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Due-date assignment and machine scheduling in a low machine-rate situation with stochastic processing times
Computers & Operations Research 40 (2013) 1100–1108
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Due-date assignment and machine scheduling in a low machine-rate situation with stochastic processing times Mehdi Iranpoor a, S.M.T. Fatemi Ghomi a,*, M. Zandieh b a b
Department of Industrial Engineering, Amirkabir University of Technology, 424 Hafez Avenue, 1591634311 Tehran, Iran Department of Industrial Management, Management and Accounting Faculty, Shahid Beheshti University, G.C., Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Available online 27 November 2012
Due date assignment combined with shop floor scheduling has attracted enormous amount of research, in recent years. In many make-to-order situations, the processing times are not known exactly in advance. Further, machine rates are not constant at different times. We assume the case where the machine rate is low at the beginning of the scheduling horizon. However, it can be brought back to normal rate by performing a maintenance activity. The problem includes assigning due-dates and scheduling the jobs and maintenance activity on a single machine where the processing times are stochastic. The objective is minimizing the total cost of lengths of quoted due-dates and expected deviations of completion times from declared due-dates. The optimal solutions of medium-sized problems are found by solving some nonlinear programming models. For larger problems, robust metaheuristics are developed and their performances are statistically analyzed. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Due-date assignment Stochastic processing times Maintenance activity Nonlinear programming Robust metaheuristics
1. Introduction Traditionally, the maintenance plan was usually considered as a set of constraints of machine unavailability for shop floor scheduling task. Kacem et al. developed optimal solution approaches minimizing total weighted completion time of machine scheduling problems with a given period of machine unavailability [1]. Ji et al. studied a machine scheduling problem with periodic maintenance minimizing the makespan [2]. They proved that the longest processing time priority rule generates the best approximation of optimal solution. On the other hand, numerous studies have considered random machine breakdowns in the context of scheduling problems. Liu et al. considered a single machine scheduling problem with dynamic job arrival and multiple random breakdowns [3]. They developed a multi-population genetic algorithm trying to minimize total weighted tardiness and a stability measure. Cai et al. studied a machine scheduling problem with random occurrence of breakdowns and deteriorating jobs and makespan criteria [4]. Tang and Zhao developed a dynamic programming algorithm for machine scheduling minimizing squared deviation of completion times from a common due-date [5]. The studied situation includes both random breakdowns and stochastic processing times.
*
Corresponding author. Tel.: þ 98 2164545381; fax: þ98 2166954569. E-mail address: [email protected] (S.M.T. Fatemi Ghomi).
0305-0548/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cor.2012.11.013
In the last decade, rate-modifying maintenance activity (maintenance activity, for short) has emerged. This type of maintenance is performed at low-rate processing situations and upgrades the machine processing rate. Maintenance activity has been integrated with machine scheduling problems. Examples include common due-date assignment and maintenance scheduling minimizing total earliness, tardiness, and due-date assignment costs [6] and common due-window assignment and maintenance scheduling with/without deteriorating jobs ([7] and [8], respectively). All of such studies regard that the processing times are deterministic. From the customer’s point of view, order lead-time and on time delivery are among the most important suppliers’ competence criteria [9]. This has brought about numerous studies considering the integrated problem of scheduling and due-date quotation. Shabtay [10] considered a machine scheduling minimizing holding, batch delivery, long lead time, and tardiness costs. Li et al. [11] studied machine scheduling in group technology environment minimizing total due-date, earliness, tardiness, and flow time costs. Gordon and Strusevich [12] assumed that the processing times are position-dependent and the objective is minimizing total cost of due-date, earliness, and number of tardy jobs. Shabtay et al. [13] studied several machine scheduling problems with/without controllable processing times and group technology assumption minimizing total cost of due-dates, earliness, tardiness, and resource usage. The integrated problem of due-date assignment and scheduling becomes more challenging when the processing times are
M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
stochastic. However, most of the stochastic models either work with completion-time based criteria (e.g. [14]) or assume that the due-dates are exogenous (e.g. [15,16]). A few studies have considered setting due-dates in stochastic environments, some of which are presented as follows. Portougal and Trietsch [17] studied this problem on a single machine minimizing total expected earliness and tardiness costs. The due-dates were assigned following the given service levels. They proved that some heuristic is asymptotically optimal. Baker and Trietsch [18] studied several problems with due-date cost and service level consideration. For some cases, asymptotically optimal heuristics were developed. Xia et al. [19] worked over the problem with the criterion composed of penalties on earliness, tardiness, and long assigned due-dates. They approximated the objective function and developed a heuristic to find nearly optimal robust solutions. None of these studies consider any type of maintenance. In the current paper, the integrated problem of due-date quotation and scheduling of jobs and maintenance activity on a single machine is studied. The processing times of jobs are stochastic variables with known mean and variance. In practice, long due-dates can result in sales opportunity loss [10]. On the other hand, declaring unreliable short due-dates can cause tardiness, loss of future sales, or holding costs [10]. The objective of this paper is minimizing total penalties of quoted due-dates and their expected discrepancies with completion times. Clearly, this objective is along with the practical concerns of declaring due-dates which are both short and reliable. The penalty on declared due-dates encourages declaring short lead-times, while the penalty on expected discrepancies precludes quoting unreliable due-dates. The objective function is non-regular. So, the optimal schedules may include machine idle times. So, either the optimal values of such idle intervals should be calculated (like [20]) or it should be proved that no optimal solution contains machine idle time (like [21]). For the studied problem, it is proved that inserting idle times worsens the objective value of any arbitrary solution. The paper has the following structure. Section 2 explains the problem formally and introduces the notations. Section 3 derives the optimal terms of due-dates for any specific schedule of jobs and maintenance activity. Section 4 develops an approach to find the global optimal schedules. Except for one special case, finding the optimal solutions seems unlikely in a reasonable time. Hence, in Section 5, two powerful metaheuristics are developed and tuned by a novel robust design method. In Section 6, the computational analysis evaluates both the efficiency and effectiveness of metaheuristics. Finally, conclusion and the interesting areas of future relevant research are overviewed in Section 7.
2. Problem definition and notations This paper studies a machine scheduling problem in which the processing times of jobs are stochastic variables with known values of mean and variance. However, there is no knowledge about their entire distribution. At the beginning of the scheduling horizon, the machine is in a low rate situation and can be brought back to normal state by performing a maintenance activity. Both before and after performing the maintenance, the stochastic variables of processing times are statistically independent. The machine can process one job at a time. However, during the maintenance, the machine is turned off. Moreover, no preemption is allowed. During the low machine rate, the mean and variance of processing time for job j are tj and vj, respectively. Performing the
1101
maintenance activity enhances the machine rate by factor l. The required time to perform the maintenance is a constant denoted by R. Let [j] denote the index of job scheduled as the jth job. Further, r denotes the position of maintenance activity. Setting maintenance activity at position r means it is scheduled immediately before job [r]. Other parameters are listed here: n
g y E[Y] V[Y] s[Y]
Number of jobs Unit cost of promised value of due-date Unit cost of deviation from promised due-date Expected value of arbitrary stochastic variable Y Variance of arbitrary stochastic variable Y Standard deviation of arbitrary stochastic variable Y
Further, the decision variables, together with r, include ( 1; if job i is scheduled as the jth job xij ¼ 0; otherwise
dj Cj
Due-date assigned to job j Completion time of job j
As shown in (1), the objective function is the total due-date assignment cost and expectation of squared bias from due-date. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X gdj þ y E½ C j dj 2 ð1Þ j¼1
The square root in (1) seems necessary for the sake of dimensional homogeneity. The term under the square root can be written as E2[Cj dj] þV[Cj dj]. It is similar to the meansquare-error (MSE) which is very common criterion in the literature [22,23].
3. Derivation of optimal due-date values
Theorem 1. Given the schedule of jobs and maintenance activity, the optimal values of due-dates are calculated through (2). 8 < 0; g dj ¼ EC pffiffiffiffiffiffiffiffiffiffi s Cj ; j 2 : 2 n
y g
yrg y4g
j ¼ 1,. . .,n
ð2Þ
Proof. Let f(dj) be the component of (1) for job j (that is, f dj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gdj þ y E½ C j dj 2 ). Clearly, (1) is separable with respect to djs. So, the optimal values of due-dates can be found, independently. S The domain of f(dj) is R þ {0}. The first derivative of f(dj) is shown in (3). y dj E½C j @f ð3Þ dj ¼ g þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @dj 2 dj 2E½C j dj þ E½C j 2 n o @f Hence, the critical points of f(dj) are 0 and dj @d dj ¼ 0 . j The second derivative of f(dj) is calculated in (4). yV C j @2 f ¼ d ð4Þ
3=2 j 2 2 @dj dj 2E½C j dj þ E½C 2j Since expression (4) is positive, f(dj) is convex. Therefore, n @f dj @d dj ¼ j
ð@f =@dj Þðdj Þ ¼ 0 has at most one root and either 0 or
0g is global optimum [24]. The solutions of ð@f =@dj Þðdj Þ¼0
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M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
Given r, the problem of scheduling jobs can be represented by the mathematical programming model (6–14). n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X Min y Ej 2 þ V j ð6Þ
compose a subset of the roots of the following equation:
g2 d2j 2E½C j dj þ E½C 2j ¼ y2 dj E½C j 2
This equation can be simplified as shown in (5).
g2 y2 d2j 2 g2 y2 E½C j dj þ g2 E½C 2j y2 E2 ½C j ¼ 0
j¼1
ð5Þ
n X
The number of roots of the above quadratic equation depends on the value of its discriminant. Taking account of E[C2j ]¼ E2[Cj]þV[Cj], the discriminant of (5) would be as follows:
D ¼ 4 y2 g2 g2 V½C j
i¼1
There are three cases with y and g. The case where y o g. In this case, D o0. Thus, ð@f =@dj Þðdj Þ ¼ 0 has no roots. Hence, ð@f =@dj Þðdj Þ is either positive or negative. Since
yE½C j @f ð0Þ ¼ g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 40, @dj 2 E ½C j þ V½C j
n X
xij ¼ 1;
j ¼ 1,. . .,n
ð7Þ
xij ¼ 1;
i ¼ 1,. . .,n
ð8Þ
j¼1 j n X X
n X r 1 X
g dj,1 ¼ E C j þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi s C j y2 g2 g dj,2 ¼ E C j qffiffiffiffiffiffiffiffiffiffiffiffiffiffi s C j 2 y g2 Substituting these values in ð@f =@dj Þðdj Þ results that dj,2 is the only root of ð@f =@dj Þðdj Þ ¼ 0. The convexity of f(dj) concludes that this local minimum is global, too [24]. Hence in this case, g n dj ¼ E C j qffiffiffiffiffiffiffiffiffiffiffiffiffiffi s C j : 2 y g2
j n X X
4.1. The case where y r g Firstly, suppose that maintenance is placed at position r. P According to (2), dnj ¼0, 8j. Hence, (1) is reduced to y nj¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2 ½C j þ V½C j . Theorem 2. Any dominant schedule excludes machine idle-time. Proof. Consider any arbitrary sequence of jobs and maintenance activity. Inserting idle-time just increases the value of E[Cj] for some jobs. Hence, the resultant schedule is dominated by the original sequence.
t i xij0 Ej ¼ R;
j ¼ r,. . .,n
ð10Þ
i ¼ 1 j0 ¼ r
vi xij0 V j ¼ 0;
n X r 1 X i ¼ 1 j0 ¼ 1
2
t i xij0 þ l
j ¼ 1,. . .,r1
ð11Þ
j n X X
t i xij0 V j ¼ 0;
j ¼ r,. . .,n
ð12Þ
i ¼ 1 j0 ¼ r
Ej ,V j Z0;
j ¼ 1,. . .,n
ð13Þ
xij ¼ 0,1;
i ¼ 1,. . .,n, j ¼ 1,. . .,n
ð14Þ
Constraints (7) and (8) state that each job is assigned to one distinct position. Ej and Vj are short forms for E[C[j]] and V[C[j]] respectively, where 8 j X > > > t ½i ; jor > > X X > > > t þ Rþ l t ½i ; j Z r > ½i : i¼1
i¼r
8 j X > > > v½i ; > >
j or
j r 1 > X X > 2 > > v½i þ l v½i ; > : i¼1
This section develops an approach to find the optimal schedule of jobs and maintenance activity. The proposed approach solves the job scheduling problem optimally for every position of maintenance activity. By comparing the objective values of these schedules, the globally optimal schedule of jobs and maintenance activity is given.
ð9Þ
i ¼ 1 j0 ¼ 1
Vj ¼ 4. Scheduling jobs and maintenance activity
j ¼ 1,. . .,r1
j n X X
t i xij0 þ l
i ¼ 1 j0 ¼ 1
n
ð@f =@dj Þðdj Þ is positive and dj ¼0. The case where y ¼ g. The coefficients of d2j and dj are both zero. In this case, Eq. (5) is simplified to V[Cj] ¼0. Assuming that the processing times are stochastic, this equation is never true. Similar to the case discussed above, it is proved that dnj ¼ 0. The case where y 4 g. In this case, (5) has two distinct roots denoted by dj,1 and dj,2, where
t i xij0 Ej ¼ 0;
i ¼ 1 j0 ¼ 1
j Zr
i¼r
Constraints (9)–(12) calculate the values of Ej and Vj. At the end of the model, non-negative and binary variables are declared. The problem is nonlinear mixed integer programming. The objective function is nor convex neither concave. Thus, the problem is unlikely to be solvable in polynomial time [25]. Despite including the binary variables, the problem can be solved as a nonlinear programming (NLP) model. In other words, it is proved that constraint (14) can be replaced by non-negativity constraints while any optimal xij remains integral. Theorem 3. The solution of relaxed problem (6–13) attached with non-negativity constraints of xijs yields zero-one values for all xij s. Proof. Consider the artificial problem (15–17). 2
2
2
Min z ¼ ðCons1 þ l1 t 1 þ l2 t 2 Þ þ Cons2 þ l1 v1 þ l2 v2
ð15Þ
l1 þ l2 ¼ 1
ð16Þ
l1 , l2 Z0
ð17Þ
M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
where Cons1, Cons2, t1, and t2 are non-negative constants. The objective function is convex. The gradient of the objective function is as follows: rz ¼ 2t1 2 l1 þ 2Cons1 t1 þ 2t1 t2 l2 þ2v1 l1 2t2 2 l2 þ 2Cons1 t 2 þ 2t 1 t 2 l1 þ2v2 l2 Since rz¼ 0 has no feasible solution, no interior point can be optimal [24]. Hence, the optimal solution will never be a partial assignment (that includes assigning some jobs to multiple places partially) and the optimal values of l1 and l2 will always be integral. Therefore, in any schedule of the original problem, if any two jobs are assigned partially, adjusting the assignment so that each job is exactly assigned to one position improves the objective function. Hence, solutions with partial assignments are dominated. By utilizing a branch-and-reduce approach, GAMS/BARON solves problems which are not necessarily convex but factorable. So, we reformulate problem (6–14) as a factorable problem (18–27). Min
n X
zj
ð18Þ
j¼1 n X
1103
4.2. The case where y 4 g Suppose that maintenance is placed at position r. In this case, g n dj ¼ E C j qffiffiffiffiffiffiffiffiffiffiffiffiffiffi s C j : 2 y g2 Theorem 5. Any dominant schedule excludes machine idle-time. Proof. The proof is similar to proof of Theorem 2. The problem can be represented by the following mixedinteger nonlinear programming model: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r
n X Min gEj þ y2 g2 V j ð28Þ j¼1 n X
xij ¼ 1;
j ¼ 1,. . .,n
ð29Þ
xij ¼ 1;
i ¼ 1,. . .,n
ð30Þ
i¼1 n X j¼1
xij ¼ 1;
j ¼ 1,. . .,n
ð19Þ
i¼1
j n X X
t i xij0 Ej ¼ 0;
j ¼ 1,. . .,r1
ð31Þ
i ¼ 1 j0 ¼ 1 n X
xij ¼ 1;
i ¼ 1,. . .,n
ð20Þ
j¼1
n X r1 X
t i xij0 þ l
i ¼ 1 j0 ¼ 1 j n X X
t i xij0 Ej ¼ 0;
j ¼ 1,. . .,r1
ð21Þ
i ¼ 1 j0 ¼ 1
j n X X
j n X X
t i xij0 Ej ¼ R;
j ¼ r,. . .,n
ð32Þ
i ¼ 1 j0 ¼ r
vi xij0 V j ¼ 0;
j ¼ 1,. . .,r1
ð33Þ
i ¼ 1 j0 ¼ 1 n X r1 X
t i xij0 þ l
i ¼ 1 j0 ¼ 1 j n X X
j n X X
t i xij0 Ej ¼ R;
j ¼ r,. . .,n
ð22Þ
i ¼ 1 j0 ¼ r
vi xij0 V j ¼ 0;
2
t i xij0 þ l
i ¼ 1 j0 ¼ 1
j ¼ 1,. . .,r1
ð23Þ
j n X X
j ¼ r,. . .,n
ð24Þ
Ej ,V j ,zj Z0;
j ¼ 1,. . .,n
j ¼ 1,. . .,n
i, j ¼ 1,. . .,n
ð25Þ ð26Þ ð27Þ
where the objective function components are defined in (25). The problem is called mean-variance agreeable whenever for any pair of arbitrary jobs j and k, if tj Ztk, then vj Zvk [18]. Theorem 4 shows that in such a case, the problem can be solved in polynomial time. Theorem 4. When the processing times are mean-variance agreeable, scheduling the jobs in the non-decreasing order of tj þvj is optimal, regardless of the position of maintenance activity. Proof. The proof is based on the pairwise interchange argument [26]. Suppose that in an arbitrary schedule, job k is scheduled immediately before job k0 while tk rtk0 and vk rvk0 . It is obvious that interchanging the position of such jobs could never improve the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P value of nj¼ 1 E2j þ V j .
j n X X
t i xij0 V j ¼ 0;
j ¼ r,. . .,n
ð34Þ
i ¼ 1 j0 ¼ r
Ej , V j Z0; xij ¼ 0,1;
t i xij0 V j ¼ 0;
i ¼ 1 j0 ¼ r
Ej 2 þV j zj 2 ¼ 0;
xij Z 0;
2
t i xij0 þ l
i ¼ 1 j0 ¼ 1
i ¼ 1 j0 ¼ 1 n X r1 X
n X r 1 X
j ¼ 1,. . .,n i ¼ 1,. . .,n, j ¼ 1,. . .,n
ð35Þ ð36Þ
Constraints (29) and (30) are well-known assignment constraints. Constraints (31) through (34) set the values of Ej and Vj. The last two constraints declare non-negative and binary variables, respectively. The objective function is concave. Thus, the problem is unlikely to be solvable in polynomial time [27]. Expressions (29,30) compose the constraints of linear assignment problem (LAP). It is well-known that the coefficient matrix of LAP is totally unimodular [28]. Therefore, every extreme point of LAP is integer-valued. In addition, constraints (31–34) calculate the values of Ej and Vj and do not affect the feasible region of xijs. Thus, the extreme points of feasible region have integral xijs. Further, expression (28) is concave. Concave minimization over a compact convex set attains its minimum at an extreme point [25]. Hence, constraint (36) can be replaced by non-negativity constraints. Similar to Section 4.1, the problem is reformulated to a factorable structure as follows: Min
n X
gEj þzj
ð37Þ
j¼1 n X i¼1
xij ¼ 1;
j ¼ 1,. . .,n
ð38Þ
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M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
xij ¼ 1;
i ¼ 1,. . .,n
ð39Þ
j¼1 n X
j X
t i xij0 Ej ¼ 0;
j ¼ 1,. . .,r1
ð40Þ
i ¼ 1 j0 ¼ 1 n X r1 X
t i xij0 þ l
i ¼ 1 j0 ¼ 1 j n X X
j n X X
t i xij0 Ej ¼ R;
j ¼ r,. . .,n
ð41Þ
i ¼ 1 j0 ¼ r
vi xij0 V j ¼ 0;
j ¼ 1,. . .,r1
ð42Þ
i ¼ 1 j0 ¼ 1 n X r1 X
2
t i xij0 þ l
i ¼ 1 j0 ¼ 1
j n X X
y2 g2 V j zj 2 ¼ 0;
Ej , V j , zj Z0; xij Z 0;
t i xij0 V j ¼ 0;
j ¼ r,. . .,n
ð43Þ
i ¼ 1 j0 ¼ r
j ¼ 1,. . .,n
j ¼ 1,. . .,n
i, j ¼ 1,. . .,n
ð44Þ ð45Þ ð46Þ
Theorem 6. When the processing times are mean-variance agreeable, scheduling the jobs in the non-decreasing order of tj þvj is optimal regardless of the position of maintenance activity. Proof. The proof is based on the pairwise interchange argument [26] and is performed similar to the proof of Theorem 4.
5. Metaheuristics Generally, non-convex optimization problems are NP-hard. For the current specific problem, two special cases were presented for which the polynomial time algorithms can be developed. In other cases, the problem can be solved by NLP solvers like BARON. However, finding the globally optimal solutions is limited to medium-sized problems. For larger problems, we propose two rigorous metaheuristic approaches called Ant Colony Optimization (ACO) and Greedy Randomized Adaptive Search Procedure (GRASP). The selection of metaheuristics is based on an evaluation of prosperity and wide-spread use of various metaheuristics in the field of scheduling in recent years. Some published examples of successful use of GRASP in the scheduling context in the last two years include [29–31]. The use of ACO comparatively seems more well-liked than GRASP. Sample papers of using ACO in the scheduling area in the current year include [32–34].
5.1. Pseudo-codes of proposed metaheuristics The first metaheuristic used in this paper is the ACO [35]. In this algorithm each solution is constructed step by step while artificial ants pass through the solution elements. In the studied problem, each solution has nþ1 elements. Element 0 displays the position of maintenance activity, while elements 1 through n determine the index of jobs scheduled at positions 1 to n, respectively. Two criteria direct the ants selecting elements: heuristic value and pheromone value. Pheromone value summarizes the quality of solutions found by past ants which have traversed the elements during solution construction. However, heuristic values are problem specific. In this paper, the heuristic value for each position of maintenance activity is the objective value of the best known solution with maintenance activity located at that position. Further, the heuristic value for other components is the increment in the objective function resulted from adding a new component to the partial schedule of the last step. ACO has many variations. In this paper, a combination of rankbased and max–min ant systems [35] is used. Accordingly, only a subset of ants lay down pheromone. The value of pheromone laid by such ants is relevant to their objective values. Further, pheromone values are allowed to change in a predetermined interval. The ACO is equipped with a simple 2-swap local search algorithm. We call this metaheuristic as ACOh—an acronym for ACO and heuristic. The pseudo-code of the ACOh is shown in Fig. 1. Another metaheuristic applied in this paper is Greedy Randomized Adaptive Search (GRASP) [36]. Similar to the ACOh, this metaheuristic uses a combination of local search (LS) and constructive procedures. In the construction phase, GRASP uses a semigreedy approach. It ranks the alternative elements, based on their greedy values. The well-ranked elements are put into a candidate list. The selection of elements is made from candidate list based on probabilities. Local search is applied on any constructed solution. Encoding of the solutions for GRASP is performed in the same way as performed for ACOh. In this paper, the basic GRASP is equipped with elite set solutions, partial local search, and path relinking. Elite solutions are some of the best-found solutions. During the construction phase, they affect the probabilities of selecting each element based on a similarity index. Further, after each solution is generated by local search, a path is created between it and each of the elite solutions aiming at finding better solutions. This can be done by iteratively changing the position of two jobs. The pseudo-code of the proposed algorithm is shown in Fig. 2. 5.2. Simulation-based robust parameter setting The designer of experiments sometimes faces with nuisance factors. Despite being uncontrollable in practice, these factors can be controlled during the experiment. Such factors are called
Fig. 1. The ACOh metaheuristic.
M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
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Fig. 2. The proposed GRASP metaheuristic.
noises [37]. Robust design is the approved method of designing experiments when some noise factors exist. Robust parameter design was developed by Taguchi. The Taguchi method uses a crossed-array design. This method itself is inefficient, includes aliasing two-factor interactions with main effects, and there are some critiques to the way it does the analysis of variance [37]. Because of these limitations, utilizing the principles of Taguchi’s experimental design approach with integrated combined array design and dual response surface method seems both more efficient and more flexible [22]. Both of the aforementioned approaches (i.e. Taguchi and response surface methods) consider only a few levels of noise factors. The reason is that controlling such factors in physical settings is very difficult. However, if the experiments are run by a simulation on a computer, this difficulty disappears. Giovagnoli and Romano [38] proposed that in such cases, a simulation on noise factors is performed. Then the calculated sample mean and sample variance can be used by dual response surface methodology. Obviously, in the design of metaheuristics, the problem features are noise factors. In the current studied problem, the noise factors are size of the problem (n), coefficients of cost function (y, g), mean and variance of processing times of jobs (tj, vj ), the required time to perform maintenance activity (R), and rate of reduction of processing times (l). Let U[.,.] denote the Uniform distribution. During the simulation, we select n from U[20,100], y and g from U[1,10], tj from U[60,100], vj from U[5,20], R from U[200,1000], and l from U[0.3,0.7]. The dual response surface method is used which includes two responses of mean and variance to design a robust parameter setting. Each combination of controllable factors is run for 50 randomly generated set of noise factor settings. Mean square error (MSE) is used to combine the responses [22]. The stopping criterion of metaheuristics, in this section and in Section 6, is CPU time limit of n/5 seconds. All the test problems of Sections 5 and 6 are run on a laptop computer with Core2 Duo 2.4 GHz of Intel CPU and 2 GB of RAM.
5.2.1. Parameter setting for ACOh In this subsection, the parameters of ACOh algorithm are tuned. The design factors and their selected operating zones are shown in Table 1. The initial value of pheromone is held constant at the best practice level reported in [35]. In addition, the weight of pheromone trail is held fixed at value 1 [35]. Further, all pheromone values are reset to initial value if no improvement happens during 3 iterations. To fit a response surface, Box-Behnken design [37] is used resulting in 46 experimental runs. For each run, 50 randomly
Table 1 Low–high levels of design factors for ACOh. Factor name
Low value
High value
Number of ants Quotient of division Evaporation rate Weight of heuristic Number of pheromone-laying ants
10 3 0.1 1 2
60 10 0.8 5 6
Table 2 Low–high levels of design factors for GRASP. Factor name
Low value
High value
Number of partial LS Number of final LS Number of steps of LS Number of elite solutions Weight of similarity Threshold of dissimilarity
2 5 3 5 0.1 0.7
10 15 9 15 0.5 0.9
generated test problems were solved. For each run, the meansquare error (MSE) was calculated as the response of interest. Then, Minitab 16 is used to analyze the MSE values. The value of lack-of-fit test (p ¼0.148) indicates that the full quadratic model adequately fits the response surface at level of significance 0.1. In order to keep the text concise, the details of this test are not presented. Now, it is the time to optimize the response surface. Response optimizer of Minitab 16 is used in order to find the optimal levels of control factors. Running this optimizer finds the best design configuration as: Number of ants¼60 Division¼10 Evaporation rate ¼0.8 Weight of heuristic¼5 Number of pheromone laying ants ¼6
5.2.2. Parameter setting for GRASP The design factors and their selected operating zones are shown in Table 2. Box-Behnken design [37] results in 54 experimental runs. For each run, 50 randomly generated test problems were solved and the MSE measure was calculated. Minitab 16 is used to analyze the MSE values. The value of lack-of-fit test (p ¼0.017) indicates that the full quadratic model adequately fits the
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M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
response surface at level of significance 0.01.By running the Minitab optimizer, the best design configuration is found as: Number of partial local search¼8 Number of final local search¼ 11 Number of steps to perform local search¼3 Number of elite solutions ¼11 Weight of similarity index ¼0.471717 Threshold of dissimilarity¼0.746465
6. Computational analysis This section is dedicated to the performance evaluation of proposed metaheuristics. The characteristics of test problems are determined randomly in a manner discussed in Section 5.2 except for the problem size. Optimal solutions can be found by BARON for small-tomedium problem sizes. Remind that for a 70-job problem, 71 models should be solved for various positions of maintenance activity, each of which takes around 8500 s (see Table 4) which is around a total of 7 days. However, if the metaheuristics can predict the probable optimal position of maintenance activity, BARON can be run for only that position, hoping to find optimal solutions in reasonable times. Table 3 shows the percent of times that metaheuristics find the optimal position of maintenance activity for some randomly generated small to medium-sized problems. As shown in Table 3, both of the metaheuristics could find the optimal position of maintenance activity for all test problems. Hence, in the rest of the paper, for each problem instance only the model with the best-guess position of maintenance is solved by BARON. For each problem, such a position is derived by comparing the solutions found by metaheuristics. If both metaheuristic solutions contain the same maintenance position, this position is chosen as the best-guess. On the other hand, if the solutions have different maintenance position, the solution with better objective
Table 3 The Best-guess positions of maintenance vs. optimal positions. Problem size
# Instances
15 20 25 30
% True guesses
10 10 10 10
ACOh
GRASP
100 100 100 100
100 100 100 100
Table 4 Fixed and 95% confidence intervals of run times. Problem size
20 30 40 50 70 n
4 6 8 10 14
GRASP
4 6 8 10 14
Left endpoints of confidence intervals. Right endpoints of confidence intervals.
nn
value determines the best-guess position. The BARON’s solution for best-guess position is called best-guess optimal. In the remaining part of this subsection, the metaheuristic approaches are compared with BARON solver with respect to run times and objective values. Twenty test problems are generated randomly for every problem size of 20, 30, 40, 50, and 70 jobs. Table 4 summarizes the run times. Note that as mentioned in Section 5.2, the run times of metaheuristics are n/5. Hence they remain constant for various instances of each problem size. Obviously, the metaheuristics surpass BARON in run-time competition even for small problem sizes. Another measure of performance is the relative percentage deviation of each metaheuristic (RPDMet), which is computed through RPDMet ¼
ObjMet Opt 100% Opt
where ObjMet and Opt are the objective value of solution found by the corresponding metaheuristic and best-guess optimal solution, respectively. The average of RPD%s is depicted in Fig. 3. Both of the proposed metaheuristics are satisfactory since their average deviations from best-guess optima are less than 4%. However among them, GRASP seems more effective. Fig. 3 shows that GRASP is superior to ACOh on average. A statistical analysis is required to check the significance of this depicted dominance. The well-known paired t-test is suitable for paired values of RPD%s. One test is performed for each problem size. Table 5 shows the results of such tests. For all the problem sizes, the confidence intervals do not include zero, meaning that GRASP is statistically more effective than ACOh.
7. Conclusion and future directions
Run time (seconds) ACOh
Fig. 3. Average of RPD% as a measure of performance.
GAMS t1*
Mean
t2**
6.6 51.7 594.1 1089 5902
8.1 79.0 474.3 1672 8590
9.5 106.3 354.5 2255 11,279
Many manufacturing and service environments usually face with the challenging situation in which the processing times of some jobs are not known precisely in advance. The current paper studied a sequencing and due-date quotation problem in such an environment. Moreover, it was assumed that the processing rate of machine is increasable by a maintenance activity. Hence, the maintenance activity is also required to be scheduled. The scheduling criterion considered both delivery speed and leadtime reliability. Except for one special case, the problem required an NLP solver and took a long time to solve a large problem. Thus, two efficient
M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
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Table 5 Paired t-test and Confidence Interval: ACOh; GRASP. Problem size
Statistical analysis
20
N Mean ACOh 20 0.03777 GRASP 20 0.00000 Difference 20 0.03777 95% CI for mean difference: (0.02904, 0.04650) t-test of mean difference ¼0 (vs not¼ 0):
St Dev 0.01865 0.00001 0.01865 t-value ¼ 9.06 p-value ¼0.000
SE Mean 0.00417 0.00000 0.00417
30
N Mean ACOh 20 0.01751 GRASP 20 0.00006 Difference 20 0.01745 95% CI for mean difference: (0.01251, 0.02238) t-test of mean difference¼ 0 (vs not¼ 0):
St Dev 0.01055 0.00006 0.01055 t-value ¼7.40 p-value ¼ 0.000
SE Mean 0.00236 0.00001 0.00236
40
N Mean ACOh 20 0.01751 GRASP 20 0.00006 Difference 20 0.01745 95% CI for mean difference: (0.01251, 0.02238) t-test of mean difference¼ 0 (vs not¼ 0):
St Dev 0.01055 0.00006 0.01055 t-value ¼7.40 p-value ¼ 0.000
SE Mean 0.00236 0.00001 0.00236
50
N Mean ACOh 20 0.02010 GRASP 20 0.00011 Difference 20 0.01999 95% CI for mean difference: (0.01587, 0.02410) t-test of mean difference ¼ 0 (vs not¼ 0):
St Dev 0.00880 0.00007 0.00880 t-value¼ 10.16 p-value ¼ 0.000
SE Mean 0.00197 0.00002 0.00197
70
N Mean ACOh 20 0.02223 GRASP 20 0.00026 Difference 20 0.02197 95% CI for mean difference: (0.01973, 0.02421) t-test of mean difference¼ 0 (vs not¼ 0):
St Dev 0.00447 0.00047 0.00479 t-value ¼20.51 p-value¼ 0.000
SE Mean 0.00100 0.00011 0.00107
metaheuristics—ACOh and GRASP—were proposed. Statistical analysis indicated that both approaches are efficient, however GRASP outperforms ACOh. This paper is one of the few studies working on due-date quotation problem in stochastic environments. There is a lack-ofstudies considering this problem in multi-stage environments. Further, the stochastic environment composed of a combination of jobs with either exogenous or endogenous due-dates is another problem that is both worthy of study and common in practice.
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[12] Gordon VS, Strusevich VA. Single machine scheduling and due date assignment with positionally dependent processing times. European Journal of Operational Research 2009;198(1):57–62. [13] Shabtay D, Itskovich Y, Yedidsion L, Oron D. Optimal due date assignment and resource allocation in a group technology scheduling environment. Computers and Operations Research 2010;37(12):2218–28. [14] Wu CW, Brown KN, Beck JC. Scheduling with uncertain durations: modeling B-robust scheduling with constraints. Computers and Operations Research 2009;36(8):2348–56. [15] Soroush HM. Solving a stochastic single machine problem with initial idle time and quadratic objective. Computers and Operations Research 2010;37(7): 1328–47. [16] Soroush HM. Minimizing the weighted number of early and tardy jobs in a stochastic single machine scheduling problem. European Journal of Operational Research 2007;181(1):266–87. [17] Portougal V, Trietsch D. Setting due dates in a stochastic single machine environment. Computers and Operations Research 2006;33(6):1681–94. [18] Baker KR, Trietsch D. Safe scheduling: setting due dates in single-machine problems. European Journal of Operational Research 2009;196(1):69–77. [19] Xia Y, Chen B, Yue J. Job sequencing and due date assignment in a single machine shop with uncertain processing times. European Journal of Operational Research 2008;184(1):63–75. [20] Colin EC, Quinino RC. An algorithm for insertion of idle time in the singlemachine scheduling problem with convex cost functions. Computers and Operations Research 2005;32(9):2285–96. [21] Shabtay D. Due date assignments and scheduling a single machine with a general earliness/tardiness cost function. Computers and Operations Research 2008;35(5):1539–45. [22] Chen L, Integrated Robust Design using Response Surface Methodology and Constrained Optimization. Ontario: University of Waterloo; 2008. ¨ [23] Koksoy O. A nonlinear programming solution to robust multi-response quality problem. Applied Mathematics and Computation 2008;196(2):603–12. [24] Bazaraa MS, Sherali HD, Shetty CM. Nonlinear Programming Theory and Algorithms. 3rd ed. Hoboken, New Jersey: John Wiley and Sons; 2006. [25] Tuy H. Convex Analysis and Global Optimization. London: Kluwer Academic Publishers; 1998. [26] Pinedo ML. Scheduling: Theory, Algorithms and Systems. 3rd ed. New York: Springer; 2008. [27] Chinchuluun A, Rentsen E, Pardalos PM. A numerical method for concave programming problems. In: Jeyakumar V, Rubinov A, editors. Continuous Optimization. New York: Springer; 2005. p. 251–73. [28] Burkard R, Dell’Amico M, Martello S. Assignment Problems. Philadelphia: SIAM; 2009. [29] Damodaran P, Ve´lez-Gallego MC, Maya JA. GRASP approach for makespan minimization on parallel batch processing machines. Journal of Intelligent Manufacturing 2011;22(5):767–77.
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[30] Arroyo JEC, Pereira AAdS. A GRASP heuristic for the multi-objective permutation flowshop scheduling problem. International Journal of Advanced Manufacturing Technology 2011;55(5–8):741–53. [31] Witkowski T, Antczak A, Antczak P. Evaluation of the influence of selected production factors on makespan for different routes in FJSP problem. In: Zhu M, editor. Business, Economics, Financial Sciences, and Management. 1st ed. Berlin: Springer-Verlag; 2012. p. 373–82. [32] Keskinturk T, Yildirim MB, Barut M. An ant colony optimization algorithm for load balancing in parallel machines with sequence-dependent setup times. Computers and Operations Research 2012;39(6):1225–35. [33] Guo C, Zhibin J, Zhang H, Li N. Decomposition-based classified ant colony optimization algorithm for scheduling semiconductor wafer fabrication system. Computers and Industrial Engineering 2012;62(1):141–51.
[34] Xu R, Chen H, Li X. Makespan minimization on single batch-processing machine via ant colony optimization. Computers and Operations Research 2012;39(3):582–93. ¨ [35] Dorigo M, Stutzle T. Ant Colony Optimisation. 1st ed. Cambridge, Massachusetts: The MIT Press; 2004. [36] Resende MGC, Ribeiro CC. Greedy randomized adaptive search procedures: advances, hybridizations, and applications. In: Gendreau M, Potvin J-Y, editors. Handbook of Metaheuristics. 2nd ed. New York: Springer; 2010. p. 283–319. [37] Montgomery DC. Design and analysis of experiments. 7th ed. New York: Wiley; 2008. [38] Giovagnoli A, Romano D. Robust design via simulation experiments: a modified dual response surface approach. Quality and Reliability Engineering International 2008;24(4):401–16.
Contents lists available at SciVerse ScienceDirect
Computers & Operations Research journal homepage: www.elsevier.com/locate/caor
Due-date assignment and machine scheduling in a low machine-rate situation with stochastic processing times Mehdi Iranpoor a, S.M.T. Fatemi Ghomi a,*, M. Zandieh b a b
Department of Industrial Engineering, Amirkabir University of Technology, 424 Hafez Avenue, 1591634311 Tehran, Iran Department of Industrial Management, Management and Accounting Faculty, Shahid Beheshti University, G.C., Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Available online 27 November 2012
Due date assignment combined with shop floor scheduling has attracted enormous amount of research, in recent years. In many make-to-order situations, the processing times are not known exactly in advance. Further, machine rates are not constant at different times. We assume the case where the machine rate is low at the beginning of the scheduling horizon. However, it can be brought back to normal rate by performing a maintenance activity. The problem includes assigning due-dates and scheduling the jobs and maintenance activity on a single machine where the processing times are stochastic. The objective is minimizing the total cost of lengths of quoted due-dates and expected deviations of completion times from declared due-dates. The optimal solutions of medium-sized problems are found by solving some nonlinear programming models. For larger problems, robust metaheuristics are developed and their performances are statistically analyzed. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Due-date assignment Stochastic processing times Maintenance activity Nonlinear programming Robust metaheuristics
1. Introduction Traditionally, the maintenance plan was usually considered as a set of constraints of machine unavailability for shop floor scheduling task. Kacem et al. developed optimal solution approaches minimizing total weighted completion time of machine scheduling problems with a given period of machine unavailability [1]. Ji et al. studied a machine scheduling problem with periodic maintenance minimizing the makespan [2]. They proved that the longest processing time priority rule generates the best approximation of optimal solution. On the other hand, numerous studies have considered random machine breakdowns in the context of scheduling problems. Liu et al. considered a single machine scheduling problem with dynamic job arrival and multiple random breakdowns [3]. They developed a multi-population genetic algorithm trying to minimize total weighted tardiness and a stability measure. Cai et al. studied a machine scheduling problem with random occurrence of breakdowns and deteriorating jobs and makespan criteria [4]. Tang and Zhao developed a dynamic programming algorithm for machine scheduling minimizing squared deviation of completion times from a common due-date [5]. The studied situation includes both random breakdowns and stochastic processing times.
*
Corresponding author. Tel.: þ 98 2164545381; fax: þ98 2166954569. E-mail address: [email protected] (S.M.T. Fatemi Ghomi).
0305-0548/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cor.2012.11.013
In the last decade, rate-modifying maintenance activity (maintenance activity, for short) has emerged. This type of maintenance is performed at low-rate processing situations and upgrades the machine processing rate. Maintenance activity has been integrated with machine scheduling problems. Examples include common due-date assignment and maintenance scheduling minimizing total earliness, tardiness, and due-date assignment costs [6] and common due-window assignment and maintenance scheduling with/without deteriorating jobs ([7] and [8], respectively). All of such studies regard that the processing times are deterministic. From the customer’s point of view, order lead-time and on time delivery are among the most important suppliers’ competence criteria [9]. This has brought about numerous studies considering the integrated problem of scheduling and due-date quotation. Shabtay [10] considered a machine scheduling minimizing holding, batch delivery, long lead time, and tardiness costs. Li et al. [11] studied machine scheduling in group technology environment minimizing total due-date, earliness, tardiness, and flow time costs. Gordon and Strusevich [12] assumed that the processing times are position-dependent and the objective is minimizing total cost of due-date, earliness, and number of tardy jobs. Shabtay et al. [13] studied several machine scheduling problems with/without controllable processing times and group technology assumption minimizing total cost of due-dates, earliness, tardiness, and resource usage. The integrated problem of due-date assignment and scheduling becomes more challenging when the processing times are
M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
stochastic. However, most of the stochastic models either work with completion-time based criteria (e.g. [14]) or assume that the due-dates are exogenous (e.g. [15,16]). A few studies have considered setting due-dates in stochastic environments, some of which are presented as follows. Portougal and Trietsch [17] studied this problem on a single machine minimizing total expected earliness and tardiness costs. The due-dates were assigned following the given service levels. They proved that some heuristic is asymptotically optimal. Baker and Trietsch [18] studied several problems with due-date cost and service level consideration. For some cases, asymptotically optimal heuristics were developed. Xia et al. [19] worked over the problem with the criterion composed of penalties on earliness, tardiness, and long assigned due-dates. They approximated the objective function and developed a heuristic to find nearly optimal robust solutions. None of these studies consider any type of maintenance. In the current paper, the integrated problem of due-date quotation and scheduling of jobs and maintenance activity on a single machine is studied. The processing times of jobs are stochastic variables with known mean and variance. In practice, long due-dates can result in sales opportunity loss [10]. On the other hand, declaring unreliable short due-dates can cause tardiness, loss of future sales, or holding costs [10]. The objective of this paper is minimizing total penalties of quoted due-dates and their expected discrepancies with completion times. Clearly, this objective is along with the practical concerns of declaring due-dates which are both short and reliable. The penalty on declared due-dates encourages declaring short lead-times, while the penalty on expected discrepancies precludes quoting unreliable due-dates. The objective function is non-regular. So, the optimal schedules may include machine idle times. So, either the optimal values of such idle intervals should be calculated (like [20]) or it should be proved that no optimal solution contains machine idle time (like [21]). For the studied problem, it is proved that inserting idle times worsens the objective value of any arbitrary solution. The paper has the following structure. Section 2 explains the problem formally and introduces the notations. Section 3 derives the optimal terms of due-dates for any specific schedule of jobs and maintenance activity. Section 4 develops an approach to find the global optimal schedules. Except for one special case, finding the optimal solutions seems unlikely in a reasonable time. Hence, in Section 5, two powerful metaheuristics are developed and tuned by a novel robust design method. In Section 6, the computational analysis evaluates both the efficiency and effectiveness of metaheuristics. Finally, conclusion and the interesting areas of future relevant research are overviewed in Section 7.
2. Problem definition and notations This paper studies a machine scheduling problem in which the processing times of jobs are stochastic variables with known values of mean and variance. However, there is no knowledge about their entire distribution. At the beginning of the scheduling horizon, the machine is in a low rate situation and can be brought back to normal state by performing a maintenance activity. Both before and after performing the maintenance, the stochastic variables of processing times are statistically independent. The machine can process one job at a time. However, during the maintenance, the machine is turned off. Moreover, no preemption is allowed. During the low machine rate, the mean and variance of processing time for job j are tj and vj, respectively. Performing the
1101
maintenance activity enhances the machine rate by factor l. The required time to perform the maintenance is a constant denoted by R. Let [j] denote the index of job scheduled as the jth job. Further, r denotes the position of maintenance activity. Setting maintenance activity at position r means it is scheduled immediately before job [r]. Other parameters are listed here: n
g y E[Y] V[Y] s[Y]
Number of jobs Unit cost of promised value of due-date Unit cost of deviation from promised due-date Expected value of arbitrary stochastic variable Y Variance of arbitrary stochastic variable Y Standard deviation of arbitrary stochastic variable Y
Further, the decision variables, together with r, include ( 1; if job i is scheduled as the jth job xij ¼ 0; otherwise
dj Cj
Due-date assigned to job j Completion time of job j
As shown in (1), the objective function is the total due-date assignment cost and expectation of squared bias from due-date. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X gdj þ y E½ C j dj 2 ð1Þ j¼1
The square root in (1) seems necessary for the sake of dimensional homogeneity. The term under the square root can be written as E2[Cj dj] þV[Cj dj]. It is similar to the meansquare-error (MSE) which is very common criterion in the literature [22,23].
3. Derivation of optimal due-date values
Theorem 1. Given the schedule of jobs and maintenance activity, the optimal values of due-dates are calculated through (2). 8 < 0; g dj ¼ EC pffiffiffiffiffiffiffiffiffiffi s Cj ; j 2 : 2 n
y g
yrg y4g
j ¼ 1,. . .,n
ð2Þ
Proof. Let f(dj) be the component of (1) for job j (that is, f dj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gdj þ y E½ C j dj 2 ). Clearly, (1) is separable with respect to djs. So, the optimal values of due-dates can be found, independently. S The domain of f(dj) is R þ {0}. The first derivative of f(dj) is shown in (3). y dj E½C j @f ð3Þ dj ¼ g þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @dj 2 dj 2E½C j dj þ E½C j 2 n o @f Hence, the critical points of f(dj) are 0 and dj @d dj ¼ 0 . j The second derivative of f(dj) is calculated in (4). yV C j @2 f ¼ d ð4Þ
3=2 j 2 2 @dj dj 2E½C j dj þ E½C 2j Since expression (4) is positive, f(dj) is convex. Therefore, n @f dj @d dj ¼ j
ð@f =@dj Þðdj Þ ¼ 0 has at most one root and either 0 or
0g is global optimum [24]. The solutions of ð@f =@dj Þðdj Þ¼0
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M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
Given r, the problem of scheduling jobs can be represented by the mathematical programming model (6–14). n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X Min y Ej 2 þ V j ð6Þ
compose a subset of the roots of the following equation:
g2 d2j 2E½C j dj þ E½C 2j ¼ y2 dj E½C j 2
This equation can be simplified as shown in (5).
g2 y2 d2j 2 g2 y2 E½C j dj þ g2 E½C 2j y2 E2 ½C j ¼ 0
j¼1
ð5Þ
n X
The number of roots of the above quadratic equation depends on the value of its discriminant. Taking account of E[C2j ]¼ E2[Cj]þV[Cj], the discriminant of (5) would be as follows:
D ¼ 4 y2 g2 g2 V½C j
i¼1
There are three cases with y and g. The case where y o g. In this case, D o0. Thus, ð@f =@dj Þðdj Þ ¼ 0 has no roots. Hence, ð@f =@dj Þðdj Þ is either positive or negative. Since
yE½C j @f ð0Þ ¼ g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 40, @dj 2 E ½C j þ V½C j
n X
xij ¼ 1;
j ¼ 1,. . .,n
ð7Þ
xij ¼ 1;
i ¼ 1,. . .,n
ð8Þ
j¼1 j n X X
n X r 1 X
g dj,1 ¼ E C j þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi s C j y2 g2 g dj,2 ¼ E C j qffiffiffiffiffiffiffiffiffiffiffiffiffiffi s C j 2 y g2 Substituting these values in ð@f =@dj Þðdj Þ results that dj,2 is the only root of ð@f =@dj Þðdj Þ ¼ 0. The convexity of f(dj) concludes that this local minimum is global, too [24]. Hence in this case, g n dj ¼ E C j qffiffiffiffiffiffiffiffiffiffiffiffiffiffi s C j : 2 y g2
j n X X
4.1. The case where y r g Firstly, suppose that maintenance is placed at position r. P According to (2), dnj ¼0, 8j. Hence, (1) is reduced to y nj¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2 ½C j þ V½C j . Theorem 2. Any dominant schedule excludes machine idle-time. Proof. Consider any arbitrary sequence of jobs and maintenance activity. Inserting idle-time just increases the value of E[Cj] for some jobs. Hence, the resultant schedule is dominated by the original sequence.
t i xij0 Ej ¼ R;
j ¼ r,. . .,n
ð10Þ
i ¼ 1 j0 ¼ r
vi xij0 V j ¼ 0;
n X r 1 X i ¼ 1 j0 ¼ 1
2
t i xij0 þ l
j ¼ 1,. . .,r1
ð11Þ
j n X X
t i xij0 V j ¼ 0;
j ¼ r,. . .,n
ð12Þ
i ¼ 1 j0 ¼ r
Ej ,V j Z0;
j ¼ 1,. . .,n
ð13Þ
xij ¼ 0,1;
i ¼ 1,. . .,n, j ¼ 1,. . .,n
ð14Þ
Constraints (7) and (8) state that each job is assigned to one distinct position. Ej and Vj are short forms for E[C[j]] and V[C[j]] respectively, where 8 j X > > > t ½i ; jor > >
i¼r
8 j X > > > v½i ; > >
j or
j r 1 > X X > 2 > > v½i þ l v½i ; > : i¼1
This section develops an approach to find the optimal schedule of jobs and maintenance activity. The proposed approach solves the job scheduling problem optimally for every position of maintenance activity. By comparing the objective values of these schedules, the globally optimal schedule of jobs and maintenance activity is given.
ð9Þ
i ¼ 1 j0 ¼ 1
Vj ¼ 4. Scheduling jobs and maintenance activity
j ¼ 1,. . .,r1
j n X X
t i xij0 þ l
i ¼ 1 j0 ¼ 1
n
ð@f =@dj Þðdj Þ is positive and dj ¼0. The case where y ¼ g. The coefficients of d2j and dj are both zero. In this case, Eq. (5) is simplified to V[Cj] ¼0. Assuming that the processing times are stochastic, this equation is never true. Similar to the case discussed above, it is proved that dnj ¼ 0. The case where y 4 g. In this case, (5) has two distinct roots denoted by dj,1 and dj,2, where
t i xij0 Ej ¼ 0;
i ¼ 1 j0 ¼ 1
j Zr
i¼r
Constraints (9)–(12) calculate the values of Ej and Vj. At the end of the model, non-negative and binary variables are declared. The problem is nonlinear mixed integer programming. The objective function is nor convex neither concave. Thus, the problem is unlikely to be solvable in polynomial time [25]. Despite including the binary variables, the problem can be solved as a nonlinear programming (NLP) model. In other words, it is proved that constraint (14) can be replaced by non-negativity constraints while any optimal xij remains integral. Theorem 3. The solution of relaxed problem (6–13) attached with non-negativity constraints of xijs yields zero-one values for all xij s. Proof. Consider the artificial problem (15–17). 2
2
2
Min z ¼ ðCons1 þ l1 t 1 þ l2 t 2 Þ þ Cons2 þ l1 v1 þ l2 v2
ð15Þ
l1 þ l2 ¼ 1
ð16Þ
l1 , l2 Z0
ð17Þ
M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
where Cons1, Cons2, t1, and t2 are non-negative constants. The objective function is convex. The gradient of the objective function is as follows: rz ¼ 2t1 2 l1 þ 2Cons1 t1 þ 2t1 t2 l2 þ2v1 l1 2t2 2 l2 þ 2Cons1 t 2 þ 2t 1 t 2 l1 þ2v2 l2 Since rz¼ 0 has no feasible solution, no interior point can be optimal [24]. Hence, the optimal solution will never be a partial assignment (that includes assigning some jobs to multiple places partially) and the optimal values of l1 and l2 will always be integral. Therefore, in any schedule of the original problem, if any two jobs are assigned partially, adjusting the assignment so that each job is exactly assigned to one position improves the objective function. Hence, solutions with partial assignments are dominated. By utilizing a branch-and-reduce approach, GAMS/BARON solves problems which are not necessarily convex but factorable. So, we reformulate problem (6–14) as a factorable problem (18–27). Min
n X
zj
ð18Þ
j¼1 n X
1103
4.2. The case where y 4 g Suppose that maintenance is placed at position r. In this case, g n dj ¼ E C j qffiffiffiffiffiffiffiffiffiffiffiffiffiffi s C j : 2 y g2 Theorem 5. Any dominant schedule excludes machine idle-time. Proof. The proof is similar to proof of Theorem 2. The problem can be represented by the following mixedinteger nonlinear programming model: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r
n X Min gEj þ y2 g2 V j ð28Þ j¼1 n X
xij ¼ 1;
j ¼ 1,. . .,n
ð29Þ
xij ¼ 1;
i ¼ 1,. . .,n
ð30Þ
i¼1 n X j¼1
xij ¼ 1;
j ¼ 1,. . .,n
ð19Þ
i¼1
j n X X
t i xij0 Ej ¼ 0;
j ¼ 1,. . .,r1
ð31Þ
i ¼ 1 j0 ¼ 1 n X
xij ¼ 1;
i ¼ 1,. . .,n
ð20Þ
j¼1
n X r1 X
t i xij0 þ l
i ¼ 1 j0 ¼ 1 j n X X
t i xij0 Ej ¼ 0;
j ¼ 1,. . .,r1
ð21Þ
i ¼ 1 j0 ¼ 1
j n X X
j n X X
t i xij0 Ej ¼ R;
j ¼ r,. . .,n
ð32Þ
i ¼ 1 j0 ¼ r
vi xij0 V j ¼ 0;
j ¼ 1,. . .,r1
ð33Þ
i ¼ 1 j0 ¼ 1 n X r1 X
t i xij0 þ l
i ¼ 1 j0 ¼ 1 j n X X
j n X X
t i xij0 Ej ¼ R;
j ¼ r,. . .,n
ð22Þ
i ¼ 1 j0 ¼ r
vi xij0 V j ¼ 0;
2
t i xij0 þ l
i ¼ 1 j0 ¼ 1
j ¼ 1,. . .,r1
ð23Þ
j n X X
j ¼ r,. . .,n
ð24Þ
Ej ,V j ,zj Z0;
j ¼ 1,. . .,n
j ¼ 1,. . .,n
i, j ¼ 1,. . .,n
ð25Þ ð26Þ ð27Þ
where the objective function components are defined in (25). The problem is called mean-variance agreeable whenever for any pair of arbitrary jobs j and k, if tj Ztk, then vj Zvk [18]. Theorem 4 shows that in such a case, the problem can be solved in polynomial time. Theorem 4. When the processing times are mean-variance agreeable, scheduling the jobs in the non-decreasing order of tj þvj is optimal, regardless of the position of maintenance activity. Proof. The proof is based on the pairwise interchange argument [26]. Suppose that in an arbitrary schedule, job k is scheduled immediately before job k0 while tk rtk0 and vk rvk0 . It is obvious that interchanging the position of such jobs could never improve the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P value of nj¼ 1 E2j þ V j .
j n X X
t i xij0 V j ¼ 0;
j ¼ r,. . .,n
ð34Þ
i ¼ 1 j0 ¼ r
Ej , V j Z0; xij ¼ 0,1;
t i xij0 V j ¼ 0;
i ¼ 1 j0 ¼ r
Ej 2 þV j zj 2 ¼ 0;
xij Z 0;
2
t i xij0 þ l
i ¼ 1 j0 ¼ 1
i ¼ 1 j0 ¼ 1 n X r1 X
n X r 1 X
j ¼ 1,. . .,n i ¼ 1,. . .,n, j ¼ 1,. . .,n
ð35Þ ð36Þ
Constraints (29) and (30) are well-known assignment constraints. Constraints (31) through (34) set the values of Ej and Vj. The last two constraints declare non-negative and binary variables, respectively. The objective function is concave. Thus, the problem is unlikely to be solvable in polynomial time [27]. Expressions (29,30) compose the constraints of linear assignment problem (LAP). It is well-known that the coefficient matrix of LAP is totally unimodular [28]. Therefore, every extreme point of LAP is integer-valued. In addition, constraints (31–34) calculate the values of Ej and Vj and do not affect the feasible region of xijs. Thus, the extreme points of feasible region have integral xijs. Further, expression (28) is concave. Concave minimization over a compact convex set attains its minimum at an extreme point [25]. Hence, constraint (36) can be replaced by non-negativity constraints. Similar to Section 4.1, the problem is reformulated to a factorable structure as follows: Min
n X
gEj þzj
ð37Þ
j¼1 n X i¼1
xij ¼ 1;
j ¼ 1,. . .,n
ð38Þ
1104 n X
M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
xij ¼ 1;
i ¼ 1,. . .,n
ð39Þ
j¼1 n X
j X
t i xij0 Ej ¼ 0;
j ¼ 1,. . .,r1
ð40Þ
i ¼ 1 j0 ¼ 1 n X r1 X
t i xij0 þ l
i ¼ 1 j0 ¼ 1 j n X X
j n X X
t i xij0 Ej ¼ R;
j ¼ r,. . .,n
ð41Þ
i ¼ 1 j0 ¼ r
vi xij0 V j ¼ 0;
j ¼ 1,. . .,r1
ð42Þ
i ¼ 1 j0 ¼ 1 n X r1 X
2
t i xij0 þ l
i ¼ 1 j0 ¼ 1
j n X X
y2 g2 V j zj 2 ¼ 0;
Ej , V j , zj Z0; xij Z 0;
t i xij0 V j ¼ 0;
j ¼ r,. . .,n
ð43Þ
i ¼ 1 j0 ¼ r
j ¼ 1,. . .,n
j ¼ 1,. . .,n
i, j ¼ 1,. . .,n
ð44Þ ð45Þ ð46Þ
Theorem 6. When the processing times are mean-variance agreeable, scheduling the jobs in the non-decreasing order of tj þvj is optimal regardless of the position of maintenance activity. Proof. The proof is based on the pairwise interchange argument [26] and is performed similar to the proof of Theorem 4.
5. Metaheuristics Generally, non-convex optimization problems are NP-hard. For the current specific problem, two special cases were presented for which the polynomial time algorithms can be developed. In other cases, the problem can be solved by NLP solvers like BARON. However, finding the globally optimal solutions is limited to medium-sized problems. For larger problems, we propose two rigorous metaheuristic approaches called Ant Colony Optimization (ACO) and Greedy Randomized Adaptive Search Procedure (GRASP). The selection of metaheuristics is based on an evaluation of prosperity and wide-spread use of various metaheuristics in the field of scheduling in recent years. Some published examples of successful use of GRASP in the scheduling context in the last two years include [29–31]. The use of ACO comparatively seems more well-liked than GRASP. Sample papers of using ACO in the scheduling area in the current year include [32–34].
5.1. Pseudo-codes of proposed metaheuristics The first metaheuristic used in this paper is the ACO [35]. In this algorithm each solution is constructed step by step while artificial ants pass through the solution elements. In the studied problem, each solution has nþ1 elements. Element 0 displays the position of maintenance activity, while elements 1 through n determine the index of jobs scheduled at positions 1 to n, respectively. Two criteria direct the ants selecting elements: heuristic value and pheromone value. Pheromone value summarizes the quality of solutions found by past ants which have traversed the elements during solution construction. However, heuristic values are problem specific. In this paper, the heuristic value for each position of maintenance activity is the objective value of the best known solution with maintenance activity located at that position. Further, the heuristic value for other components is the increment in the objective function resulted from adding a new component to the partial schedule of the last step. ACO has many variations. In this paper, a combination of rankbased and max–min ant systems [35] is used. Accordingly, only a subset of ants lay down pheromone. The value of pheromone laid by such ants is relevant to their objective values. Further, pheromone values are allowed to change in a predetermined interval. The ACO is equipped with a simple 2-swap local search algorithm. We call this metaheuristic as ACOh—an acronym for ACO and heuristic. The pseudo-code of the ACOh is shown in Fig. 1. Another metaheuristic applied in this paper is Greedy Randomized Adaptive Search (GRASP) [36]. Similar to the ACOh, this metaheuristic uses a combination of local search (LS) and constructive procedures. In the construction phase, GRASP uses a semigreedy approach. It ranks the alternative elements, based on their greedy values. The well-ranked elements are put into a candidate list. The selection of elements is made from candidate list based on probabilities. Local search is applied on any constructed solution. Encoding of the solutions for GRASP is performed in the same way as performed for ACOh. In this paper, the basic GRASP is equipped with elite set solutions, partial local search, and path relinking. Elite solutions are some of the best-found solutions. During the construction phase, they affect the probabilities of selecting each element based on a similarity index. Further, after each solution is generated by local search, a path is created between it and each of the elite solutions aiming at finding better solutions. This can be done by iteratively changing the position of two jobs. The pseudo-code of the proposed algorithm is shown in Fig. 2. 5.2. Simulation-based robust parameter setting The designer of experiments sometimes faces with nuisance factors. Despite being uncontrollable in practice, these factors can be controlled during the experiment. Such factors are called
Fig. 1. The ACOh metaheuristic.
M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
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Fig. 2. The proposed GRASP metaheuristic.
noises [37]. Robust design is the approved method of designing experiments when some noise factors exist. Robust parameter design was developed by Taguchi. The Taguchi method uses a crossed-array design. This method itself is inefficient, includes aliasing two-factor interactions with main effects, and there are some critiques to the way it does the analysis of variance [37]. Because of these limitations, utilizing the principles of Taguchi’s experimental design approach with integrated combined array design and dual response surface method seems both more efficient and more flexible [22]. Both of the aforementioned approaches (i.e. Taguchi and response surface methods) consider only a few levels of noise factors. The reason is that controlling such factors in physical settings is very difficult. However, if the experiments are run by a simulation on a computer, this difficulty disappears. Giovagnoli and Romano [38] proposed that in such cases, a simulation on noise factors is performed. Then the calculated sample mean and sample variance can be used by dual response surface methodology. Obviously, in the design of metaheuristics, the problem features are noise factors. In the current studied problem, the noise factors are size of the problem (n), coefficients of cost function (y, g), mean and variance of processing times of jobs (tj, vj ), the required time to perform maintenance activity (R), and rate of reduction of processing times (l). Let U[.,.] denote the Uniform distribution. During the simulation, we select n from U[20,100], y and g from U[1,10], tj from U[60,100], vj from U[5,20], R from U[200,1000], and l from U[0.3,0.7]. The dual response surface method is used which includes two responses of mean and variance to design a robust parameter setting. Each combination of controllable factors is run for 50 randomly generated set of noise factor settings. Mean square error (MSE) is used to combine the responses [22]. The stopping criterion of metaheuristics, in this section and in Section 6, is CPU time limit of n/5 seconds. All the test problems of Sections 5 and 6 are run on a laptop computer with Core2 Duo 2.4 GHz of Intel CPU and 2 GB of RAM.
5.2.1. Parameter setting for ACOh In this subsection, the parameters of ACOh algorithm are tuned. The design factors and their selected operating zones are shown in Table 1. The initial value of pheromone is held constant at the best practice level reported in [35]. In addition, the weight of pheromone trail is held fixed at value 1 [35]. Further, all pheromone values are reset to initial value if no improvement happens during 3 iterations. To fit a response surface, Box-Behnken design [37] is used resulting in 46 experimental runs. For each run, 50 randomly
Table 1 Low–high levels of design factors for ACOh. Factor name
Low value
High value
Number of ants Quotient of division Evaporation rate Weight of heuristic Number of pheromone-laying ants
10 3 0.1 1 2
60 10 0.8 5 6
Table 2 Low–high levels of design factors for GRASP. Factor name
Low value
High value
Number of partial LS Number of final LS Number of steps of LS Number of elite solutions Weight of similarity Threshold of dissimilarity
2 5 3 5 0.1 0.7
10 15 9 15 0.5 0.9
generated test problems were solved. For each run, the meansquare error (MSE) was calculated as the response of interest. Then, Minitab 16 is used to analyze the MSE values. The value of lack-of-fit test (p ¼0.148) indicates that the full quadratic model adequately fits the response surface at level of significance 0.1. In order to keep the text concise, the details of this test are not presented. Now, it is the time to optimize the response surface. Response optimizer of Minitab 16 is used in order to find the optimal levels of control factors. Running this optimizer finds the best design configuration as: Number of ants¼60 Division¼10 Evaporation rate ¼0.8 Weight of heuristic¼5 Number of pheromone laying ants ¼6
5.2.2. Parameter setting for GRASP The design factors and their selected operating zones are shown in Table 2. Box-Behnken design [37] results in 54 experimental runs. For each run, 50 randomly generated test problems were solved and the MSE measure was calculated. Minitab 16 is used to analyze the MSE values. The value of lack-of-fit test (p ¼0.017) indicates that the full quadratic model adequately fits the
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M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
response surface at level of significance 0.01.By running the Minitab optimizer, the best design configuration is found as: Number of partial local search¼8 Number of final local search¼ 11 Number of steps to perform local search¼3 Number of elite solutions ¼11 Weight of similarity index ¼0.471717 Threshold of dissimilarity¼0.746465
6. Computational analysis This section is dedicated to the performance evaluation of proposed metaheuristics. The characteristics of test problems are determined randomly in a manner discussed in Section 5.2 except for the problem size. Optimal solutions can be found by BARON for small-tomedium problem sizes. Remind that for a 70-job problem, 71 models should be solved for various positions of maintenance activity, each of which takes around 8500 s (see Table 4) which is around a total of 7 days. However, if the metaheuristics can predict the probable optimal position of maintenance activity, BARON can be run for only that position, hoping to find optimal solutions in reasonable times. Table 3 shows the percent of times that metaheuristics find the optimal position of maintenance activity for some randomly generated small to medium-sized problems. As shown in Table 3, both of the metaheuristics could find the optimal position of maintenance activity for all test problems. Hence, in the rest of the paper, for each problem instance only the model with the best-guess position of maintenance is solved by BARON. For each problem, such a position is derived by comparing the solutions found by metaheuristics. If both metaheuristic solutions contain the same maintenance position, this position is chosen as the best-guess. On the other hand, if the solutions have different maintenance position, the solution with better objective
Table 3 The Best-guess positions of maintenance vs. optimal positions. Problem size
# Instances
15 20 25 30
% True guesses
10 10 10 10
ACOh
GRASP
100 100 100 100
100 100 100 100
Table 4 Fixed and 95% confidence intervals of run times. Problem size
20 30 40 50 70 n
4 6 8 10 14
GRASP
4 6 8 10 14
Left endpoints of confidence intervals. Right endpoints of confidence intervals.
nn
value determines the best-guess position. The BARON’s solution for best-guess position is called best-guess optimal. In the remaining part of this subsection, the metaheuristic approaches are compared with BARON solver with respect to run times and objective values. Twenty test problems are generated randomly for every problem size of 20, 30, 40, 50, and 70 jobs. Table 4 summarizes the run times. Note that as mentioned in Section 5.2, the run times of metaheuristics are n/5. Hence they remain constant for various instances of each problem size. Obviously, the metaheuristics surpass BARON in run-time competition even for small problem sizes. Another measure of performance is the relative percentage deviation of each metaheuristic (RPDMet), which is computed through RPDMet ¼
ObjMet Opt 100% Opt
where ObjMet and Opt are the objective value of solution found by the corresponding metaheuristic and best-guess optimal solution, respectively. The average of RPD%s is depicted in Fig. 3. Both of the proposed metaheuristics are satisfactory since their average deviations from best-guess optima are less than 4%. However among them, GRASP seems more effective. Fig. 3 shows that GRASP is superior to ACOh on average. A statistical analysis is required to check the significance of this depicted dominance. The well-known paired t-test is suitable for paired values of RPD%s. One test is performed for each problem size. Table 5 shows the results of such tests. For all the problem sizes, the confidence intervals do not include zero, meaning that GRASP is statistically more effective than ACOh.
7. Conclusion and future directions
Run time (seconds) ACOh
Fig. 3. Average of RPD% as a measure of performance.
GAMS t1*
Mean
t2**
6.6 51.7 594.1 1089 5902
8.1 79.0 474.3 1672 8590
9.5 106.3 354.5 2255 11,279
Many manufacturing and service environments usually face with the challenging situation in which the processing times of some jobs are not known precisely in advance. The current paper studied a sequencing and due-date quotation problem in such an environment. Moreover, it was assumed that the processing rate of machine is increasable by a maintenance activity. Hence, the maintenance activity is also required to be scheduled. The scheduling criterion considered both delivery speed and leadtime reliability. Except for one special case, the problem required an NLP solver and took a long time to solve a large problem. Thus, two efficient
M. Iranpoor et al. / Computers & Operations Research 40 (2013) 1100–1108
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Table 5 Paired t-test and Confidence Interval: ACOh; GRASP. Problem size
Statistical analysis
20
N Mean ACOh 20 0.03777 GRASP 20 0.00000 Difference 20 0.03777 95% CI for mean difference: (0.02904, 0.04650) t-test of mean difference ¼0 (vs not¼ 0):
St Dev 0.01865 0.00001 0.01865 t-value ¼ 9.06 p-value ¼0.000
SE Mean 0.00417 0.00000 0.00417
30
N Mean ACOh 20 0.01751 GRASP 20 0.00006 Difference 20 0.01745 95% CI for mean difference: (0.01251, 0.02238) t-test of mean difference¼ 0 (vs not¼ 0):
St Dev 0.01055 0.00006 0.01055 t-value ¼7.40 p-value ¼ 0.000
SE Mean 0.00236 0.00001 0.00236
40
N Mean ACOh 20 0.01751 GRASP 20 0.00006 Difference 20 0.01745 95% CI for mean difference: (0.01251, 0.02238) t-test of mean difference¼ 0 (vs not¼ 0):
St Dev 0.01055 0.00006 0.01055 t-value ¼7.40 p-value ¼ 0.000
SE Mean 0.00236 0.00001 0.00236
50
N Mean ACOh 20 0.02010 GRASP 20 0.00011 Difference 20 0.01999 95% CI for mean difference: (0.01587, 0.02410) t-test of mean difference ¼ 0 (vs not¼ 0):
St Dev 0.00880 0.00007 0.00880 t-value¼ 10.16 p-value ¼ 0.000
SE Mean 0.00197 0.00002 0.00197
70
N Mean ACOh 20 0.02223 GRASP 20 0.00026 Difference 20 0.02197 95% CI for mean difference: (0.01973, 0.02421) t-test of mean difference¼ 0 (vs not¼ 0):
St Dev 0.00447 0.00047 0.00479 t-value ¼20.51 p-value¼ 0.000
SE Mean 0.00100 0.00011 0.00107
metaheuristics—ACOh and GRASP—were proposed. Statistical analysis indicated that both approaches are efficient, however GRASP outperforms ACOh. This paper is one of the few studies working on due-date quotation problem in stochastic environments. There is a lack-ofstudies considering this problem in multi-stage environments. Further, the stochastic environment composed of a combination of jobs with either exogenous or endogenous due-dates is another problem that is both worthy of study and common in practice.
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