 14 15(i) (ii ) (iii) (iv) (i) (ii ) (iii) (iv)
--I 1 41 0 0 41 0 0 0 0 0 0 41 0 02 2 0 0 0 0 40 0 40 0 40 40 0 40 . 403 3 0 58 0 58 0 0 0 0 58 58 0 0 04 4 (i) 0 128 0 128 116 0 116 0 128 128 0 0 116
(ii) 116 128 116 128 0 116 0 116 128 128 0 116 05 4 (i) 0 128 0 128 116 0 116 0 128 128 0 0 116
(ii) 116 128 116 128 0 116 0 116 128 128 0 116 06 5 (i) 0 28 28 0 28 28 28 28 0 0 28 0 0
(ii) 28 0 0 28 0 28 28 0 0 0 28 0 07 5 (i) 0 28 28 0 28 28 28 28 0 0 28 0 0
(ii) 28 0 0 28 0 28 28 0 0 0 28 0 08 6 (i) 0 68 0 68 68 0 68 0 68 68 0 0 68
(ii) 68 68 68 68 0 68 0 68 68 68 0 68 09 6 (i) 0 68 0 68 68 0 68 0 68 68 0 0 68
(ii) 68 . 68 68 68 0 68 0 68 68 68 0 68 010 7 (i) - - - - 0 58 0 58 0 0 58 58 0
(ii) - - - - 58 0 0 58 58 58 0 0 0(iii ) - - - - 58 58 0 58 0 0 0 58 0(iv) - - - - 0 0 0 0 58 58 58 0 0
11 8 (i) 0 52 52 0 - - - - 52 52 0 0 52(ii) 52 0 52 0 - - - - 0 0 52 52 23(iii) 0 0 0 0 - - - - 52 52 52 0 52(iv) 52 52 52 0 - - - - 0 0 0 52 0
12 9 124 64 0 124 64 0 64 0 - 60 124 0 6413 9 124 64 0 124 64 0 64 0 60 - 124 0 6414 12 64 0 0 64 0 64 64 0 4 4 - 0 35III 14 40 0 40 0 0 40 0 40 0 0 0 - 016 15 0 0 0 0 33 46 46 0 33 33 46 0 -
Table 4 (Oontinued).
~
E~.
1s.;;:.
L~.
';::
~;:,;...~
'"...,
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68 A. P. Muhlemann and A. G. Lockett
The product types in the above table refer to the products whose processrequirements are given in Table 1. The batches are split into individual jobs,and renumbered from 1 to 16. (Hence job 1 is type 1, job 2 type 2, job 3type 3, job 4 type 4, job 5 type 4 and so on.) Selecting the appropriateelements from the basic delay matrix (in Table 2) gives the delay matrix forthe above set of products in Table 4.
Step 1 gives job number 12 (product type 9) as the first job (since thesmallest element is zero, and the row corresponding to product type 9 contains the least number of zeros).
Step 2 gives job number II (product type 8 version (ii)) as next job. Thisis because this job gives the least delay of all the remaining jobs when itfollows product type 9 (this delay is zero), and ties have arbitrarily beenbroken. These delays (for all remaining jobs) are simply given in the rowcorresponding to product type 9 in the 'delay matrix, and are 11(L).
Step 3 gives job number 4 (product type 4 version (ii)) as the next job.We require the job which minimized the sum of the delays between producttype 9 and the job and product type 8 version (ii) and the job. These aresimply 11(L) .plus the row corresponding to product type 8 version (ii) in thedelay matrix, call these 12(L), and product type 4 version (ii) has the smallestvalue in ML) of 52.
The rest of the steps, together with the value for the recurrence formulaI;(L) (obtained by adding an appropriate row of the delay matrix to 1;_I(L))are summarized in Table 5.
Clearly we need only calculate I;(L) for those jobs still requiring sequencing, i.e. the vertical lines correspond to jobs that have been sequenced (Sindicates that a particular version was selected, unmarked vertical linescorrespond to alternative, not selected versions).
A basis lor comparison
Because of the difficulty in evaluating a particular heuristic, four of thecommon loading rules are used as a basis for comparison with the heuristicdeveloped here. (Loading rules are probably the most widely developedheuristics in the area of scheduling, and some of these are discussed in detailin Gere (1966).) These are shortest processing time (SPT), SPT per operation(SPT/O), longest processing time (LPT) and LPT per operation (LPT/O).
However, having decided to schedule the products in a particular sequenceon the basis of one of the above rules, a further problem arises. Where thereare alternative methods of production, which method is selected? The ruleused here is that the alternative (version) which minimizes the makespan atthat stage is selected. Consider producing one batch of product 2, one batchof product 7, two batches of product 8 and using SPT as a scheduling procedure. Product 2 is scheduled first; for the first batch of product 8, version(ii) is selected and scheduled next; for the second batch of product 8, version(i) is selected; and finally for product 7, version (iv) is selected and scheduled last.
Results
I nitial results
Historical data showed that the number of batches for individual productsvaried from zero to ten, with an average of four. Forty sets of typical quarterly
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l~.
~Q.:~...~~
:...
i~.
1~~.;;.
I,(L)I.(L)I.(L)I.(L)I.(L)I.(L)I,(L)f.(L)fo(L)I.o(L)I,,(L)I,.(L)113(L)I..(L)I..(L)
64116244244244244372372412412470470528596
I
,---"-., I L /6 (ii) /~
s
I9
(i)
6464
192192232232360360360393393393451515
s
I8~
6\i) (ii)
64 6464 116
192 244192 244232 244232 244360 372360 372360 412393 412393 470393 470451 528
I7
s
.-"----,
5(i) (ii)
o 12452 17652 17676 20476 204
s
~r--"----,
5(i) (ii)
o 12452 17652 176
s
I5~
4(i) (ii)
64 064 5280 16480 164
120 164120 164
s
I4
ss
64 6464 6464 19264 192
192192320320320320320320
~
4
-----l __I (i) (ii)
64 064 52
s
124124124124124124124
12 911 8 (ii)4 4 (ii)6 5 (i)2 27 5 (i)5 4 (i)1 1
15 1416 1510 7 (iii)14 123 38 6 (i)9 6 (i)
13 9
Job selected
No. I Type__I 1__1_ 2 3
Table 5. Summary of the steps in the calculation for the heuristic.
0'<:J:)
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I I I I I I ±1j16Job selected 10 11 12
~I TypeL/
7 8 9 9 12 14 15/~(i) (ii) (iii) (iv) (i) (ii) (iii) (iv)
12 9 124 64 0 124 64 0 64 0 60 124 0 64 fl(L)11 8 (ii) 176 64 52 124 60· 176 52 87 f.(L)4 4 (ii) 292 192 168 252 188 176 168 87 fa(L)6 5 (i) 292 220 196 252 188 204 168 87 I.(L)2 2 292 220 196 252 228 204 208 127 I.(L)7 5 (i) 292 248 224 252 228 232 208 127 fs(L)5 4 (i) 292 376 224 380 356 232 208 243 '1(L)1 1 333 376 224 421 356 273 208 243 fs(L)
15 14 373 376 264 421 356 273 243 f.(L)16 15 373 376 264 421 389 319 Ilo(L)10 7 (iii) 389 319 fu(L)14 12 393 'ulL)3 3 451 'ulL)8 6 (i) 519 'ulL)9 6 (i) 587 fu(L)
13 9I
...o
~
~
f~~
~
is S
Table 5 (Continued).
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A 8cheduling problem with machine flexibility 71
requirements were scheduled using the heuristics outlined, and results arepresented in Table 6. Along with the heuristics, the time taken for a singlerandomly scheduled sequence is given for comparison. The results show thatthe best schedule is given by either SPT/O or H in each case. None of theother methods produce a best schedule, and were usually fur worse than thebest.
QuarterNo. Rand SPT SPTjO LPT LPTjO H Best
1 2240 2199 2121 2162 2569 1704 H2 2272 2043 1700 2374 2367 1745 SPTjO3 2450 2268 1795 1902 2236 1714 H4 1914 1905 1476 1617 2059 1415 H5 3020 2165 1718 1931 2334 1851 SPTjO6 2652 2338 1606 1775 2122 1755 SPTjO7 2194 2167 1474 1784 2093 1734 SPTjO8 1872 1470 1321 1548 1940 1405 SPTjO9 2339 1743 1678 1687 2389 1464 H
10 1986 1388 1194 1416 1743 1267 SPTjO11 1979 2005 1735 1872 2377 1690 H12 2233 1690 1575 1787 2243 1753 SPTjO13 2415 2211 1696 1942 2417 1637 H14 2878 2299 2222 2269 3048 2098 H15 2887 2095 1791 1967 2484 1794 SPTjO16 2298 2093 1745 1861 2261 1477 H17 2582 2139 1733 2015 2488 1877 SPTjO18 2548 2211 2140 1932 2507 1761 H19 2247 2173 1958 2046 2571 1861l H20 2049 1737 1328 1766 1692 1302 H21 2731 2573 2070 2324 2778 1988 H22 2354 2083 1562 1943 2024 1820 SPT/O23 2216 2015 1707 1704 1798 1684 H24 2110 1763 1679 1770 2219 1815 SPT/O25 2112 1539 1337 1508 1941 15711 SPT/O26 1770 1428 1365 1459 1846 1525 SPT/O27 2615 2129 11120 2130 2521 1892 H28 1933 1773 1639 1724 2099 16711 SPT/O29 2397 1981 1583 1796 2310 1680 SPT/O30 2153 1687 1369 1610 1940 1477 SPT/O31 2294 2125 1649 1924 2337 1461l H32 2762 1925 1681 1917 2456 1781 SPT/O33 2361 1693 1385 1741 2140 1580 SPT/O34 2253 1749 1674 1756 2401 1594 H35 2664 1797 1693 1830 2415 1778 . SPT/O36 2509 1934 2017 1879 2605 1638 H37 2910 2261 2242 2228 2865 1931 H38 2487 1729 1651 1826 2241 1715 SPT/O39 2724 2409 1824 2088 2692 1680 H40 2605 2235 2027 2160 2597 1755 H
Key: Rand = random ; SPT=shortest process time; SPT/O=shortest processtime per operation; LPT=longest.process time; LPTjO=longest process time peroperation; H = the heuristic.
Table 6. Makespan in hours for typical quarterly requirements.
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72 A. P. Muhlemann and A. G. Lockett
Other problems
From these results, the heuristic appeared to be worthy of further consideration. In order to test whether the results were due to the nature of theproblem or to the particular data, other tests of the heuristic were required.
To do this three more problems were generated, which were all based onthe same fundamental structure as the original problem. The product versionshad at most two stages, each stage could involve at most two vessels. Thevariables were number of stages for each type, number of vessels for eachstage, the process time and the clean-out times. These were determinedrandomly and the results are presented in Table 7. For each of the fourproblems a summary of the results of scheduling forty sets of typical jobs isgiven.
Heuristic
Problem Number of time best solution obtained
8PT 8PT/O LPT LPT/O H
Original 0 20 0 0 20Random I 0 15 4 II 10Random 2 I 1 3 5 30Random 3 0 12 20 I 7
Total 1 48 27 17 67
Table 7. Comparative results for the heuristics.
Heuristic
Problem Average position
Rand 8PT 8PT/O LPT LPT/O H
Original 6·9 4·1 1-6 3·2 . 6·5 1·7Random I 7·4 4·4 2·0 3·9 2·5 2·7Random 2 7·6 5·1 3·6 2·8 2·8 1·6Random 3 7-l 4·3 2·2 1·8 4·2 3·7
Overallaverage 7·2 4·5 2·4 2·9 4·1 2·4
Table 8. Average positions for the heuristics and the four problems.
It can be clearly seen that 8PT/O, Hand LPT are the best heuristics forthc random sets I, 2 and 3, respectively. Overall H is the most successful.Moreover, even when H does not produce the best solution, it produces agood solution. This is evident from Table 8 which gives the average positions (first, second third, etc.) of all the heuristics for the different problems.
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A scheduling problem with machine flexibility 73
Conclusions
It is possible to improve the sequences even more by exploiting the actualnumerical data, in each particular case. This involves a detailed analysis ofthe problem structure and its interaction with the pattern of productionrequirements in each period. Although this has been done, it is not reportedhere because there is no generally applicable procedure. However, a hybridheuristic of the form' schedule using H, schedule using SPT/O and select thesequence with the smallest makespan " has much to offer. Over all problemsH or SPT/O produced the best sequence 82% of the time, and moreover, Hor SPT/O failed to produce the best or second best sequence only II % ofthe time. The times taken to compute the heuristics vary considerably buton average H is 20% faster than SPT/O. The calculations for both thesemethods are easy to implement, and use little computer time; hence use ofthe hybrid heuristic is perfectly feasible.
On preaente Ie problems pose par la repartition de la production de produitaphermeceut.iquea dena un nombre limite de recipients. Ce problems poesede leatra.its caracteriatiquea sui venta : Ie moment de substitution depend de Is repartition.180 production est effectuee en plusieurs sta.d.es, l'utilisation parallele des recipients
.est. employee at il y a. de 180 Jlexibilite dans Ie choix des recipien.ts.On discute une technique d'optimisa.tion mais on Is rejette comme etant impossible
A operer sur ordinateur. Par consequent, on developpe une heuristique et on enevnlue le fonctionnement. On rend compte egelemerrt des resulta.ts typiquea decette heuristique.
Ein Planungaproblem fUr Produktion von pharmazeutischen Erzeugnissen ineiner begrenaten Zahl von Behii.ltem wird dargelegt. EB hat die Charakteriatikevon eequeneabhengigen Oberga.ngszeiten, Vielstufenproduktion, Parallelverwendungvon Behaltern und Flexibilitat in der Behalterwehl. Ein cptimalea Verfa.hren wirddiskutiert, doch els rechnerisch undurchftihrbar verworfen. Daher wird ein heuristisches Verfa.hren entwickelt und seine Leistungsfahigkeit ausgewertet. Es wirdauch tiber typiache Resultate berichtet.
REFERENCESCONWAY, R. W., MAXWELL, W. J., and MILLER, L. W., 1967, Theory 01 Scheduling
(Reading: Addison-Wesley).GERE, W. S., 1966, Heuristics in job-shop scheduling, Mgmt Sci., 13, 167.LITTLE, J. D. C., MURTY, K. G., SWEENEY, D. W., and KAREL, C., 1963,An algorithm
for the travelling salesman problem, Ops. Res., 11, 972.LOCKETT, A. G., and MUHLEMANN, A. P., 1972, A scheduling problem involving
sequence dependent changeover times, Ops. Res., 20, 895.MANNE, A. S., 1960, On the job-shop scheduling problem, Ope. Res., 8, 219.MUHLEMANN, A. P., 1972, Some scheduling problems with sequence dependent
changeover times, Ph.D. Thesis, Manchester University.SCHWARTZ, E. S., 1964, A heuristic procedure for parallel sequencing with choice of
machines, Mgmt Sci., 10, 767.SRIVASTAVA, S., KUMAR, S., GARG, R., and SEN, P., 1969, Generalised travelling
salesman problem through n sets of nodes, Can. Oper. Res. Soc. J., 7, 97.VON LANZENAUER, C., 1970, A production scheduling model by bivalent linear
programming, Mgmt Sci., 17, 105.
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