- Email: [email protected]
Heuristics for parallel-machine flexible-resource scheduling problems with unspecified job assignment
Computers & Operations Research 26 (1999) 143—155
Heuristics for parallel-machine flexible-resource scheduling problems with unspecified job assignment Richard L. Daniels!,1, Stella Y. Hua",*,2, Scott Webster#,3 ! School of Management, Georgia Institute of Technology, Atlanta, GA, USA " School of Business, University of Wisconsin-Madison, Madison, WI 53706-1323, USA # School of Management, Syracuse University, NY, USA Received March 1998; received in revised form June 1998
Scope and purpose This paper considers a parallel-machine scheduling problem in which the operational impact of resource flexibility is explored through machine flexibility as well as labor flexibility. The main contribution of this paper is the identification of the existence of the problem in practice, the development of two efficient heuristics, and the comparison and analysis of the proposed heuristics. Abstract In this paper, we examine the parallel-machine flexible-resource scheduling problem in which job assignment to machines are not specified (UPMFRS). The UPMFRS problem is NP-hard, motivating the
* Corresponding author. Fax: (608)263-3142; e-mail: [email protected]. 1 Richard L. Daniels is an Associate Professor of Operations Management in the DuPree School of Management at the Georgia Institute of Technology. He received his Ph.D. in Operations Management from the Anderson Graduate School of Management at UCLA. His research interests include manufacturing planning and control systems, resource flexibility and its impact on operational efficiency, and decision making under uncertainty. His articles have appeared in a number of journals, including Management Science, Operations Research, Naval Research Logistics, and IIE Transactions. 2 Stella Y. Hua is a Ph.D. candidate of Operations and Information Management in the School of Business at the University of Wisconsin-Madison. Her research interests include production planning and scheduling, integrated supply chain management, and new product development. 3 Scott Webster is an Associate Professor of Quantitative Methods in the School of Management at Syracuse University. He received a Ph.D. in Operations Management and Decision Sciences from Indiana University. Prior to pursuing his doctorate, he worked in the departments of Management Science and Finance at Whirlpool Corporation. His research focuses on a variety of logistics issues including production scheduling, design of distribution networks, and tactical policies for managing logistics systems. 0305-0548/98/$19.00#0.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 8 ) 0 0 0 5 4 - 9
144
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
development of effective heuristics that approximately determine the job assignment to machines and the allocation of resources to jobs. The paper compares a decomposition heuristic and a tabu-search heuristic, and concludes that the tabu-search heuristic is cost and quality effective in locating the near-optimal solutions. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Production scheduling; Flexible resource allocation; Heuristics.
1. Introduction In manufacturing environments, resource flexibility has become an important strategic tool. More and more manufacturers have begun to focus on controlling job processing times by allocating scarce and flexible resources to improve system performance as measured by schedule makespan. Resource flexibility provides an almost immediate return on investment, by breaking processing bottlenecks, improving system throughput rates, and lowering inventory levels. Labor flexibility is a common example of resource flexibility that can be achieved by cross-training workers to develop the skills required to perform different tasks associated with different machines. Daniels et al. [1] illustrate how labor flexibility can be utilized in a parallel-machine manufacturing environment, providing mathematical formulations and heuristic approachs for the parallelmachine flexible-resource scheduling problem (PMFRS). The results demonstrate that substantial improvements in operational performance can be achieved through the deployment of labor flexibility. This paper further investigates the operational impact of resource flexibility by incorporating machine flexibility as well as labor flexibility in the parallel-machine manufacturing environment. With machine flexibility, manufacturers can assign a wide range of products or parts to a group of identical machines. The pooling effect enables manufacturers to potentially reduce delivery time and operating costs. We explore the impact of resource flexibility by developing and analyzing heuristics for the identical parallel-machine flexible-resource scheduling problem with unspecified job assignment (UPMFRS). Existing resource allocation models can be categorized by the number of resources involved. Models in which job processing times are controlled by a single limited resource are presented in Vickson [2, 3], Van Wassenhove and Baker [4], and Panwalkar and Rajagopalan [5]. These models use variable linear costs, and balance the tradeoff between performance improvement and resource allocation cost. Daniels and Sarin [6] and Daniels [7] extend these resource allocation models to incorporate multiple resources. Theoretical results are provided in balancing the tradeoff between job sequencing or tardiness and the total amount of allocated resource. Resource flexibility has been explored in different operating environments. Daniels and Mazzola [8] address a flow shop scheduling problem with flexible resources (FRFS) and prove that FRFS is unary NP-hard. A heuristic procedure is developed to obtain approximate solutions to FRFS when job processing times are a function of the amount and mix of resources dedicated to an operation. Most of the existing models in the parallel machine scheduling environment deal with nonidentical machines, such as Hariri and Potts [9], Trick [10], and Daniels et al. [11]. Daniels et al. [11] identify the benefits of resource flexibility for nonidentical parallel machine scheduling problems by
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
145
controlling processing times through labor flexibility. The structure of the processing times inherently specifies a job-machine assignment. Research in the identical parallel-machine manufacturing environment has been restricted to the different service rates or setup times. For examples, Adiri and Yehudai [12] consider the scheduling of identical parallel processors whose service rates can vary among jobs. In So [13], several heuristics for the identical processors scheduling problem with setup times are developed. Resource flexibility issues in the identical parallel-machine manufacturing environment have not been adequately addressed. One setting consistent with Daniels et al. [1, 11] can be observed in the production of printed circuit boards, where n families of circuit boards are processed on a set of m manufacturing cells. The formulation of the problem assumes that different families of boards are assigned to specific cells due to their unique production requirements. In many actual production facilities, the manufacturing cells are identical. With minimum setup and changeover time, the cells can be configured to process different families of boards. Therefore, in these environments the assignment of families of boards to cells is not specified, but rather is a product of an independent assignment decision. This paper contributes to further understanding the operational impact of resource flexibility by relaxing the assumption of predetermined job assignments. In the resulting problem, job scheduling, resource allocation, and job assignment decisions are jointly determined to optimize system performance. Two versions of the UPMFRS problem can be formulated. In the dynamic UPMFRS problem, resources can be reassigned to cells any time a job is completed. The complexity of the dynamic problem motivates investigation of a static version of the UPMFRS problem, where resource allocation decisions remain fixed throughout the scheduling horizon. Thus, once a unit of resource such as a worker is assigned to a specific cell, that assignment becomes fixed for the duration of the schedule. In this paper, we model the problem in a static environment to further understand the joint contribution and implication of job assignment and resource flexibility. Our objective is to identify an efficient heuristic to determine the job assignment and resource allocation decisions that minimize system makespan. In the next section, we formulate the static version of the UPMFRS problem, and define an equivalent formulation based on a decomposition of the problem. A decomposition heuristic and a tabu-search heuristic for the UPMFRS problem are discussed in Sections 3 and 4. Computational experience with the heuristics over a large set of test problems is reported in Section 5. Conclusions are discussed in the final section.
2. Problem formulation Assume that there exists a set of independent single-operation jobs denoted by N"M1, 2, 2 , nN available for processing at time t"0. Each job can be processed on any one of m identical machines, denoted by M"M1, 2, 2 , mN. Each machine can process at most one job at a time, and job preemption is not allowed. The objective is to minimize the maximum completion time of the jobs across machines. Let C represent the completion time of the last job processed on machine j; then system makespan can j be specified as C "max MC N. .!9 j j
146
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
Since the assignment of jobs to machines is not specified, let N denote the set of jobs that are j ultimately assigned to machine j; thus, mn assignments are possible. Of these, the number of unique assignments is a Stirling number of the second kind, which can be computed by the following recursive equation [14]:
G HA B
n m~1 mn! + k! n k k/2 " m m!
GH
m !m k
.
The number of unique assignments ignores the sequence within each machine and assumes that each machine will process at least one job when n*m. A single resource with capacity R is shared among the m machines. Each job can be processed in any one of K modes: k"1, 2, 2 , K . The processing time of job i is denoted as p when r units of i i ik ik resource are allocated to machine j, and job i is also assigned to machine j. Once resource is assigned to a specific machine, this assignment remains fixed for the duration of the planning horizon. The amount of resource assigned to each machine j is denoted as r , where + r )R, j j|M j and each job on machine j is processed in some mode such that r )r . With the job assignment ik j and the allocation of the resource across machines specified, the set of jobs on each machine can be processed in any sequence without affecting system makespan. The static version of identical parallel machine scheduling problem with job assignment unspecified can be formulated as the following mixed-integer program. Minimize C "maxj3M + + P x y .!9 ik ik ij i k Subject to + + P x )C , j3M, ik ik .!9 i3N k3K j
(1)
i
+ r x )r , j3M, i3N , ik ik j j
(2)
+ r )R, j
(3)
+ x "1, i3N, ik
(4)
+ y "1, i3N, ik
(5)
x 3M0, 1N, y 3M0, 1N, r *0, i3N, k3K , j3M, ik ij j i
(6)
k3K j
j3M
k3K i
j3M
where
G
1 if job i is processed in mode k, x " ik 0 otherwise, and
G
1 if job i is processed in mode j, y " ij 0 otherwise.
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
147
In this formulation, constraints (1) define schedule makespan, constraints (2) assume that the operating mode of each job requires no more resource than is allocated to the associated machine, constraints (3) ensure that the total amount of resource allocated among machines does not exceed the total amount of available resource, constraints (4) guarantee that exactly one processing mode is selected for each job, and constraints (5) specify that each job is assigned to one and only one machine. Constraints (1)—(6) can be used to identify feasible solutions during the solution process. Ozdamar and Ulusoy [15] survey resource-constrained project scheduling problems, in which required resources are available in limited amounts. The problem formulated here differs from the general resource-constrained project scheduling problem because UPMFRS requires simultaneous optimization of two interrelated subproblems: job assignment and resource allocation. Job assignment determines the specific jobs associated with each machine. For each job assignment, a polynomial-time algorithm developed by Daniels et al. [11] can be used to allocate the available resource optimally. Job assignment can be represented by an assignment vector a, whose i5) component a(i)"j if and only if job i is assigned to machine j. Resource-allocation vector r defines the amount of resource allocated to the machines, i.e. r"(r , r , 2 , r ). Notice that the resource-allocation vector r uniquely defines the corresponding 1 2 m processing modes and processing times of the individual jobs. Therefore, any solution to the UPMFRS problem can be expressed as (a, r). Let C (a, r) represent the system makespan .!9 associated with this solution. Let A denote the set the feasible job assignments, and let P be the set of all resource-allocation policies. A solution (a, r) is therefore feasible to the UPMFRS problem if the corresponding solution satisfies constraints (1)—(6). An equivalent formulation of the UPMFRS problem is given by
G
H
C* "min min MC (a, r)N "min MC* (a)N. .!9 .!9 .!9 a3A r3P a3A It is well known that the m parallel-machine scheduling problem to minimize system makespan is NP-hard [16]. Therefore, the parallel-machine scheduling problem with unspecified job assignment is also NP-hard. The problem complexity motivates the development of heuristic approaches for solving the problem. In the next section, we describe a decomposition heuristic for the problem, and in Section 4, we present a tabu search approach.
3. A decomposition heuristic for the UPMFRS problem The decomposition heuristic (DH) described in this section is designed to generate approximate solutions to the static UPMFRS problem with modest computational effort. Initially, all jobs are unassigned and the minimum amount of resource is allocated to each machine. A longest process time (LPT) priority index is used to assign n jobs to m machines by sequentially assigning jobs to the first available machine in nonincreasing order of job processing time. Given this initial assignment, the polynomial-time algorithm of Daniels et al. [12] is applied to allocate the available resource to yield minimum makespan.
148
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
The resource-allocation algorithm begins by allocating the minimum amount of resource to each machine so that each job can be processed in its slowest mode. The makespan C and the .!9 machine j* with the longest completion time are then identified, i.e., C "C . The minimum j* .!9 amount of resource necessary to increase the mode of at least one job assigned to machine j* is determined and denoted * . Then an additional amount of resource * is allocated to machine j*, r r the operating modes of related jobs are increased, and the amount of available resource is decreased by * units. The process is repeated until the minimum amount of additional resource r * exceeds the amount of resource remaining, or until all jobs on machine j* are assigned their r fastest processing modes. Based on the updated job processing times, the LPT index is applied to generate a job-tomachine assignment, and the resource-allocation algorithm is then reapplied to generate the next resource-allocation policy. The heuristic terminates once a job-to-machine assignment encountered has been previously evaluated. The detailed heuristic procedure is summarized below. 3.1. Decomposition Heuristic (DH) Input: N jobs, M parallel identical machines, resource requirement r and associated processing ik times p for each job i and each operating mode k, and a total of R units of available resource. ik Output: N jobs assigned to each machine j with r units of resource allocated, satisfying the j j resource constraint and yielding a solution (aDH, rDH) with corresponding makespan CDH that .!9 approximates the minimum makespan. Step 0—Initialization. Let rL be the minimum required resource on each machine j, and set j CDH "R. .!9 Step 1—Define initial assignment a . Use the LPT index to determine an initial assignment a . 0 0 Step2—Solve the static PMFRS problem. Solve the static PMFRS problem with the current assignment, yielding C* Ma N and resource allocation vector r. If C* Ma N(CDH , then .!9 0 .!9 0 .!9 CDH "C* Ma N, aDH"a , and rDH"r. .!9 .!9 0 0 Step 3—Define the incumbent job assignment a. Set r "r for all jobs assigned to machine j, ik j and identify corresponding job processing times p . Again, use an LPT index to obtain new ik assignment a. Step 4—Compare the new assignment with the previous assignments. If the new assignment has been previously evaluated, then stop. Otherwise, a "a and go to Step 2. 0 The following example illustrates the mechanics of the decomposition heuristic. Consider 10 jobs to be processed on 3 identical machines with a total of 4 units of a flexible resource. The assignment of jobs to machines is unknown and job processing times depend on the amount of resource allocated to each job as shown in Table 1. The decomposition heuristic starts with a single unit of resource on each machine. The processing time of each job with one unit of resource is ranked in nonincreasing order to obtain an initial assignment a . For initial assignment a , jobs M1, 9, 2N are assigned to machine 1, jobs 0 0 M5, 7, 10N to machine 2, and jobs M4, 3, 6, 8N to machine 3. Given this assignment, machine 3 will complete processing at the latest time. Therefore, the remaining unit of resource is allocated to machine 3. Solving the static PMFRS problem yields C Ma , rN"78. .!9 0 The processing times of all jobs are updated according to the incumbent allocation of resource, and then ranked in nonincreasing order to yield new job assignment a, in which jobs M1, 9, 2N are
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
149
Table 1 Example problem Job
1 unit of resource
Processing time with 2 units resource
Processing time with 3 units of resource
1 2 3 4 5 6 7 8 9 10
40 15 32 33 35 7 28 6 20 15
30 11 24 25 26 5 21 5 15 11
27 10 21 22 23 5 19 4 13 10
assigned to machine 1, jobs M5, 3, 6, 8N to machine 2, and jobs M7, 4, 10N to machine 3. The new static PMFRS problem is solved, resulting in a makespan of 76. The process is repeated with the updated job processing times until this same job assignment is reached, terminating the process.
4. A tabu-search heuristic for the UPMFRS problem Tabu search is a neighborhood search procedure for solving optimization problems that is characterized by its ability to escape the trap of local optimality [17—20]. Computational experience on a wide variety of problems has demonstrated the method’s effectiveness in generating near-optimal solutions for complex problems. Skorin-Kapov [21] and Taillard [22] apply tabu search for the Quadratic Assignment Problem, yielding excellent solutions with modest computational effort. Daniels and Mazzola [23] employ a nested tabu-search heuristic for the FlexibleResource Flow Shop Scheduling problem. Hubscher and Glover [24] apply tabu search to the Parallel-Machine Scheduling problem. Daniels et al. [11] extend these applications to the ParallelMachine Flexible Resource Scheduling problem. We define a tabu-search heuristic (TSH) for the UPMFRS problem that searches among the set of possible job assignments, and evaluates alternative assignments using the static PMFRS algorithm of Daniels et al. [11]. Initially, we use the LPT index to construct an initial assignment of the jobs, and apply the static PMFRS algorithm to determine the allocation of resource to machines that minimize makespan given the incumbent job assignment. Alternative assignments are generated by considering all pairwise exchanges of jobs on different machines. Let a(x, y) be the assignment resulted from the exchange of job x and job y. For each job assignment, the static PMFRS algorithm is applied to determine the corresponding optimal resource-allocation policy. If an improvement in makespan is observed, then the heuristic solution is updated and the associated pairwise exchange is defined as a tabu move. Define ¹ to be the set of tabu moves, with ¹ the rth r tabu move, and ¹"M¹ , ¹ , 2 , ¹ N, i.e. a tabu move is discarded from the tabu list after 0 1 t t additions to the list. The tabu list is used to temporarily exclude certain elements of the
150
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
neighborhood from the search, and by varying the length of this list, the search can be diversified and intensified. Consistent with Glover [18], a tabu list length of seven is chosen for this application; however, we note that, as in Glover [18] and Taillard [22], the best length for the tabu list may depend on problem dimension. The tabu search is run for a total of ¹I iterations. The tabu search heuristic is detailed below. 4.1. Tabu-Search Heuristic (TSH) Input: N jobs, M parallel identical machines, resource requirement r and associated processing ik times p for each job i and each operating mode k, and a total of R units of available resource, and ik tabu list length t. Output: N jobs assigned to each machine j with r units of resource allocated, satisfying the j j resource constraint and yielding a solution (aDH, rDH) with corresponding makespan CDH that .!9 approximates the minimum makespan. Step 0—Initialization. Let rL be the minimum required resource on each machine j, and set j CTSH"R. Let ¹"0; q"0; ¹I"N2; counter"0, and COUNTER"N2/2. .!9 Step 1—Define initial assignment a. Use the LPT index to determine initial assignment a. Step2—Solve the static PMFRS problem for initial assignment a. Solve the static PMFRS problem given initial assignment, a, yielding C* Ma N and resource allocation vector r. Set .!9 0 CTSH"C* Ma N, aTSH"a, and rTSH"r. .!9 .!9 0 Step 3—Pairwise exchange. For current job assignment a, define ¼(a) as the set of all possible pairwise exchanges among jobs assigned to different machines in job assignment a. Step 4—Solve the static PMFRS problem for the job assignment in ¼ (a). For job assignment a@3¼(a), apply the static PMFRS algorithm to obtain resource-allocation policy r(a@) and corresponding makespan C* Ma@N. If C* Ma@N(CTSH , then CTSH"C* Ma@N, aTSH"a@, .!9 .!9 .!9 .!9 .!9 rTSH"r(a@), and counter"0. Otherwise, counter"counter#1. If counter is greater than COUNTER, then stop. Step 5—Modify tabu list. If q)t!1, then add the pairwise exchange that differentiates job assignments a and a@ to position q in tabu list ¹ and set q"q#1; otherwise, discard the pairwise exchange in position ¹ , move each remaining element of the tabu list up one position, and place 0 the pairwise exchange that differentiates job assignment a and a@ into position q in tabu list. Go to Step 4 if there are pairwise exchanges in ¼(a) that have not been considered. Set iteration" iteration#1. If iteration'¹I, stop; otherwise, go to Step 3. To illustrate the mechanics of the tabu search heuristic, we again use the example problem in Table 1. The procedure begins by ranking the jobs in nonincreasing order of their slowest processing times and assigning jobs to available machines according to this index. As in Section 3, this process yields a solution with system makespan C Ma, rN of 78. Given this solution, 33 unique .!9 pairwise exchanges between jobs on different machines are possible. The pairwise exchanges are chosen randomly at each iteration. Assume we first exchange job 1 and job 5, which leads to a new job assignment a (1, 5). The static PMFRS problem is then solved and the makespan is still 78. So we eliminate the exchange and keep the original job assignment a. We go through the next few exchanges, and the results remain same. The makespan is reduced to 76 after we solve the static PMFRS problem with the exchange of job 5 and job 4. We define the exchange of job 5 and job 4 as a tabu-move and add it to the tabu list. The process is repeated until no improved solution is
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
151
identified for COUNTER number of consecutive iterations, or until a total of TI iterations is completed.
5. Computational results The decomposition and tabu-search heuristics for the UPMFRS problem were coded in C and implemented on a Compaq Pentium 120 MHz computer with 24 meg memory. More than 800 test problems were used to compare the performance of the two heuristics. The key design variables included in the study are the number of jobs and machines, the available resource and its impact on operation processing times, and the location of processing bottlenecks. With the varying design variables, the test problems reflect a wide range of applications in parallel-machine manufacturing environments. In addition, the experiment is a natural extension of the computational work presented in Daniels et al. [11]. The test results thus provide insights into the impact of resource flexibility on parallel machine environments where both job assignment and resource allocation are important, and provide evidence of the computational performance of the proposed heuristics. Each test problem type is defined by specifying values for the number of jobs n, the number of machines m, the total amount of available resource R, and the sensitivity of job processing times to resource allocation, which is controlled by parameter b, with 0)b)1. The processing time of job i when processed in mode k3K is given by: i
C A BD
1 pL " 1!b ik k
PK . i1
Hence, for large values of b, the assignment of additional resource to a job has a greater effect on reducing the associated processing time than when b is small. This approach for modeling the impact of resource allocation on job processing times was first used in Daniels and Mazzola [23]. Three values of b, b"0.2, 0.5 and 0.8 are reflected in the experimental design. Ten replications of each problem type are included in the experiment design. Two sets of experiments were run. The first set of experiments were conducted on test problems consisting of n"10 and 15 jobs scheduled over m"3 and 4 machines with R"3, 4, and 5 units of resource available. The slowest processing times for each job are generated randomly from a uniform distribution of integers U [R, 50]. These problems are solved using both the decomposition and the tabu-search heuristic, and these approximate solutions are then compared with the optimal solutions obtained by an exhaustive search approach that considers every possible combination of job assignment and resource allocation. The second set of the test problems consists of 540 larger UPMFRS problems where n"20, 30, and 50 jobs must be scheduled on m"3, 4 and 5 machines with R"m#1 and m#2 units of available resource. The slowest processing times for each job in the larger problems are generated randomly from a uniform distribution of integers U[R, 250]. Since most of these problems are too large to be solved to optimality, only solutions from the tabu-search heuristic and the decomposition heuristic are compared. The computational performance of the procedures for the 10- and 15-job problems are presented in Tables 2 and 3 for m"3 and 4, respectively. The average and maximum deviation from the
152
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
Table 2 Heuristic performance on 3-machine problem n
R
Beta
Tabu search % above C* max Avg. (%)
Max. (%)
Decomposition % above C* max CPU
Avg. (%)
Max. (%)
Exhaustive search CPU
CPU
10 10 10
3 3 3
0.2 0.5 0.8
0.71 0.80 1.60
1.41 1.49 5.10
0.09 0.078 0.053
4.54 3.21 2.04
9.78 6.61 5.10
0.066 0.131 0.138
1.834 2.465 2.63
10 10 10
4 4 4
0.2 0.5 0.8
1.13 2.61 9.84
3.09 7.89 21.43
0.04 0.253 0.149
1.76 8.71 19.25
3.98 10.44 24.32
0.068 0.169 0.122
2.345 3.274 3.581
10 10 10
5 5 5
0.2 0.5 0.8
0.35 2.03 1.77
2.53 7.83 6.59
0.105 0.273 0.237
3.28 15.13 35.80
4.94 18.84 40.26
0.078 0.166 0.133
2.8 4.027 4.256
15 15 15
3 3 3
0.2 0.5 0.8
0.15 0.42 0.29
0.81 1.28 2.21
0.35 0.245 0.187
2.17 1.75 0.59
3.85 3.16 2.21
0.121 0.276 0.222
485.519 764.862 795.354
15 15 15
4 4 4
0.2 0.5 0.8
0.94 4.33 15.51
3.41 10.68 21.59
0.22 0.403 0.504
2.92 10.00 19.62
3.74 11.80 20.60
0.148 0.104 0.291
697.583 882.46 1057.393
15 15 15
5 5 5
0.2 0.5 0.8
0.39 1.19 3.13
1.78 4.73 8.51
0.3367 0.555 1.439
4.50 18.17 38.81
5.93 20.63 40.59
0.176 0.104 0.297
834.792 1117.091 1248.995
Table 3 Heuristic performance on 4-machine problem n
R
Beta
Tabu search % above C* max Avg. (%)
Max. (%)
Decomposition % above C* max CPU
Avg. (%)
Exhaustive search
Max. (%)
CPU
CPU
10 10 10
4 4 4
0.2 0.5 0.8
0.54 0.82 1.30
3.77 6.17 8.06
0.049 0.072 0.107
3.79 3.51 6.12
6.58 9.21 10.53
0.086 0.18 0.117
22.651 38.073 39.509
10 10 10
5 5 5
0.2 0.5 0.8
1.22 1.24 9.75
4.55 5.88 17.89
0.052 0.231 0.105
3.35 8.01 14.46
8.47 12.00 18.75
0.139 0.186 0.155
37.107 50.548 60.073
15 15 15
4 4 4
0.2 0.5 0.8
1.36 1.73 1.80
2.88 4.95 3.40
0.1505 0.204 0.22
3.95 3.45 4.32
8.47 4.95 7.32
0.1171 0.1471 0.09
36284.45 49006.02 48864.63
15 15 15
5 5 5
0.2 0.5 0.8
1.72 6.08 10.84
5.26 16.00 18.04
4.17 10.03 15.07
8.19 16.33 18.07
0.12 0.08 0.22
45371.65 48032.25 47989.30
0.2 0.33 0.27
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
153
Table 4 Heuristic performance of larger problem n
m
R
Beta
DH % above TSH
n
m
R
Beta
DH % above TSH
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5
4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7
0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8
6.25 13.90 27.22 8.54 25.54 47.93 3.14 4.36 22.20 7.40 10.89 31.54 3.60 6.88 10.43 4.48 6.84 15.11
30 30 30 30 30 30 30 30 30
4 4 4 5 5 5 5 5 5
6 6 6 6 6 6 7 7 7
0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8
5.38 15.20 24.21 2.52 3.71 6.31 3.24 6.44 11.11
30 30 30 30 30 30 30 30 30
3 3 3 3 3 3 4 4 4
4 4 4 5 5 5 5 5 5
0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8
5.69 13.15 18.65 9.24 24.62 45.77 2.03 6.75 13.08
50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50
3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5
4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7
0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8
2.22 5.19 17.26 8.90 17.90 46.11 2.78 2.28 1.14 3.33 5.22 10.96 0.84 1.28 1.47 3.34 1.16 4.05
optimal system makespan, and the average CPU times in seconds for both heuristics, as well as the exhaustive search, are reported. The results indicate that both the tabu-search heuristic and decomposition heuristic are effective in obtaining approximate solutions, as reflected by the average and maximum deviations from the optimal system makespan. The approximation errors associated with the decomposition heuristic average 8.24% with a maximum of 40.59%. The tabu-search heuristic generates solutions that are 2.31% above the optimal solution on average, with a maximum deviation of 21.59%. Therefore, the tabu-search procedure appears to outperform the decomposition procedure, since the computation times for the two approaches are not significantly different (the average computation times for the tabu-search procedure and the decomposition procedure are 0.23 seconds and 0.14 seconds, respectively, over the 300 problems). The computation times for the heuristics increase with the number of jobs and the number of resources. However, these solution times are extremely small compared to the computation time of
154
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
the exhaustive search, which averaged over 2.5 hours. These results clearly illustrate the cost/quality trade-off that underlies the UPMFRS problem. Table 4 show the computational performance of the heuristic on the set of larger UPMFRS problems. The improvement in solution quality of the tabu search heuristic over the decomposition heuristic is captured by the percentage measure (CDH !CTSH )/CTSH , where CTSH and CDH are the .!9 .!9 .!9 .!9 .!9 corresponding solutions for the tabu search and decomposition heuristics. This percentage improvement averages 11.27% with a maximum of 47.93% over the 540 problems. In addition, Table 4 show that the percentage of improvement increases as the total units of available resource R increase, and as the parameter b increases. This provides additional evidence of the effectiveness of tabu search in obtaining good solutions with reasonable computational effect.
6. Conclusions This paper presents and computationally investigates a decomposition heuristic and a tabusearch heuristic for the parallel-machine flexible-resource scheduling problem with unspecified job assignment. Both heuristics presented are based on a decomposition of the problem into job assignment and resource allocation subproblems. Over 800 problems were tested to compare the performance of the heuristics. This computational experience suggests that the tabu-search heuristic is most effective in obtaining close approximations of the optimal solution with modest computational effort. More research on the strategic significance of resource flexibility, and how this flexibility can be utilized to improve operational performance, is clearly needed. Future research should explore the dynamic version of the UPMFRS problem, where jobs and resources are reallocated over time in response to shifting bottlenecks within the system. Models that help determine how jobs should be released and assigned to production centers, and how resource should be deployed dynamically to optimize production efficiency and responsiveness would yield important contributions to this increasingly important area of production scheduling.
References [1] Daniels RL, Hoopes BJ, Mazzola JB. An analysis of heuristics for the parallel-machine flexible-resource scheduling problem. Fuqua chool of Business Working Paper, Duke University, USA, 1994. [2] Vickson RG. Two single-machine sequencing problem involving controllable job processing times. AIEE Transactions 1980;12:258—62. [3] Vickson RG. Choosing the job sequence and processing times to minimize processing plus flow cost on a single machine. Operations Research 1980;28:1155—67. [4] Van Wassenhove LN, Baker KR. A bicriterion approach to time cost trade-offs in sequencing. European Journal of Operational Research 1982;11:48—54. [5] Panwalker SS, Rajagopalan R. Single-machine sequencing with controllable processing times. European Journal of Operational Research 1992;59:298—302. [6] Daniels RL, Sarin RK. Single machine scheduling with controllable processing times and number of job tardy. Operations Research 1989;37:981—4. [7] Daniels RL, A multi-objective approach to resource allocation in single machine scheduling. European Journal of Operational Research 1990;48:226—41.
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
155
[8] Daniels RL, Mazzola JB. Flow shop scheduling with resource flexibility. Operations Research 1994;42:504—22. [9] Hariri AMA, Potts CN. Heuristics for scheduling unrelated parallel machines. Computers and Operations Research 1991;18:323—31. [10] Trick MA. Scheduling multiple variable-speed machines. Operations Research 1994;42:234—48. [11] Daniels RL, Hoopes BJ, Mazzola JB. Scheduling parallel manufacturing cells with resource flexibility. Management Science 1996;42:1260—76. [12] Adiri I, Yehudai Z. Scheduling on machines with variable service rates. Computers and Operations Research 1987;14:289—97. [13] So KC. Some heuristics for scheduling jobs on parallel machines with setups. Management Science 1990;36:467—75. [14] Ho JC, Chang YL. Minimizing the number of tardy jobs for m parallel machines. European Journal of Operational Research 1995;84:343—55. [15] Ozdamar L, Ulusoy G. A survey on the resource-constrained project scheduling problem. IIE Transactions 1995;27:574—86. [16] Lenstra JK, Rinnooy Kan AHG, Brucker P. Complexity of machine scheduling problems. Annals of Discrete Mathematics 1977;1:343—62. [17] Anderson EJ, Glass CA, Potts CN. Machine scheduling, in: Aarts E, Lenstra J, editors, Local search in combinatorial optimization, Chap 11 Wiley, Chichester, 1997. [18] Glover F. Tabu search: a tutorial. Interfaces 1990; 20:74—94. [19] Glover F. Tabu search — Part I. ORSA Journal on Computing 1989;1:190—206. [20] Glover F. Tabu search — Part II, ORSA Journal on Computing 1990;2:4—32. [21] Skorin-Kapov J. Tabu search applied to the quadratic assignment problem. ORSA Journal on Computing 1990;2:33—45. [22] Taillard E, Robust taboo search for the quadratic assignment problem. Parallel Computing 1991;17:443—55. [23] Daniels RL, Mazzola JB. A tabu-search heuristic for the flexible-resource flow shop scheduling problem. Annals of Operations Research 1993;41:207—30. [24] Hubscher R, Glover F. Applying tabu search with influential diversification to multiprocessor scheduling. Computers and Operations Research 1994;21:877—84. [25] Cheng TCE, Sin CCS. A state-of-the-art review of parallel-machine scheduling research. European Journal of Operational Research 1990;48:226—41.
Heuristics for parallel-machine flexible-resource scheduling problems with unspecified job assignment Richard L. Daniels!,1, Stella Y. Hua",*,2, Scott Webster#,3 ! School of Management, Georgia Institute of Technology, Atlanta, GA, USA " School of Business, University of Wisconsin-Madison, Madison, WI 53706-1323, USA # School of Management, Syracuse University, NY, USA Received March 1998; received in revised form June 1998
Scope and purpose This paper considers a parallel-machine scheduling problem in which the operational impact of resource flexibility is explored through machine flexibility as well as labor flexibility. The main contribution of this paper is the identification of the existence of the problem in practice, the development of two efficient heuristics, and the comparison and analysis of the proposed heuristics. Abstract In this paper, we examine the parallel-machine flexible-resource scheduling problem in which job assignment to machines are not specified (UPMFRS). The UPMFRS problem is NP-hard, motivating the
* Corresponding author. Fax: (608)263-3142; e-mail: [email protected]. 1 Richard L. Daniels is an Associate Professor of Operations Management in the DuPree School of Management at the Georgia Institute of Technology. He received his Ph.D. in Operations Management from the Anderson Graduate School of Management at UCLA. His research interests include manufacturing planning and control systems, resource flexibility and its impact on operational efficiency, and decision making under uncertainty. His articles have appeared in a number of journals, including Management Science, Operations Research, Naval Research Logistics, and IIE Transactions. 2 Stella Y. Hua is a Ph.D. candidate of Operations and Information Management in the School of Business at the University of Wisconsin-Madison. Her research interests include production planning and scheduling, integrated supply chain management, and new product development. 3 Scott Webster is an Associate Professor of Quantitative Methods in the School of Management at Syracuse University. He received a Ph.D. in Operations Management and Decision Sciences from Indiana University. Prior to pursuing his doctorate, he worked in the departments of Management Science and Finance at Whirlpool Corporation. His research focuses on a variety of logistics issues including production scheduling, design of distribution networks, and tactical policies for managing logistics systems. 0305-0548/98/$19.00#0.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 8 ) 0 0 0 5 4 - 9
144
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
development of effective heuristics that approximately determine the job assignment to machines and the allocation of resources to jobs. The paper compares a decomposition heuristic and a tabu-search heuristic, and concludes that the tabu-search heuristic is cost and quality effective in locating the near-optimal solutions. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Production scheduling; Flexible resource allocation; Heuristics.
1. Introduction In manufacturing environments, resource flexibility has become an important strategic tool. More and more manufacturers have begun to focus on controlling job processing times by allocating scarce and flexible resources to improve system performance as measured by schedule makespan. Resource flexibility provides an almost immediate return on investment, by breaking processing bottlenecks, improving system throughput rates, and lowering inventory levels. Labor flexibility is a common example of resource flexibility that can be achieved by cross-training workers to develop the skills required to perform different tasks associated with different machines. Daniels et al. [1] illustrate how labor flexibility can be utilized in a parallel-machine manufacturing environment, providing mathematical formulations and heuristic approachs for the parallelmachine flexible-resource scheduling problem (PMFRS). The results demonstrate that substantial improvements in operational performance can be achieved through the deployment of labor flexibility. This paper further investigates the operational impact of resource flexibility by incorporating machine flexibility as well as labor flexibility in the parallel-machine manufacturing environment. With machine flexibility, manufacturers can assign a wide range of products or parts to a group of identical machines. The pooling effect enables manufacturers to potentially reduce delivery time and operating costs. We explore the impact of resource flexibility by developing and analyzing heuristics for the identical parallel-machine flexible-resource scheduling problem with unspecified job assignment (UPMFRS). Existing resource allocation models can be categorized by the number of resources involved. Models in which job processing times are controlled by a single limited resource are presented in Vickson [2, 3], Van Wassenhove and Baker [4], and Panwalkar and Rajagopalan [5]. These models use variable linear costs, and balance the tradeoff between performance improvement and resource allocation cost. Daniels and Sarin [6] and Daniels [7] extend these resource allocation models to incorporate multiple resources. Theoretical results are provided in balancing the tradeoff between job sequencing or tardiness and the total amount of allocated resource. Resource flexibility has been explored in different operating environments. Daniels and Mazzola [8] address a flow shop scheduling problem with flexible resources (FRFS) and prove that FRFS is unary NP-hard. A heuristic procedure is developed to obtain approximate solutions to FRFS when job processing times are a function of the amount and mix of resources dedicated to an operation. Most of the existing models in the parallel machine scheduling environment deal with nonidentical machines, such as Hariri and Potts [9], Trick [10], and Daniels et al. [11]. Daniels et al. [11] identify the benefits of resource flexibility for nonidentical parallel machine scheduling problems by
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
145
controlling processing times through labor flexibility. The structure of the processing times inherently specifies a job-machine assignment. Research in the identical parallel-machine manufacturing environment has been restricted to the different service rates or setup times. For examples, Adiri and Yehudai [12] consider the scheduling of identical parallel processors whose service rates can vary among jobs. In So [13], several heuristics for the identical processors scheduling problem with setup times are developed. Resource flexibility issues in the identical parallel-machine manufacturing environment have not been adequately addressed. One setting consistent with Daniels et al. [1, 11] can be observed in the production of printed circuit boards, where n families of circuit boards are processed on a set of m manufacturing cells. The formulation of the problem assumes that different families of boards are assigned to specific cells due to their unique production requirements. In many actual production facilities, the manufacturing cells are identical. With minimum setup and changeover time, the cells can be configured to process different families of boards. Therefore, in these environments the assignment of families of boards to cells is not specified, but rather is a product of an independent assignment decision. This paper contributes to further understanding the operational impact of resource flexibility by relaxing the assumption of predetermined job assignments. In the resulting problem, job scheduling, resource allocation, and job assignment decisions are jointly determined to optimize system performance. Two versions of the UPMFRS problem can be formulated. In the dynamic UPMFRS problem, resources can be reassigned to cells any time a job is completed. The complexity of the dynamic problem motivates investigation of a static version of the UPMFRS problem, where resource allocation decisions remain fixed throughout the scheduling horizon. Thus, once a unit of resource such as a worker is assigned to a specific cell, that assignment becomes fixed for the duration of the schedule. In this paper, we model the problem in a static environment to further understand the joint contribution and implication of job assignment and resource flexibility. Our objective is to identify an efficient heuristic to determine the job assignment and resource allocation decisions that minimize system makespan. In the next section, we formulate the static version of the UPMFRS problem, and define an equivalent formulation based on a decomposition of the problem. A decomposition heuristic and a tabu-search heuristic for the UPMFRS problem are discussed in Sections 3 and 4. Computational experience with the heuristics over a large set of test problems is reported in Section 5. Conclusions are discussed in the final section.
2. Problem formulation Assume that there exists a set of independent single-operation jobs denoted by N"M1, 2, 2 , nN available for processing at time t"0. Each job can be processed on any one of m identical machines, denoted by M"M1, 2, 2 , mN. Each machine can process at most one job at a time, and job preemption is not allowed. The objective is to minimize the maximum completion time of the jobs across machines. Let C represent the completion time of the last job processed on machine j; then system makespan can j be specified as C "max MC N. .!9 j j
146
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
Since the assignment of jobs to machines is not specified, let N denote the set of jobs that are j ultimately assigned to machine j; thus, mn assignments are possible. Of these, the number of unique assignments is a Stirling number of the second kind, which can be computed by the following recursive equation [14]:
G HA B
n m~1 mn! + k! n k k/2 " m m!
GH
m !m k
.
The number of unique assignments ignores the sequence within each machine and assumes that each machine will process at least one job when n*m. A single resource with capacity R is shared among the m machines. Each job can be processed in any one of K modes: k"1, 2, 2 , K . The processing time of job i is denoted as p when r units of i i ik ik resource are allocated to machine j, and job i is also assigned to machine j. Once resource is assigned to a specific machine, this assignment remains fixed for the duration of the planning horizon. The amount of resource assigned to each machine j is denoted as r , where + r )R, j j|M j and each job on machine j is processed in some mode such that r )r . With the job assignment ik j and the allocation of the resource across machines specified, the set of jobs on each machine can be processed in any sequence without affecting system makespan. The static version of identical parallel machine scheduling problem with job assignment unspecified can be formulated as the following mixed-integer program. Minimize C "maxj3M + + P x y .!9 ik ik ij i k Subject to + + P x )C , j3M, ik ik .!9 i3N k3K j
(1)
i
+ r x )r , j3M, i3N , ik ik j j
(2)
+ r )R, j
(3)
+ x "1, i3N, ik
(4)
+ y "1, i3N, ik
(5)
x 3M0, 1N, y 3M0, 1N, r *0, i3N, k3K , j3M, ik ij j i
(6)
k3K j
j3M
k3K i
j3M
where
G
1 if job i is processed in mode k, x " ik 0 otherwise, and
G
1 if job i is processed in mode j, y " ij 0 otherwise.
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
147
In this formulation, constraints (1) define schedule makespan, constraints (2) assume that the operating mode of each job requires no more resource than is allocated to the associated machine, constraints (3) ensure that the total amount of resource allocated among machines does not exceed the total amount of available resource, constraints (4) guarantee that exactly one processing mode is selected for each job, and constraints (5) specify that each job is assigned to one and only one machine. Constraints (1)—(6) can be used to identify feasible solutions during the solution process. Ozdamar and Ulusoy [15] survey resource-constrained project scheduling problems, in which required resources are available in limited amounts. The problem formulated here differs from the general resource-constrained project scheduling problem because UPMFRS requires simultaneous optimization of two interrelated subproblems: job assignment and resource allocation. Job assignment determines the specific jobs associated with each machine. For each job assignment, a polynomial-time algorithm developed by Daniels et al. [11] can be used to allocate the available resource optimally. Job assignment can be represented by an assignment vector a, whose i5) component a(i)"j if and only if job i is assigned to machine j. Resource-allocation vector r defines the amount of resource allocated to the machines, i.e. r"(r , r , 2 , r ). Notice that the resource-allocation vector r uniquely defines the corresponding 1 2 m processing modes and processing times of the individual jobs. Therefore, any solution to the UPMFRS problem can be expressed as (a, r). Let C (a, r) represent the system makespan .!9 associated with this solution. Let A denote the set the feasible job assignments, and let P be the set of all resource-allocation policies. A solution (a, r) is therefore feasible to the UPMFRS problem if the corresponding solution satisfies constraints (1)—(6). An equivalent formulation of the UPMFRS problem is given by
G
H
C* "min min MC (a, r)N "min MC* (a)N. .!9 .!9 .!9 a3A r3P a3A It is well known that the m parallel-machine scheduling problem to minimize system makespan is NP-hard [16]. Therefore, the parallel-machine scheduling problem with unspecified job assignment is also NP-hard. The problem complexity motivates the development of heuristic approaches for solving the problem. In the next section, we describe a decomposition heuristic for the problem, and in Section 4, we present a tabu search approach.
3. A decomposition heuristic for the UPMFRS problem The decomposition heuristic (DH) described in this section is designed to generate approximate solutions to the static UPMFRS problem with modest computational effort. Initially, all jobs are unassigned and the minimum amount of resource is allocated to each machine. A longest process time (LPT) priority index is used to assign n jobs to m machines by sequentially assigning jobs to the first available machine in nonincreasing order of job processing time. Given this initial assignment, the polynomial-time algorithm of Daniels et al. [12] is applied to allocate the available resource to yield minimum makespan.
148
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
The resource-allocation algorithm begins by allocating the minimum amount of resource to each machine so that each job can be processed in its slowest mode. The makespan C and the .!9 machine j* with the longest completion time are then identified, i.e., C "C . The minimum j* .!9 amount of resource necessary to increase the mode of at least one job assigned to machine j* is determined and denoted * . Then an additional amount of resource * is allocated to machine j*, r r the operating modes of related jobs are increased, and the amount of available resource is decreased by * units. The process is repeated until the minimum amount of additional resource r * exceeds the amount of resource remaining, or until all jobs on machine j* are assigned their r fastest processing modes. Based on the updated job processing times, the LPT index is applied to generate a job-tomachine assignment, and the resource-allocation algorithm is then reapplied to generate the next resource-allocation policy. The heuristic terminates once a job-to-machine assignment encountered has been previously evaluated. The detailed heuristic procedure is summarized below. 3.1. Decomposition Heuristic (DH) Input: N jobs, M parallel identical machines, resource requirement r and associated processing ik times p for each job i and each operating mode k, and a total of R units of available resource. ik Output: N jobs assigned to each machine j with r units of resource allocated, satisfying the j j resource constraint and yielding a solution (aDH, rDH) with corresponding makespan CDH that .!9 approximates the minimum makespan. Step 0—Initialization. Let rL be the minimum required resource on each machine j, and set j CDH "R. .!9 Step 1—Define initial assignment a . Use the LPT index to determine an initial assignment a . 0 0 Step2—Solve the static PMFRS problem. Solve the static PMFRS problem with the current assignment, yielding C* Ma N and resource allocation vector r. If C* Ma N(CDH , then .!9 0 .!9 0 .!9 CDH "C* Ma N, aDH"a , and rDH"r. .!9 .!9 0 0 Step 3—Define the incumbent job assignment a. Set r "r for all jobs assigned to machine j, ik j and identify corresponding job processing times p . Again, use an LPT index to obtain new ik assignment a. Step 4—Compare the new assignment with the previous assignments. If the new assignment has been previously evaluated, then stop. Otherwise, a "a and go to Step 2. 0 The following example illustrates the mechanics of the decomposition heuristic. Consider 10 jobs to be processed on 3 identical machines with a total of 4 units of a flexible resource. The assignment of jobs to machines is unknown and job processing times depend on the amount of resource allocated to each job as shown in Table 1. The decomposition heuristic starts with a single unit of resource on each machine. The processing time of each job with one unit of resource is ranked in nonincreasing order to obtain an initial assignment a . For initial assignment a , jobs M1, 9, 2N are assigned to machine 1, jobs 0 0 M5, 7, 10N to machine 2, and jobs M4, 3, 6, 8N to machine 3. Given this assignment, machine 3 will complete processing at the latest time. Therefore, the remaining unit of resource is allocated to machine 3. Solving the static PMFRS problem yields C Ma , rN"78. .!9 0 The processing times of all jobs are updated according to the incumbent allocation of resource, and then ranked in nonincreasing order to yield new job assignment a, in which jobs M1, 9, 2N are
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
149
Table 1 Example problem Job
1 unit of resource
Processing time with 2 units resource
Processing time with 3 units of resource
1 2 3 4 5 6 7 8 9 10
40 15 32 33 35 7 28 6 20 15
30 11 24 25 26 5 21 5 15 11
27 10 21 22 23 5 19 4 13 10
assigned to machine 1, jobs M5, 3, 6, 8N to machine 2, and jobs M7, 4, 10N to machine 3. The new static PMFRS problem is solved, resulting in a makespan of 76. The process is repeated with the updated job processing times until this same job assignment is reached, terminating the process.
4. A tabu-search heuristic for the UPMFRS problem Tabu search is a neighborhood search procedure for solving optimization problems that is characterized by its ability to escape the trap of local optimality [17—20]. Computational experience on a wide variety of problems has demonstrated the method’s effectiveness in generating near-optimal solutions for complex problems. Skorin-Kapov [21] and Taillard [22] apply tabu search for the Quadratic Assignment Problem, yielding excellent solutions with modest computational effort. Daniels and Mazzola [23] employ a nested tabu-search heuristic for the FlexibleResource Flow Shop Scheduling problem. Hubscher and Glover [24] apply tabu search to the Parallel-Machine Scheduling problem. Daniels et al. [11] extend these applications to the ParallelMachine Flexible Resource Scheduling problem. We define a tabu-search heuristic (TSH) for the UPMFRS problem that searches among the set of possible job assignments, and evaluates alternative assignments using the static PMFRS algorithm of Daniels et al. [11]. Initially, we use the LPT index to construct an initial assignment of the jobs, and apply the static PMFRS algorithm to determine the allocation of resource to machines that minimize makespan given the incumbent job assignment. Alternative assignments are generated by considering all pairwise exchanges of jobs on different machines. Let a(x, y) be the assignment resulted from the exchange of job x and job y. For each job assignment, the static PMFRS algorithm is applied to determine the corresponding optimal resource-allocation policy. If an improvement in makespan is observed, then the heuristic solution is updated and the associated pairwise exchange is defined as a tabu move. Define ¹ to be the set of tabu moves, with ¹ the rth r tabu move, and ¹"M¹ , ¹ , 2 , ¹ N, i.e. a tabu move is discarded from the tabu list after 0 1 t t additions to the list. The tabu list is used to temporarily exclude certain elements of the
150
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
neighborhood from the search, and by varying the length of this list, the search can be diversified and intensified. Consistent with Glover [18], a tabu list length of seven is chosen for this application; however, we note that, as in Glover [18] and Taillard [22], the best length for the tabu list may depend on problem dimension. The tabu search is run for a total of ¹I iterations. The tabu search heuristic is detailed below. 4.1. Tabu-Search Heuristic (TSH) Input: N jobs, M parallel identical machines, resource requirement r and associated processing ik times p for each job i and each operating mode k, and a total of R units of available resource, and ik tabu list length t. Output: N jobs assigned to each machine j with r units of resource allocated, satisfying the j j resource constraint and yielding a solution (aDH, rDH) with corresponding makespan CDH that .!9 approximates the minimum makespan. Step 0—Initialization. Let rL be the minimum required resource on each machine j, and set j CTSH"R. Let ¹"0; q"0; ¹I"N2; counter"0, and COUNTER"N2/2. .!9 Step 1—Define initial assignment a. Use the LPT index to determine initial assignment a. Step2—Solve the static PMFRS problem for initial assignment a. Solve the static PMFRS problem given initial assignment, a, yielding C* Ma N and resource allocation vector r. Set .!9 0 CTSH"C* Ma N, aTSH"a, and rTSH"r. .!9 .!9 0 Step 3—Pairwise exchange. For current job assignment a, define ¼(a) as the set of all possible pairwise exchanges among jobs assigned to different machines in job assignment a. Step 4—Solve the static PMFRS problem for the job assignment in ¼ (a). For job assignment a@3¼(a), apply the static PMFRS algorithm to obtain resource-allocation policy r(a@) and corresponding makespan C* Ma@N. If C* Ma@N(CTSH , then CTSH"C* Ma@N, aTSH"a@, .!9 .!9 .!9 .!9 .!9 rTSH"r(a@), and counter"0. Otherwise, counter"counter#1. If counter is greater than COUNTER, then stop. Step 5—Modify tabu list. If q)t!1, then add the pairwise exchange that differentiates job assignments a and a@ to position q in tabu list ¹ and set q"q#1; otherwise, discard the pairwise exchange in position ¹ , move each remaining element of the tabu list up one position, and place 0 the pairwise exchange that differentiates job assignment a and a@ into position q in tabu list. Go to Step 4 if there are pairwise exchanges in ¼(a) that have not been considered. Set iteration" iteration#1. If iteration'¹I, stop; otherwise, go to Step 3. To illustrate the mechanics of the tabu search heuristic, we again use the example problem in Table 1. The procedure begins by ranking the jobs in nonincreasing order of their slowest processing times and assigning jobs to available machines according to this index. As in Section 3, this process yields a solution with system makespan C Ma, rN of 78. Given this solution, 33 unique .!9 pairwise exchanges between jobs on different machines are possible. The pairwise exchanges are chosen randomly at each iteration. Assume we first exchange job 1 and job 5, which leads to a new job assignment a (1, 5). The static PMFRS problem is then solved and the makespan is still 78. So we eliminate the exchange and keep the original job assignment a. We go through the next few exchanges, and the results remain same. The makespan is reduced to 76 after we solve the static PMFRS problem with the exchange of job 5 and job 4. We define the exchange of job 5 and job 4 as a tabu-move and add it to the tabu list. The process is repeated until no improved solution is
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
151
identified for COUNTER number of consecutive iterations, or until a total of TI iterations is completed.
5. Computational results The decomposition and tabu-search heuristics for the UPMFRS problem were coded in C and implemented on a Compaq Pentium 120 MHz computer with 24 meg memory. More than 800 test problems were used to compare the performance of the two heuristics. The key design variables included in the study are the number of jobs and machines, the available resource and its impact on operation processing times, and the location of processing bottlenecks. With the varying design variables, the test problems reflect a wide range of applications in parallel-machine manufacturing environments. In addition, the experiment is a natural extension of the computational work presented in Daniels et al. [11]. The test results thus provide insights into the impact of resource flexibility on parallel machine environments where both job assignment and resource allocation are important, and provide evidence of the computational performance of the proposed heuristics. Each test problem type is defined by specifying values for the number of jobs n, the number of machines m, the total amount of available resource R, and the sensitivity of job processing times to resource allocation, which is controlled by parameter b, with 0)b)1. The processing time of job i when processed in mode k3K is given by: i
C A BD
1 pL " 1!b ik k
PK . i1
Hence, for large values of b, the assignment of additional resource to a job has a greater effect on reducing the associated processing time than when b is small. This approach for modeling the impact of resource allocation on job processing times was first used in Daniels and Mazzola [23]. Three values of b, b"0.2, 0.5 and 0.8 are reflected in the experimental design. Ten replications of each problem type are included in the experiment design. Two sets of experiments were run. The first set of experiments were conducted on test problems consisting of n"10 and 15 jobs scheduled over m"3 and 4 machines with R"3, 4, and 5 units of resource available. The slowest processing times for each job are generated randomly from a uniform distribution of integers U [R, 50]. These problems are solved using both the decomposition and the tabu-search heuristic, and these approximate solutions are then compared with the optimal solutions obtained by an exhaustive search approach that considers every possible combination of job assignment and resource allocation. The second set of the test problems consists of 540 larger UPMFRS problems where n"20, 30, and 50 jobs must be scheduled on m"3, 4 and 5 machines with R"m#1 and m#2 units of available resource. The slowest processing times for each job in the larger problems are generated randomly from a uniform distribution of integers U[R, 250]. Since most of these problems are too large to be solved to optimality, only solutions from the tabu-search heuristic and the decomposition heuristic are compared. The computational performance of the procedures for the 10- and 15-job problems are presented in Tables 2 and 3 for m"3 and 4, respectively. The average and maximum deviation from the
152
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
Table 2 Heuristic performance on 3-machine problem n
R
Beta
Tabu search % above C* max Avg. (%)
Max. (%)
Decomposition % above C* max CPU
Avg. (%)
Max. (%)
Exhaustive search CPU
CPU
10 10 10
3 3 3
0.2 0.5 0.8
0.71 0.80 1.60
1.41 1.49 5.10
0.09 0.078 0.053
4.54 3.21 2.04
9.78 6.61 5.10
0.066 0.131 0.138
1.834 2.465 2.63
10 10 10
4 4 4
0.2 0.5 0.8
1.13 2.61 9.84
3.09 7.89 21.43
0.04 0.253 0.149
1.76 8.71 19.25
3.98 10.44 24.32
0.068 0.169 0.122
2.345 3.274 3.581
10 10 10
5 5 5
0.2 0.5 0.8
0.35 2.03 1.77
2.53 7.83 6.59
0.105 0.273 0.237
3.28 15.13 35.80
4.94 18.84 40.26
0.078 0.166 0.133
2.8 4.027 4.256
15 15 15
3 3 3
0.2 0.5 0.8
0.15 0.42 0.29
0.81 1.28 2.21
0.35 0.245 0.187
2.17 1.75 0.59
3.85 3.16 2.21
0.121 0.276 0.222
485.519 764.862 795.354
15 15 15
4 4 4
0.2 0.5 0.8
0.94 4.33 15.51
3.41 10.68 21.59
0.22 0.403 0.504
2.92 10.00 19.62
3.74 11.80 20.60
0.148 0.104 0.291
697.583 882.46 1057.393
15 15 15
5 5 5
0.2 0.5 0.8
0.39 1.19 3.13
1.78 4.73 8.51
0.3367 0.555 1.439
4.50 18.17 38.81
5.93 20.63 40.59
0.176 0.104 0.297
834.792 1117.091 1248.995
Table 3 Heuristic performance on 4-machine problem n
R
Beta
Tabu search % above C* max Avg. (%)
Max. (%)
Decomposition % above C* max CPU
Avg. (%)
Exhaustive search
Max. (%)
CPU
CPU
10 10 10
4 4 4
0.2 0.5 0.8
0.54 0.82 1.30
3.77 6.17 8.06
0.049 0.072 0.107
3.79 3.51 6.12
6.58 9.21 10.53
0.086 0.18 0.117
22.651 38.073 39.509
10 10 10
5 5 5
0.2 0.5 0.8
1.22 1.24 9.75
4.55 5.88 17.89
0.052 0.231 0.105
3.35 8.01 14.46
8.47 12.00 18.75
0.139 0.186 0.155
37.107 50.548 60.073
15 15 15
4 4 4
0.2 0.5 0.8
1.36 1.73 1.80
2.88 4.95 3.40
0.1505 0.204 0.22
3.95 3.45 4.32
8.47 4.95 7.32
0.1171 0.1471 0.09
36284.45 49006.02 48864.63
15 15 15
5 5 5
0.2 0.5 0.8
1.72 6.08 10.84
5.26 16.00 18.04
4.17 10.03 15.07
8.19 16.33 18.07
0.12 0.08 0.22
45371.65 48032.25 47989.30
0.2 0.33 0.27
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
153
Table 4 Heuristic performance of larger problem n
m
R
Beta
DH % above TSH
n
m
R
Beta
DH % above TSH
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5
4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7
0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8
6.25 13.90 27.22 8.54 25.54 47.93 3.14 4.36 22.20 7.40 10.89 31.54 3.60 6.88 10.43 4.48 6.84 15.11
30 30 30 30 30 30 30 30 30
4 4 4 5 5 5 5 5 5
6 6 6 6 6 6 7 7 7
0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8
5.38 15.20 24.21 2.52 3.71 6.31 3.24 6.44 11.11
30 30 30 30 30 30 30 30 30
3 3 3 3 3 3 4 4 4
4 4 4 5 5 5 5 5 5
0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8
5.69 13.15 18.65 9.24 24.62 45.77 2.03 6.75 13.08
50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50
3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5
4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7
0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8
2.22 5.19 17.26 8.90 17.90 46.11 2.78 2.28 1.14 3.33 5.22 10.96 0.84 1.28 1.47 3.34 1.16 4.05
optimal system makespan, and the average CPU times in seconds for both heuristics, as well as the exhaustive search, are reported. The results indicate that both the tabu-search heuristic and decomposition heuristic are effective in obtaining approximate solutions, as reflected by the average and maximum deviations from the optimal system makespan. The approximation errors associated with the decomposition heuristic average 8.24% with a maximum of 40.59%. The tabu-search heuristic generates solutions that are 2.31% above the optimal solution on average, with a maximum deviation of 21.59%. Therefore, the tabu-search procedure appears to outperform the decomposition procedure, since the computation times for the two approaches are not significantly different (the average computation times for the tabu-search procedure and the decomposition procedure are 0.23 seconds and 0.14 seconds, respectively, over the 300 problems). The computation times for the heuristics increase with the number of jobs and the number of resources. However, these solution times are extremely small compared to the computation time of
154
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
the exhaustive search, which averaged over 2.5 hours. These results clearly illustrate the cost/quality trade-off that underlies the UPMFRS problem. Table 4 show the computational performance of the heuristic on the set of larger UPMFRS problems. The improvement in solution quality of the tabu search heuristic over the decomposition heuristic is captured by the percentage measure (CDH !CTSH )/CTSH , where CTSH and CDH are the .!9 .!9 .!9 .!9 .!9 corresponding solutions for the tabu search and decomposition heuristics. This percentage improvement averages 11.27% with a maximum of 47.93% over the 540 problems. In addition, Table 4 show that the percentage of improvement increases as the total units of available resource R increase, and as the parameter b increases. This provides additional evidence of the effectiveness of tabu search in obtaining good solutions with reasonable computational effect.
6. Conclusions This paper presents and computationally investigates a decomposition heuristic and a tabusearch heuristic for the parallel-machine flexible-resource scheduling problem with unspecified job assignment. Both heuristics presented are based on a decomposition of the problem into job assignment and resource allocation subproblems. Over 800 problems were tested to compare the performance of the heuristics. This computational experience suggests that the tabu-search heuristic is most effective in obtaining close approximations of the optimal solution with modest computational effort. More research on the strategic significance of resource flexibility, and how this flexibility can be utilized to improve operational performance, is clearly needed. Future research should explore the dynamic version of the UPMFRS problem, where jobs and resources are reallocated over time in response to shifting bottlenecks within the system. Models that help determine how jobs should be released and assigned to production centers, and how resource should be deployed dynamically to optimize production efficiency and responsiveness would yield important contributions to this increasingly important area of production scheduling.
References [1] Daniels RL, Hoopes BJ, Mazzola JB. An analysis of heuristics for the parallel-machine flexible-resource scheduling problem. Fuqua chool of Business Working Paper, Duke University, USA, 1994. [2] Vickson RG. Two single-machine sequencing problem involving controllable job processing times. AIEE Transactions 1980;12:258—62. [3] Vickson RG. Choosing the job sequence and processing times to minimize processing plus flow cost on a single machine. Operations Research 1980;28:1155—67. [4] Van Wassenhove LN, Baker KR. A bicriterion approach to time cost trade-offs in sequencing. European Journal of Operational Research 1982;11:48—54. [5] Panwalker SS, Rajagopalan R. Single-machine sequencing with controllable processing times. European Journal of Operational Research 1992;59:298—302. [6] Daniels RL, Sarin RK. Single machine scheduling with controllable processing times and number of job tardy. Operations Research 1989;37:981—4. [7] Daniels RL, A multi-objective approach to resource allocation in single machine scheduling. European Journal of Operational Research 1990;48:226—41.
R.L. Daniels et al. / Computers & Operations Research 26 (1999) 143—155
155
[8] Daniels RL, Mazzola JB. Flow shop scheduling with resource flexibility. Operations Research 1994;42:504—22. [9] Hariri AMA, Potts CN. Heuristics for scheduling unrelated parallel machines. Computers and Operations Research 1991;18:323—31. [10] Trick MA. Scheduling multiple variable-speed machines. Operations Research 1994;42:234—48. [11] Daniels RL, Hoopes BJ, Mazzola JB. Scheduling parallel manufacturing cells with resource flexibility. Management Science 1996;42:1260—76. [12] Adiri I, Yehudai Z. Scheduling on machines with variable service rates. Computers and Operations Research 1987;14:289—97. [13] So KC. Some heuristics for scheduling jobs on parallel machines with setups. Management Science 1990;36:467—75. [14] Ho JC, Chang YL. Minimizing the number of tardy jobs for m parallel machines. European Journal of Operational Research 1995;84:343—55. [15] Ozdamar L, Ulusoy G. A survey on the resource-constrained project scheduling problem. IIE Transactions 1995;27:574—86. [16] Lenstra JK, Rinnooy Kan AHG, Brucker P. Complexity of machine scheduling problems. Annals of Discrete Mathematics 1977;1:343—62. [17] Anderson EJ, Glass CA, Potts CN. Machine scheduling, in: Aarts E, Lenstra J, editors, Local search in combinatorial optimization, Chap 11 Wiley, Chichester, 1997. [18] Glover F. Tabu search: a tutorial. Interfaces 1990; 20:74—94. [19] Glover F. Tabu search — Part I. ORSA Journal on Computing 1989;1:190—206. [20] Glover F. Tabu search — Part II, ORSA Journal on Computing 1990;2:4—32. [21] Skorin-Kapov J. Tabu search applied to the quadratic assignment problem. ORSA Journal on Computing 1990;2:33—45. [22] Taillard E, Robust taboo search for the quadratic assignment problem. Parallel Computing 1991;17:443—55. [23] Daniels RL, Mazzola JB. A tabu-search heuristic for the flexible-resource flow shop scheduling problem. Annals of Operations Research 1993;41:207—30. [24] Hubscher R, Glover F. Applying tabu search with influential diversification to multiprocessor scheduling. Computers and Operations Research 1994;21:877—84. [25] Cheng TCE, Sin CCS. A state-of-the-art review of parallel-machine scheduling research. European Journal of Operational Research 1990;48:226—41.