- Email: [email protected]
Common due date assignment for scheduling on a single machine with jointly reducible processing times
Int. J. Production Economics 69 (2001) 317}322
Common due date assignment for scheduling on a single machine with jointly reducible processing times Dirk Biskup, Hermann Jahnke* Faculty of Economics and Business Administration, University of Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany Received 1 October 1999; accepted 15 March 2000
Abstract This paper focuses on analyzing the problem of assigning a common due date to a set of jobs and scheduling them on a single machine. The processing times of the jobs are assumed to be controllable but contrary to former approaches we consider a situation in which it is only possible to reduce all processing times by the same proportional amount. This situation, which is quite interesting from a practical point of view, has to the best of our knowledge never been under study before. Besides the assignment of the common due date we concentrate on two goals, namely minimizing the sum of earliness and tardiness penalties and minimizing the number of late jobs. 2001 Elsevier Science B.V. All rights reserved. Keywords: Scheduling; Earliness; Tardiness; Controllable processing times
1. Introduction Meeting due dates is among the most important goals in scheduling practice, see, for example, [10,13]. Therefore, we concentrate on this objective in the present paper. More speci"cally we consider the scheduling of jobs against a common due date; this means the same due date has been assigned to all jobs. There are many practical situations in which a common due date exists, e.g. in a just-intime-production environment, in the assembly schedule or for a batch delivery. Furthermore, it might be reasonable to assign a common due date
* Corresponding author. Tel.: #49-0521-106-3927; fax: #49-0521-106-6036. E-mail address: [email protected] (H. Jahnke).
to a set of jobs to treat di!erent customers equally. Generally, two situations should be distinguished: On the one hand, a common due date may be (externally) given or agreed upon, on the other, a common due date can be a decision variable with its value speci"ed by the company. The latter situation occurs, for example, at di!erent stages of the production process, where the timing of an assembly can be determined by the company. Another example due to Cheng [5] is that of a shoemaker, who issues the same due date to all customers waiting at a time. In both situations it is important to quote realistic due dates. A long due date delays the production process but an unrealistic short due date cannot speed it up (as some of the parts will not be "nished) and causes high holding costs for the parts waiting for assembly. A somewhat similar situation exists for the shoemaker. Long due dates
0925-5273/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 0 4 0 - 2
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cannot serve as a marketing instrument but the promise of a short due date which cannot be kept causes dissatisfaction and leads } in the long run } to the loss of goodwill. Thus, the assignment of realistic common due dates is an important task which will be considered in this paper. For an excellent review on common due date research see Baker and Scudder [2]. Scheduling problems with continuously compressible processing times have received considerable attention during the last years, see [1,3,6,7,9,12,14,15]. The assumption in these papers is that the normal processing time of a job can be reduced continuously to a minimum value by incurring higher processing costs. Chen et al. [4] appear to be the only researchers studying the case of discretely controllable processing times. In practice, there are situations in which processing times are not individually reducible. The only option might be to reduce the processing times of all jobs jointly, i.e. simultaneously by the same percentage. A furnace, for example, might be heated to a speci"ed temperature every day before the processing of the jobs starts. It is impossible (or a least not advantageous) to change the temperature for every single job. Note that, on the one hand, the temperature has an in#uence on the processing times of the jobs, but on the other, each temperature induces di!erent costs for using the furnace. A similar situation might occur for a machine at which speci"c tools have to be changed after a "xed period of time, say a week. Each time a new tool is mounted, a decision about the tool characteristics, that is the productive power, needs to be made. For example a drilling machine might run with a diamond drill, a high- or a low-quality steel drill. If the diamond drill is set up, the processing of the jobs can be carried out faster than with a steel drill by incurring the highest costs. On the other hand, setting up the low-quality steel drill induces the lowest costs and the highest processing times. Another example is the speed of an assembly line which usually can be chosen out of a speci"ed interval (depending, for example, on the number of workers and tools available). With respect to the new orders it may be necessary to decide on the speed of the assembly line when production starts while it is generally not possible or advantageous to
change the speed between di!erent jobs during a day. In the next section we introduce some assumptions and the notation used. Section 3 deals with the problem of jointly minimizing due date assignment costs and the sum of earliness and tardiness penalties. In Section 4 we analyze an objective function consisting of costs for assigning a due date and "nishing jobs late.
2. Assumptions and notation There are n jobs available at time zero. Each job has a normal processing time and the jobs are indexed according to the shortest (normal) processing time (SPT) sequence, i.e. p )p )2)p . If L the jobs are processed with their normal processing times, the costs K for operating the machine are induced. Note, that these operating costs are usually neglected in scheduling theory, as they are assumed to be "xed and thus have no in#uence on an optimal schedule. If the speed of the machine is continuously controllable, x will be the proportion of the processing time reduction of all jobs, 0)x)x (1. That
is, for a given x the normal processing times are reduced to p (1!x), p (1!x),2, p (1!x). For L the sake of simplicity, the operating costs k(x) with k(0)"K are given as a monotonous increasing function in x. For example, k(x)"K#ax is a linear function of operating costs, with the intercept of the ordinate K and the slope a, and k(x)"K#ax describes a quadratic operating cost function. Convex cost functions, however, seem to "t most situations in practice. For the sake of simplicity we assume K"0.
3. Minimizing earliness and tardiness penalties It has been widely known in industrial practice that "nishing jobs too early will result in holding costs. These costs could be handling-costs depending on the quantity stored and holding-costs like insurance premiums, imputed interests and taxes, which depend on the value of the inventory. Depreciations may be caused by technical progress
D. Biskup, H. Jahnke / Int. J. Production Economics 69 (2001) 317}322
and/or changing markets. Furthermore, early jobs tie up capital, so that opportunity costs have to be considered. On the other hand, the e!ects of tardy jobs are dissatis"ed customers and thus, in the long run, the loss of goodwill and reputation. Consequently, orders may be canceled and/or no more orders will be placed in the future so that tardiness can reduce the sales volume in the long run. In addition, contractual penalties for late deliveries might have been agreed upon with some of the customers. Comparing the consequences of earliness and tardiness, tardy jobs are usually considered more undesirable than early jobs. The remainder of this section mainly consists of two parts: Firstly, the problem to jointly minimize earliness, tardiness and due date assignment costs with "xed (normal) processing times is introduced and some of its theoretical results together with an optimizing algorithm are cited from the literature. Secondly, we extend the problem by the possibility of continuously reducing processing times and present a new solution procedure for this situation. Let C , E "max+0, d!C , and ¹ "max+0, G G G G C !d, be the completion time, earliness and tardiG ness of job i, i"1,2, n, respectively. Further, let a and b be the penalties for earliness and tardiness and c be the due date assignment cost per time unit representing the notion that an early due date is desirable to the customers. The general objective is to "nd a feasible schedule S and a common due date d which jointly minimize L f (S, d)" (aE #b¹ #cd). G G G
319
Lemma 2. Let [r] indicate the job scheduled at the rth position of a schedule. Assuming that the completion of one job coincides with the due date the objective function (1) can be rewritten as L f (S, d)" u p , P P
P where u "min+(r!1)a#nc, (n!r#1)b, P is the positional weight which arises if a job occupies the rth position in a schedule. The proofs of Lemmas 1 and 2 can be found in Panwalkar et al. [11], see also Baker and Scudder [2]. Problem (1) can now be solved in two steps. Firstly, a simple matching algorithm is applied to obtain an optimal sequence of the jobs: The longest job is placed at the position with the smallest u , the second longest job at the position P with the second smallest u , etc. Secondly, the due P date is determined by d" @ p and the processG G
ing of the "rst job starts at time zero. This procedure is called Algorithm K in the following, as it is originally due to Kanet [8]. Furthermore, let f H denote the objective function value minimizing (1). Including the possibility to decrease the processing times continuously by incurring higher operating costs according to k(x), the following lemma holds.
(1)
For example, with a"b"1 and c"0, the objective function (1) becomes that for the well-known problem of Kanet [8]. Clearly, an optimal schedule does not contain any idle time between any pair of consecutive jobs. Note that a reduction of the processing times results in lower due date assignment costs as well as lower earliness and tardiness costs, since all jobs are completed closer to the common due date. Lemma 1. An optimal schedule exists, in which the bth job is completed at d, where b is the smallest integer greater than or equal to n(b!c)/(a#b).
Lemma 3. If the processing times are reduced by x, the objective function value of an optimal sequence decreases to f H"(1!x) f H. V Proof. By reducing the processing times of all jobs by the same proportion, the initial SPT ordering still holds. Consequently, the "rst step of the Algorithm K leads to the same job sequence. According to Lemma 2 the new objective function value is L f H" u (1!x)p V P P
P L "(1!x) u p "(1!x) f H. P P
P
䊐
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As a consequence of Lemma 3, the optimal sequence S and the optimal reduction of the processing times can be obtained by the following procedure:
pH"2.5. The optimal due date is dH"pH#pH"2. The minimal total costs induced are 92.5.
4. Minimizing the number of tardy jobs Step 1: For the case of normal processing times determine f H and an optimal sequence of the jobs SH by the Algorithm K. Step 2: Find the optimal reduction of the processing times xH by solving the following program: Minimize
(1!x) f H#k(x)
subject to 0)x)x .
Step 3: If xH'0, reduce the processing times to pH"(1!xH)p for i"1,2, n and calculate the G G optimal due date dH by the second step of the Algorithm K. For the special case of a linear cost function k(x), the following lemma holds. Lemma 4. If k(x) is linear in x, say k(x)"ax, an optimal sequence exists, in which the compression of the job processing times is x"0 if a*f H or x"x if a)f H.
Proof. The objective function (1!x) f H#ax" f H#(a!f H)x is linear in x, hence a global minimum exists at the border of the solution space, that is at x"0 if a*f H or at x"x if a)f H. )
Example 1. Let n"5 jobs be given with the normal processing times p "1, p "2, p "3, p "4 and p "5. The earliness, tardiness and due date assignment penalties are a"3, b"4 and c"2, respectively. The processing times are continuously reducible to 60% of their normal values, thus 0)x)0.6, by inducing the operating costs k(x)"13x#106x. Step 1: With b"2 the positional penalties are: u "10, u "13, u "12, u "8, u "4. An optimal sequence is SH"(3, 1, 2, 4, 5) with f H"119. Step 2: Minimizing 119(1!x)#13x#106x subject to 0)x)0.6 gives xH"0.5. Step 3: With xH"0.5 the optimal processing times are: pH"0.5, pH"1, pH"1.5, pH"2 and
Another objective in connection with the assignment of common due dates is to minimize the number of tardy jobs, see Cheng [5]. This goal should be considered if the costs induced by "nishing a job late do not depend on the absolute value of its tardiness; if, for example, a promised delivery date is not kept, a stipulated penalty might occur or the order might be canceled. Similar to Section 3 the original problem (with normal processing times) is at "rst presented and some of its theory will be repeated. Afterwards, the possibility of reducing the processing times is included in the analysis. Let ¸(i) be the indicator function assuming the value 1 if the job i is late, i.e.
1 if C 'd, G 0 if C )d, G and g be the penalty cost for "nishing a job late. The goal is to jointly minimize the costs of assigning a long due date and completing jobs late, i.e. ¸(i)"
L f (S, d)" (g¸(i)#cd). G
(2)
Lemma 5. The optimal solution is obtained, if the jobs are sequenced according to the SPT rule and the assigned due date coincides with the completion of the bth job, d"C , so that the inequality @ p )g/nc)p holds. @ @> The proof of Lemma 5 can be found in Cheng [5]. Now the situation with jointly controllable processing times will be considered. If all processing times are reduced by the proportion x, b and hence d depend on x since g/nc is a constant. Therefore, a trade-o! between continuously increasing operating costs, stepwise decreasing lateness costs and due date assignment costs has to be taken into account. Exactly one job less will be "nished late if the processing times are reduced by the proportion
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x so that g g p (1!x)" or x"1! @> nc ncp @> holds. Note that this reduction of processing times leads to a new due date of C (1!x) altering @> the due date assignment costs by nc[C @> (1!x)!C ], with C and C representing the @ @ @> completion time of the jobs b and b#1 for the case of normal processing times, respectively. Hence the reduction of processing times should be realized, if the incurred costs are compensated for by the reduced lateness, i.e. if k(x)#nc[C (1!x)! @> C ](g holds. Of course, the second term on the @ left-hand side of this inequality might be negative. The comparison between increased operating and decreased lateness costs has to be carried out with all jobs b#1, b#2,2 until a further reduction of processing times is not possible. Note that only a subset of all possible reductions of the processing times have to be considered, namely those for which p (1!x )"g/nc for i"1, 2,2 with @>G G x )x holds. Thus, the following algorithm can G
be applied to solve problem (2) for the case that the processing times are jointly reducible: Step 1: Sequence the jobs according to the SPT rule (SH). Step 2: Find the optimal due date d"C with @ p )g/nc)p . @ @> Step 3: Find iH3+1, 2,2, n!b, so that * f H" G k(xH)#nc[C H (1!x H )!C ]!iHg for x H " G @ G G @>G 1!g/ncp H is minimal with respect to 0)x H @>G G )x .
Step 4: If * f H (0 reduce the processing times by G x H and assign the optimal due date d"C H . @>G G Example 1 (continued). Let c"2 and g"28. Step 1: SH"(1, 2, 3, 4, 5). Step 2: As p )g/nc"2.8)p it follows that d"C "3 (b"2) with f (S, d)"28 ) 3#5 ) 3 ) 2" 114. Step 3: x "1!g/ncp "1!28/(05 ) 2 ) 3)" and * f "k( )#5 ) 2[C (1! )!C ]!1 ) 28" !0.6622. x "0.3 and * f "!2.56. x " and * f "6.2416. iH"2.
321
Step 4: Since * f (0 it follows that pH" 0.7, pH"1.4, pH"2.1, pH"2.8 and pH"3.5, dH"C "7 with the minimal costs of 111.44. 5. Conclusions In this paper we studied the problem of assigning a common due date to a set of simultaneously available jobs for which the processing times are jointly reducible. Within this setting we concentrated on two kinds of objective functions. Besides considering due date assignment costs the "rst goal was to minimize the sum of earliness and tardiness penalties while the second one was to minimize the number of late jobs. For both cases polynomially solvable algorithms have been found and demonstrated by an example.
References [1] B. Alidaee, A. Ahmadian, Two parallel machine sequencing problems involving controllable job processing times, European Journal of Operational Research 70 (1993) 335}341. [2] K.R. Baker, G.D. Scudder, Sequencing with earliness and tardiness penalties: A review, Operations Research 38 (1990) 22}36. [3] D. Biskup, T.C.E. Cheng, Single-machine scheduling with controllable processing times and earliness, tardiness and completion time penalties, Engineering Optimization 31 (1999) 329}336. [4] Z.-L. Chen, Q. Lu, G. Tang, Single machine scheduling with discretely controllable processing times, Operations Research Letters 21 (1997) S 69}76. [5] T.C.E. Cheng, Common due-date assignment and scheduling for a single processor to minimize the number of tardy jobs, Engineering Optimization 16 (1990) 129}136. [6] T.C.E. Cheng, Z.L. Chen, C.-L. Li, Parallel-machine scheduling with controllable processing times, IIE Transactions 28 (1996) 177}180. [7] T.C.E. Cheng, C. Oguz, X.D. Qi, Due-date assignment and single machine scheduling with compressible processing times, International Journal of Production Economics 43 (1996) 29}35. [8] J.J. Kanet, Minimizing the average deviation of job completion times about a common due date, Naval Research Logistics Quarterly 28 (1981) 643}651. [9] E. Nowicki, S. Zdrzalka, A survey of results for sequencing problems with controllable processing times, Discrete Applied Mathematics 26 (1990) 271}287.
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[10] S.S. Panwalkar, R.A. Dudek, M.L. Smith, Sequencing research and the industrial scheduling problem, in: S.E. Elmaghraby (Ed.), Symposium on the Theory of Scheduling and its Applications, Springer, Berlin, 1973, pp. 29}38. [11] S.S. Panwalkar, M.L. Smith, A. Seidmann, Common due date assignment to minimize total penalty for the one machine scheduling problem, Operations Research 30 (1982) 391}399. [12] S.S. Panwalkar, R. Rajagopalan, Single-machine sequencing with controllable processing times, European Journal of Operational Research 59 (1992) 298}302.
[13] G. Stalk, Time } the next source of competitive advantage, Harvard Business Review 66 (1988) 41}51. [14] R.G. Vickson, Choosing the job sequence and processing times to minimize total processing plus #ow cost on a single machine, Operations Research 28 (1980) 1155}1167. [15] R.G. Vickson, Two single machine sequencing problems involving controllable job processing times, AIIE Transactions 12 (1980) 258}262.
Common due date assignment for scheduling on a single machine with jointly reducible processing times Dirk Biskup, Hermann Jahnke* Faculty of Economics and Business Administration, University of Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany Received 1 October 1999; accepted 15 March 2000
Abstract This paper focuses on analyzing the problem of assigning a common due date to a set of jobs and scheduling them on a single machine. The processing times of the jobs are assumed to be controllable but contrary to former approaches we consider a situation in which it is only possible to reduce all processing times by the same proportional amount. This situation, which is quite interesting from a practical point of view, has to the best of our knowledge never been under study before. Besides the assignment of the common due date we concentrate on two goals, namely minimizing the sum of earliness and tardiness penalties and minimizing the number of late jobs. 2001 Elsevier Science B.V. All rights reserved. Keywords: Scheduling; Earliness; Tardiness; Controllable processing times
1. Introduction Meeting due dates is among the most important goals in scheduling practice, see, for example, [10,13]. Therefore, we concentrate on this objective in the present paper. More speci"cally we consider the scheduling of jobs against a common due date; this means the same due date has been assigned to all jobs. There are many practical situations in which a common due date exists, e.g. in a just-intime-production environment, in the assembly schedule or for a batch delivery. Furthermore, it might be reasonable to assign a common due date
* Corresponding author. Tel.: #49-0521-106-3927; fax: #49-0521-106-6036. E-mail address: [email protected] (H. Jahnke).
to a set of jobs to treat di!erent customers equally. Generally, two situations should be distinguished: On the one hand, a common due date may be (externally) given or agreed upon, on the other, a common due date can be a decision variable with its value speci"ed by the company. The latter situation occurs, for example, at di!erent stages of the production process, where the timing of an assembly can be determined by the company. Another example due to Cheng [5] is that of a shoemaker, who issues the same due date to all customers waiting at a time. In both situations it is important to quote realistic due dates. A long due date delays the production process but an unrealistic short due date cannot speed it up (as some of the parts will not be "nished) and causes high holding costs for the parts waiting for assembly. A somewhat similar situation exists for the shoemaker. Long due dates
0925-5273/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 0 4 0 - 2
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cannot serve as a marketing instrument but the promise of a short due date which cannot be kept causes dissatisfaction and leads } in the long run } to the loss of goodwill. Thus, the assignment of realistic common due dates is an important task which will be considered in this paper. For an excellent review on common due date research see Baker and Scudder [2]. Scheduling problems with continuously compressible processing times have received considerable attention during the last years, see [1,3,6,7,9,12,14,15]. The assumption in these papers is that the normal processing time of a job can be reduced continuously to a minimum value by incurring higher processing costs. Chen et al. [4] appear to be the only researchers studying the case of discretely controllable processing times. In practice, there are situations in which processing times are not individually reducible. The only option might be to reduce the processing times of all jobs jointly, i.e. simultaneously by the same percentage. A furnace, for example, might be heated to a speci"ed temperature every day before the processing of the jobs starts. It is impossible (or a least not advantageous) to change the temperature for every single job. Note that, on the one hand, the temperature has an in#uence on the processing times of the jobs, but on the other, each temperature induces di!erent costs for using the furnace. A similar situation might occur for a machine at which speci"c tools have to be changed after a "xed period of time, say a week. Each time a new tool is mounted, a decision about the tool characteristics, that is the productive power, needs to be made. For example a drilling machine might run with a diamond drill, a high- or a low-quality steel drill. If the diamond drill is set up, the processing of the jobs can be carried out faster than with a steel drill by incurring the highest costs. On the other hand, setting up the low-quality steel drill induces the lowest costs and the highest processing times. Another example is the speed of an assembly line which usually can be chosen out of a speci"ed interval (depending, for example, on the number of workers and tools available). With respect to the new orders it may be necessary to decide on the speed of the assembly line when production starts while it is generally not possible or advantageous to
change the speed between di!erent jobs during a day. In the next section we introduce some assumptions and the notation used. Section 3 deals with the problem of jointly minimizing due date assignment costs and the sum of earliness and tardiness penalties. In Section 4 we analyze an objective function consisting of costs for assigning a due date and "nishing jobs late.
2. Assumptions and notation There are n jobs available at time zero. Each job has a normal processing time and the jobs are indexed according to the shortest (normal) processing time (SPT) sequence, i.e. p )p )2)p . If L the jobs are processed with their normal processing times, the costs K for operating the machine are induced. Note, that these operating costs are usually neglected in scheduling theory, as they are assumed to be "xed and thus have no in#uence on an optimal schedule. If the speed of the machine is continuously controllable, x will be the proportion of the processing time reduction of all jobs, 0)x)x (1. That
is, for a given x the normal processing times are reduced to p (1!x), p (1!x),2, p (1!x). For L the sake of simplicity, the operating costs k(x) with k(0)"K are given as a monotonous increasing function in x. For example, k(x)"K#ax is a linear function of operating costs, with the intercept of the ordinate K and the slope a, and k(x)"K#ax describes a quadratic operating cost function. Convex cost functions, however, seem to "t most situations in practice. For the sake of simplicity we assume K"0.
3. Minimizing earliness and tardiness penalties It has been widely known in industrial practice that "nishing jobs too early will result in holding costs. These costs could be handling-costs depending on the quantity stored and holding-costs like insurance premiums, imputed interests and taxes, which depend on the value of the inventory. Depreciations may be caused by technical progress
D. Biskup, H. Jahnke / Int. J. Production Economics 69 (2001) 317}322
and/or changing markets. Furthermore, early jobs tie up capital, so that opportunity costs have to be considered. On the other hand, the e!ects of tardy jobs are dissatis"ed customers and thus, in the long run, the loss of goodwill and reputation. Consequently, orders may be canceled and/or no more orders will be placed in the future so that tardiness can reduce the sales volume in the long run. In addition, contractual penalties for late deliveries might have been agreed upon with some of the customers. Comparing the consequences of earliness and tardiness, tardy jobs are usually considered more undesirable than early jobs. The remainder of this section mainly consists of two parts: Firstly, the problem to jointly minimize earliness, tardiness and due date assignment costs with "xed (normal) processing times is introduced and some of its theoretical results together with an optimizing algorithm are cited from the literature. Secondly, we extend the problem by the possibility of continuously reducing processing times and present a new solution procedure for this situation. Let C , E "max+0, d!C , and ¹ "max+0, G G G G C !d, be the completion time, earliness and tardiG ness of job i, i"1,2, n, respectively. Further, let a and b be the penalties for earliness and tardiness and c be the due date assignment cost per time unit representing the notion that an early due date is desirable to the customers. The general objective is to "nd a feasible schedule S and a common due date d which jointly minimize L f (S, d)" (aE #b¹ #cd). G G G
319
Lemma 2. Let [r] indicate the job scheduled at the rth position of a schedule. Assuming that the completion of one job coincides with the due date the objective function (1) can be rewritten as L f (S, d)" u p , P P
P where u "min+(r!1)a#nc, (n!r#1)b, P is the positional weight which arises if a job occupies the rth position in a schedule. The proofs of Lemmas 1 and 2 can be found in Panwalkar et al. [11], see also Baker and Scudder [2]. Problem (1) can now be solved in two steps. Firstly, a simple matching algorithm is applied to obtain an optimal sequence of the jobs: The longest job is placed at the position with the smallest u , the second longest job at the position P with the second smallest u , etc. Secondly, the due P date is determined by d" @ p and the processG G
ing of the "rst job starts at time zero. This procedure is called Algorithm K in the following, as it is originally due to Kanet [8]. Furthermore, let f H denote the objective function value minimizing (1). Including the possibility to decrease the processing times continuously by incurring higher operating costs according to k(x), the following lemma holds.
(1)
For example, with a"b"1 and c"0, the objective function (1) becomes that for the well-known problem of Kanet [8]. Clearly, an optimal schedule does not contain any idle time between any pair of consecutive jobs. Note that a reduction of the processing times results in lower due date assignment costs as well as lower earliness and tardiness costs, since all jobs are completed closer to the common due date. Lemma 1. An optimal schedule exists, in which the bth job is completed at d, where b is the smallest integer greater than or equal to n(b!c)/(a#b).
Lemma 3. If the processing times are reduced by x, the objective function value of an optimal sequence decreases to f H"(1!x) f H. V Proof. By reducing the processing times of all jobs by the same proportion, the initial SPT ordering still holds. Consequently, the "rst step of the Algorithm K leads to the same job sequence. According to Lemma 2 the new objective function value is L f H" u (1!x)p V P P
P L "(1!x) u p "(1!x) f H. P P
P
䊐
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D. Biskup, H. Jahnke / Int. J. Production Economics 69 (2001) 317}322
As a consequence of Lemma 3, the optimal sequence S and the optimal reduction of the processing times can be obtained by the following procedure:
pH"2.5. The optimal due date is dH"pH#pH"2. The minimal total costs induced are 92.5.
4. Minimizing the number of tardy jobs Step 1: For the case of normal processing times determine f H and an optimal sequence of the jobs SH by the Algorithm K. Step 2: Find the optimal reduction of the processing times xH by solving the following program: Minimize
(1!x) f H#k(x)
subject to 0)x)x .
Step 3: If xH'0, reduce the processing times to pH"(1!xH)p for i"1,2, n and calculate the G G optimal due date dH by the second step of the Algorithm K. For the special case of a linear cost function k(x), the following lemma holds. Lemma 4. If k(x) is linear in x, say k(x)"ax, an optimal sequence exists, in which the compression of the job processing times is x"0 if a*f H or x"x if a)f H.
Proof. The objective function (1!x) f H#ax" f H#(a!f H)x is linear in x, hence a global minimum exists at the border of the solution space, that is at x"0 if a*f H or at x"x if a)f H. )
Example 1. Let n"5 jobs be given with the normal processing times p "1, p "2, p "3, p "4 and p "5. The earliness, tardiness and due date assignment penalties are a"3, b"4 and c"2, respectively. The processing times are continuously reducible to 60% of their normal values, thus 0)x)0.6, by inducing the operating costs k(x)"13x#106x. Step 1: With b"2 the positional penalties are: u "10, u "13, u "12, u "8, u "4. An optimal sequence is SH"(3, 1, 2, 4, 5) with f H"119. Step 2: Minimizing 119(1!x)#13x#106x subject to 0)x)0.6 gives xH"0.5. Step 3: With xH"0.5 the optimal processing times are: pH"0.5, pH"1, pH"1.5, pH"2 and
Another objective in connection with the assignment of common due dates is to minimize the number of tardy jobs, see Cheng [5]. This goal should be considered if the costs induced by "nishing a job late do not depend on the absolute value of its tardiness; if, for example, a promised delivery date is not kept, a stipulated penalty might occur or the order might be canceled. Similar to Section 3 the original problem (with normal processing times) is at "rst presented and some of its theory will be repeated. Afterwards, the possibility of reducing the processing times is included in the analysis. Let ¸(i) be the indicator function assuming the value 1 if the job i is late, i.e.
1 if C 'd, G 0 if C )d, G and g be the penalty cost for "nishing a job late. The goal is to jointly minimize the costs of assigning a long due date and completing jobs late, i.e. ¸(i)"
L f (S, d)" (g¸(i)#cd). G
(2)
Lemma 5. The optimal solution is obtained, if the jobs are sequenced according to the SPT rule and the assigned due date coincides with the completion of the bth job, d"C , so that the inequality @ p )g/nc)p holds. @ @> The proof of Lemma 5 can be found in Cheng [5]. Now the situation with jointly controllable processing times will be considered. If all processing times are reduced by the proportion x, b and hence d depend on x since g/nc is a constant. Therefore, a trade-o! between continuously increasing operating costs, stepwise decreasing lateness costs and due date assignment costs has to be taken into account. Exactly one job less will be "nished late if the processing times are reduced by the proportion
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x so that g g p (1!x)" or x"1! @> nc ncp @> holds. Note that this reduction of processing times leads to a new due date of C (1!x) altering @> the due date assignment costs by nc[C @> (1!x)!C ], with C and C representing the @ @ @> completion time of the jobs b and b#1 for the case of normal processing times, respectively. Hence the reduction of processing times should be realized, if the incurred costs are compensated for by the reduced lateness, i.e. if k(x)#nc[C (1!x)! @> C ](g holds. Of course, the second term on the @ left-hand side of this inequality might be negative. The comparison between increased operating and decreased lateness costs has to be carried out with all jobs b#1, b#2,2 until a further reduction of processing times is not possible. Note that only a subset of all possible reductions of the processing times have to be considered, namely those for which p (1!x )"g/nc for i"1, 2,2 with @>G G x )x holds. Thus, the following algorithm can G
be applied to solve problem (2) for the case that the processing times are jointly reducible: Step 1: Sequence the jobs according to the SPT rule (SH). Step 2: Find the optimal due date d"C with @ p )g/nc)p . @ @> Step 3: Find iH3+1, 2,2, n!b, so that * f H" G k(xH)#nc[C H (1!x H )!C ]!iHg for x H " G @ G G @>G 1!g/ncp H is minimal with respect to 0)x H @>G G )x .
Step 4: If * f H (0 reduce the processing times by G x H and assign the optimal due date d"C H . @>G G Example 1 (continued). Let c"2 and g"28. Step 1: SH"(1, 2, 3, 4, 5). Step 2: As p )g/nc"2.8)p it follows that d"C "3 (b"2) with f (S, d)"28 ) 3#5 ) 3 ) 2" 114. Step 3: x "1!g/ncp "1!28/(05 ) 2 ) 3)" and * f "k( )#5 ) 2[C (1! )!C ]!1 ) 28" !0.6622. x "0.3 and * f "!2.56. x " and * f "6.2416. iH"2.
321
Step 4: Since * f (0 it follows that pH" 0.7, pH"1.4, pH"2.1, pH"2.8 and pH"3.5, dH"C "7 with the minimal costs of 111.44. 5. Conclusions In this paper we studied the problem of assigning a common due date to a set of simultaneously available jobs for which the processing times are jointly reducible. Within this setting we concentrated on two kinds of objective functions. Besides considering due date assignment costs the "rst goal was to minimize the sum of earliness and tardiness penalties while the second one was to minimize the number of late jobs. For both cases polynomially solvable algorithms have been found and demonstrated by an example.
References [1] B. Alidaee, A. Ahmadian, Two parallel machine sequencing problems involving controllable job processing times, European Journal of Operational Research 70 (1993) 335}341. [2] K.R. Baker, G.D. Scudder, Sequencing with earliness and tardiness penalties: A review, Operations Research 38 (1990) 22}36. [3] D. Biskup, T.C.E. Cheng, Single-machine scheduling with controllable processing times and earliness, tardiness and completion time penalties, Engineering Optimization 31 (1999) 329}336. [4] Z.-L. Chen, Q. Lu, G. Tang, Single machine scheduling with discretely controllable processing times, Operations Research Letters 21 (1997) S 69}76. [5] T.C.E. Cheng, Common due-date assignment and scheduling for a single processor to minimize the number of tardy jobs, Engineering Optimization 16 (1990) 129}136. [6] T.C.E. Cheng, Z.L. Chen, C.-L. Li, Parallel-machine scheduling with controllable processing times, IIE Transactions 28 (1996) 177}180. [7] T.C.E. Cheng, C. Oguz, X.D. Qi, Due-date assignment and single machine scheduling with compressible processing times, International Journal of Production Economics 43 (1996) 29}35. [8] J.J. Kanet, Minimizing the average deviation of job completion times about a common due date, Naval Research Logistics Quarterly 28 (1981) 643}651. [9] E. Nowicki, S. Zdrzalka, A survey of results for sequencing problems with controllable processing times, Discrete Applied Mathematics 26 (1990) 271}287.
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[10] S.S. Panwalkar, R.A. Dudek, M.L. Smith, Sequencing research and the industrial scheduling problem, in: S.E. Elmaghraby (Ed.), Symposium on the Theory of Scheduling and its Applications, Springer, Berlin, 1973, pp. 29}38. [11] S.S. Panwalkar, M.L. Smith, A. Seidmann, Common due date assignment to minimize total penalty for the one machine scheduling problem, Operations Research 30 (1982) 391}399. [12] S.S. Panwalkar, R. Rajagopalan, Single-machine sequencing with controllable processing times, European Journal of Operational Research 59 (1992) 298}302.
[13] G. Stalk, Time } the next source of competitive advantage, Harvard Business Review 66 (1988) 41}51. [14] R.G. Vickson, Choosing the job sequence and processing times to minimize total processing plus #ow cost on a single machine, Operations Research 28 (1980) 1155}1167. [15] R.G. Vickson, Two single machine sequencing problems involving controllable job processing times, AIIE Transactions 12 (1980) 258}262.