This paper studies the order acceptance and scheduling problem on unrelated parallel machines with machine eligibility constraints. Two objectives are considered to maximize total net profit and minimize the makespan, and the mathematical model of this problem is formulated as multiobjective mixed integer linear programming. Some properties with respect to the objectives are analysed, and then a classic list scheduling (LS) rule named the first available machine rule is extended, and three new LS rules are presented, which focus on the maximization of the net profit, the minimization of the makespan, and the trade-off between the two objectives, respectively. Furthermore, a list-scheduling-based multiobjective parthenogenetic algorithm (LS-MPGA) is presented with parthenogenetic operators and Pareto-ranking and selection method. Computational experiments on randomly generated instances are carried out to assess the effectiveness and efficiency of the four LS rules under the framework of LS-MPGA and discuss their application environments. Results demonstrate that the performance of the LS-MPGA developed for trade-off is superior to the other three algorithms.
In recent decades, the topic of order acceptance and scheduling (OAS) has attract considerable attention from scheduling researchers and production managers who practice it. The key issue of OAS is to make a joint decision of which orders are accepted (order acceptance decision) and how to schedule them (scheduling decision). Therefore, OAS is essentially different from traditional scheduling problem in which all jobs must be accepted, because the latter is just a special case of it. In OAS, a job has two options, to be accepted or rejected; thus the solution space can be up to
Unrelated parallel machine environment is a common workshop where processing times of orders are machine dependent. In traditional OAS on unrelated parallel machines, it is usually assumed that orders are able to be processed on any machines. However, in reality, especially in the assemble lines with multivariety production, one machine is only eligible to process specified orders, which is called the machine eligibility constraint. Recently, some scholars have considered this constraint in the unrelated parallel machine scheduling models (see [
This paper studies OAS on unrelated parallel machines with machine eligibility constraints (referred to as OAS-ME). Two objectives are considered. One is to maximize the total net profit with respect to the revenue, tardiness cost, and production cost. The other is to minimize the makespan, which is a classic scheduling criterion with respect to the productivity. Note that the two objectives are conflicting. The makespan is the completion time of the last finished orders; thus the solution with minimum makespan usually does not have the minimum total tardiness, let alone the minimum total tardiness cost (namely, total
As shown in Section
This paper considers an OAS problem on unrelated parallel machines with binary objectives. In the following, we will review the works related to the OAS on parallel machines and multiobjective unrelated parallel machine scheduling algorithms.
By now, extensive studies on OAS have been conducted in the production scheduling literature, and Slotnick [
Our problem is an extension of multiobjective unrelated parallel machine scheduling problem. Pareto-based metaheuristic is an effective approach for the multiobjective scheduling problem [
OAS-ME can be formally described as follows. Given a set of
For convenience, following notations are introduced.
Mathematical model for the OAS-ME can be formulated as a multiobjective mixed integer linear programming (MILP) as follows.
Objective (
Constraints (
Scheduling decision in OAS-ME has a special case. While
For the objective of total net profit, Theorem
If there is at least one accepted order with a negative net profit, the solution is certainly not optimal.
Denote this solution by
First, consider the total net profits of
Second, consider the makespan of
Since
Based on Theorem
The solution in which all orders are rejected is an extreme optimal solution with
If an order
For those orders that will surely be rejected in optimal solutions, they can be rejected in advance; therefore, according to Corollary
If
Moreover, from Theorem
Denote the total net profit in an optimal solution as
For an order
As aforementioned, OAS-ME is a joint decision of order acceptance and order scheduling. In the problem model in Section
OAS-ME is a NP-hard multiobjective combinatorial optimization problem, where it is practically impossible to find an optimal solution in polynomial time. For this kind of problem, constructive heuristics and metaheuristics are effective in producing good solutions in a short time. For parallel machine scheduling problem, list scheduling (LS) is a classic constructive heuristic which provides an assignment rule to schedule orders one by one in a specific order list [
Unlike the traditional LS rules which are only for scheduling problem, these LS rules for OAS-ME make one of the following decisions for an order
Hereinafter, the above two decisions are briefly described as “reject
For ease of algorithm description, some notations are defined.
One of the most popular LS rules is the first available machine (FAM) rule, which schedules the next job of the list on a machine which is available first [
According to Theorem
If
In EFAM, for one order, creating the candidate set and selecting the first available machine both take
This subsection presents a LS rule named highest profit first (HPF) rule, which prefers solutions with high net profit. HPF first finds a machine
If
HPF finds
Besides total net profit, OAS-ME has the other objective, the makespan. This subsection presents a smallest makespan first (SMF) rule. SMF schedule
If
Above LS rules either extend an existing heuristic, or only focus on one objective, which do not take into account the balance of two objectives. This subsection presents a LS rule to trade-off total net profit and the makespan, which is formalized as Rule
If
In (
For the net profit of
Therefore,
For the makespan of partial schedule
If
Therefore, (
In IFH, for each order
List scheduling supposes that the order list has been already specified; thus the difficulty of using this method is how to determine a good order list. Genetic algorithm (GA) is a popular approach to produce near-optimal solutions with flexible encoding scheme and genetic operators, and parthenogenetic algorithm (PGA) is an improved GA which is suitable for combinatorial optimization [
A chromosome in LS-MPGA corresponds to an order list; that is, it contains
In population initialization process, LS-MPGA randomly generates
A chromosome in LS-MPGA is a permutation in which all genes should be different. Traditional GA has two genetic operators as crossover and mutation, which both have a difficulty in maintaining validity of a permutation chromosome. Parthenogenetic algorithm (PGA) is an improved GA proposed by Li and Tong [
(a)
(b)
(c)
Figure
Genetic recombination operators of LS-MPGA.
Gene shift operator
Gene exchange operator
Gene inverse operator
One issue of parthenogenetic operation is how to compose these genetic operators. Insertion and swap are two of the most widely used neighbourhood structures in scheduling [
If
If
If
OAS-ME is a multiobjective optimization, and Pareto-optimal solutions are practical when considering real-life problems since the final solution of the decision-maker is always a trade-off. There have been many effective multiobjective GAs to approximate the true Pareto points, and a classic one is the fast nondominated sorting genetic algorithm (NSGA-II) proposed by Deb et al. [
LS-MPGA is terminated when the generation reaches the specified maximal generation
Randomly generate Calculate its total net profit and the makespan; Calculate its total net profit and the makespan; Calculate ranks of all chromosomes in
The advantage of LS-MPGA can be explained based on three points. The first is the simple chromosome encoding scheme with only
This section conducts an experimental study to evaluate the performance of proposed LS rules under the framework of LS-MPGA (referred to as EFAM-MPGA, HPF-MPGA, SMF-MPGA, and IFH-MPGA, respectively). Algorithms in this section are all coded in C# language and implemented on a computer with Intel Core i5-6300U/CPU 2.40 GHz 2.50 GHz and RAM 8.00 GB.
As mentioned in Section
By extensive preliminary experimentations, parameters of LS-MPGAs are set as
A number of test instances are generated with the following parameter setting. Let
Problem sizes of test instances are set as
To measure the quality of obtained Pareto front, we adopt the metrics
Metric
Metric
These metrics both contain a reference Pareto front set
Obviously, bound-points of all order acceptance solutions can construct an outer boundary line of the true Pareto front. However, it is hard to exhaustively list all order acceptance solutions in polynomial time, because for an OAS-ME with
Means (Avg.) and standard deviations (Std.) of the two metrics
Means and standard deviations of
| | EFAM-MPGA | HPF-MPGA | SMF-MPGA | IFH-MPGA | NSGA-II | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
Avg. | Std. | Avg. | Std. | Avg. | Std. | Avg. | Std. | Avg. | Std. | ||
20 | 2 | | 1.52 | 4.99 | 3.87 | 1.87 | 1.55 | 2.84 | 3.75 | 10.93 | 7.97 |
4 | 6.35 | 2.88 | 18.40 | 14.80 | | 3.13 | 8.58 | 10.96 | 32.25 | 15.78 | |
6 | 8.72 | 5.10 | 21.06 | 13.60 | 9.28 | 4.55 | | 7.46 | 55.30 | 20.74 | |
8 | | 5.67 | 26.78 | 18.27 | 14.08 | 8.39 | 11.57 | 13.49 | 65.77 | 19.53 | |
10 | 12.02 | 5.66 | 30.52 | 16.68 | 12.76 | 7.07 | | 7.95 | 66.84 | 18.27 | |
| |||||||||||
50 | 2 | | 0.83 | 5.78 | 2.67 | | 0.79 | 1.78 | 1.61 | 8.85 | 4.81 |
4 | 2.60 | 1.65 | 13.30 | 6.92 | | 1.55 | 3.42 | 3.49 | 26.35 | 8.99 | |
6 | 5.04 | 2.17 | 18.69 | 13.64 | 4.35 | 2.55 | | 2.29 | 39.77 | 11.25 | |
8 | 9.66 | 5.53 | 24.64 | 13.06 | 8.79 | 5.38 | | 2.46 | 56.88 | 14.28 | |
10 | 11.85 | 6.26 | 24.96 | 14.04 | 10.28 | 5.56 | | 2.85 | 59.37 | 13.49 | |
| |||||||||||
100 | 2 | 1.14 | 0.98 | 6.14 | 2.81 | | 0.88 | 1.45 | 1.18 | 9.44 | 2.99 |
4 | 1.95 | 1.13 | 7.45 | 3.68 | | 0.93 | 2.76 | 2.55 | 18.67 | 3.90 | |
6 | 3.95 | 2.18 | 10.67 | 7.32 | | 2.62 | 3.28 | 1.99 | 31.19 | 9.02 | |
8 | 5.60 | 2.26 | 11.80 | 6.56 | 4.06 | 2.60 | | 1.94 | 38.93 | 6.74 | |
10 | 9.91 | 10.36 | 16.38 | 12.76 | 7.69 | 9.46 | | 6.72 | 47.52 | 10.36 | |
| |||||||||||
150 | 2 | 0.91 | 0.61 | 4.95 | 2.01 | | 0.58 | 1.24 | 0.78 | 8.50 | 2.71 |
4 | 1.47 | 0.55 | 6.64 | 2.93 | | 0.53 | 1.68 | 0.99 | 16.52 | 3.59 | |
6 | 3.74 | 6.00 | 8.76 | 4.54 | | 6.54 | 3.66 | 4.45 | 26.84 | 5.56 | |
8 | 4.49 | 2.79 | 8.87 | 4.03 | | 2.33 | 3.43 | 1.97 | 36.40 | 7.08 | |
10 | 5.89 | 3.14 | 10.10 | 5.62 | | 2.53 | 3.69 | 3.43 | 45.02 | 7.29 | |
15 | 11.58 | 8.08 | 13.70 | 11.45 | 8.15 | 7.20 | | 1.71 | 53.55 | 9.53 | |
| |||||||||||
200 | 2 | 0.73 | 0.46 | 5.37 | 1.97 | | 0.39 | 0.91 | 0.52 | 9.48 | 2.61 |
4 | 1.35 | 0.65 | 6.26 | 2.83 | | 0.54 | 2.23 | 1.25 | 15.74 | 3.62 | |
6 | 2.76 | 2.27 | 8.21 | 4.26 | | 1.95 | 3.24 | 1.89 | 27.18 | 5.47 | |
8 | 4.24 | 2.25 | 7.96 | 4.90 | | 1.88 | 4.40 | 3.45 | 33.95 | 5.07 | |
10 | 5.89 | 2.87 | 9.56 | 4.91 | | 2.21 | 5.02 | 5.38 | 42.98 | 7.45 | |
15 | 9.91 | 6.72 | 9.45 | 5.76 | 7.02 | 5.50 | | 2.97 | 48.93 | 8.24 | |
20 | 13.02 | 10.43 | 8.18 | 4.98 | 9.60 | 9.46 | | 2.61 | 51.99 | 10.75 | |
| |||||||||||
Avg. | 5.68 | 12.48 | 4.77 | | 35.18 |
Means and standard deviations of
| | EFAM-MPGA | HPF-MPGA | SMF-MPGA | IFH-MPGA | NSGA-II | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
Avg. | Std. | Avg. | Std. | Avg. | Std. | Avg. | Std. | Avg. | Std. | ||
20 | 2 | | 0.205 | 0.716 | 0.185 | 0.695 | 0.207 | 0.699 | 0.221 | 0.867 | 0.134 |
4 | 0.711 | 0.165 | 0.793 | 0.115 | 0.712 | 0.148 | | 0.158 | 0.899 | 0.109 | |
6 | 0.685 | 0.197 | 0.779 | 0.135 | 0.688 | 0.165 | | 0.187 | 0.884 | 0.110 | |
8 | 0.728 | 0.122 | 0.784 | 0.144 | 0.726 | 0.112 | | 0.138 | 0.927 | 0.083 | |
10 | | 0.132 | 0.835 | 0.127 | 0.758 | 0.117 | 0.692 | 0.161 | 0.943 | 0.081 | |
| |||||||||||
50 | 2 | 0.734 | 0.119 | | 0.114 | 0.740 | 0.114 | 0.742 | 0.117 | 0.779 | 0.092 |
4 | 0.804 | 0.105 | 0.816 | 0.091 | 0.816 | 0.099 | | 0.109 | 0.815 | 0.101 | |
6 | 0.780 | 0.106 | 0.829 | 0.095 | 0.781 | 0.103 | | 0.095 | 0.862 | 0.064 | |
8 | 0.782 | 0.098 | 0.823 | 0.099 | 0.784 | 0.093 | | 0.123 | 0.881 | 0.067 | |
10 | 0.810 | 0.074 | 0.830 | 0.087 | 0.825 | 0.077 | | 0.102 | 0.908 | 0.067 | |
| |||||||||||
100 | 2 | 0.849 | 0.070 | | 0.093 | 0.854 | 0.055 | 0.851 | 0.060 | 0.808 | 0.083 |
4 | 0.895 | 0.053 | | 0.064 | 0.886 | 0.054 | 0.862 | 0.059 | 0.817 | 0.078 | |
6 | 0.873 | 0.070 | | 0.085 | 0.898 | 0.058 | 0.876 | 0.064 | 0.836 | 0.070 | |
8 | 0.874 | 0.058 | | 0.075 | 0.874 | 0.053 | 0.853 | 0.065 | 0.854 | 0.078 | |
10 | 0.890 | 0.071 | | 0.079 | 0.886 | 0.070 | 0.837 | 0.078 | 0.878 | 0.057 | |
| |||||||||||
150 | 2 | 0.867 | 0.044 | | 0.064 | 0.870 | 0.041 | 0.866 | 0.039 | 0.813 | 0.061 |
4 | 0.902 | 0.042 | | 0.049 | 0.888 | 0.057 | 0.873 | 0.036 | 0.814 | 0.059 | |
6 | 0.885 | 0.060 | | 0.071 | 0.889 | 0.082 | 0.875 | 0.056 | 0.831 | 0.066 | |
8 | 0.893 | 0.049 | | 0.073 | 0.896 | 0.067 | 0.874 | 0.055 | 0.846 | 0.055 | |
10 | 0.895 | 0.078 | | 0.064 | 0.892 | 0.061 | 0.883 | 0.065 | 0.854 | 0.048 | |
15 | 0.918 | 0.043 | | 0.065 | 0.903 | 0.058 | 0.873 | 0.053 | 0.877 | 0.051 | |
| |||||||||||
200 | 2 | 0.877 | 0.039 | | 0.046 | 0.881 | 0.037 | 0.877 | 0.041 | 0.808 | 0.054 |
4 | 0.897 | 0.038 | | 0.049 | 0.889 | 0.041 | 0.878 | 0.032 | 0.811 | 0.044 | |
6 | 0.885 | 0.056 | | 0.067 | 0.886 | 0.044 | 0.887 | 0.047 | 0.823 | 0.044 | |
8 | 0.907 | 0.046 | | 0.053 | 0.910 | 0.049 | 0.894 | 0.054 | 0.842 | 0.054 | |
10 | 0.910 | 0.055 | | 0.067 | 0.906 | 0.040 | 0.895 | 0.044 | 0.857 | 0.044 | |
15 | 0.924 | 0.054 | | 0.078 | 0.911 | 0.048 | 0.894 | 0.057 | 0.850 | 0.053 | |
20 | 0.941 | 0.039 | | 0.065 | 0.931 | 0.038 | 0.895 | 0.056 | 0.883 | 0.043 | |
| |||||||||||
Avg. | 0.839 | | 0.842 | 0.819 | 0.852 |
Results in Table
It can be observed from Table
Experimental results in Table
Average values of objectives in boundary solutions with maximal net profit.
| | EFAM-MPGA | HPF-MPGA | SMFG-MPGA | IFH-MPGA | NSGA-II | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
| | | | | | | | | | ||
20 | 2 | 4753 | 497 | | 545 | 4752 | | 4726 | 499 | 4684 | 510 |
4 | 4744 | 293 | | 369 | 4737 | | 4729 | 302 | 4598 | 342 | |
6 | 4613 | 220 | | 297 | 4602 | | 4629 | 233 | 4445 | 283 | |
8 | 4594 | 168 | | 246 | 4572 | | 4617 | 182 | 4391 | 231 | |
10 | 4785 | 156 | | 219 | 4757 | | 4809 | 162 | 4549 | 199 | |
| |||||||||||
50 | 2 | 11531 | | | 1339 | 11531 | 1147 | 11470 | 1158 | 11404 | 1266 |
4 | 11312 | 591 | | 899 | 11315 | | 11346 | 630 | 11094 | 791 | |
6 | 11588 | 422 | | 711 | 11592 | | 11740 | 482 | 11324 | 611 | |
8 | 11453 | 348 | | 574 | 11457 | | 11649 | 401 | 11121 | 518 | |
10 | 11558 | 299 | | 524 | 11567 | | 11786 | 353 | 11206 | 438 | |
| |||||||||||
100 | 2 | 23094 | 1992 | | 2628 | 23082 | | 23012 | 1993 | 22944 | 2451 |
4 | 22881 | | | 1861 | 22892 | 1061 | 23110 | 1181 | 22705 | 1587 | |
6 | 22915 | 736 | | 1397 | 22930 | | 23356 | 844 | 22656 | 1163 | |
8 | 23031 | 570 | | 1082 | 23094 | | 23663 | 668 | 22754 | 933 | |
10 | 23313 | 473 | | 961 | 23350 | | 23980 | 561 | 22982 | 803 | |
| |||||||||||
150 | 2 | 34764 | 2910 | | 4028 | 34758 | | 34628 | 2922 | 34603 | 3737 |
4 | 34032 | 1465 | | 2571 | 34061 | | 34525 | 1637 | 33959 | 2292 | |
6 | 34419 | 1072 | | 1929 | 34510 | | 35219 | 1203 | 34353 | 1680 | |
8 | 34356 | 794 | | 1471 | 34457 | | 35417 | 925 | 34242 | 1343 | |
10 | 34224 | 684 | | 1292 | 34273 | | 35421 | 795 | 33933 | 1183 | |
15 | 33892 | 487 | | 990 | 34050 | | 35411 | 582 | 33543 | 886 | |
| |||||||||||
200 | 2 | 45615 | 3781 | | 5303 | 45612 | | 45458 | 3768 | 45485 | 4963 |
4 | 45333 | 1921 | | 3226 | 45388 | | 45997 | 2133 | 45513 | 2990 | |
6 | 45051 | 1364 | | 2346 | 45131 | | 46229 | 1538 | 45072 | 2190 | |
8 | 45376 | 1031 | | 1884 | 45527 | | 46909 | 1201 | 45427 | 1698 | |
10 | 45133 | 861 | | 1619 | 45308 | | 46938 | 1007 | 45100 | 1502 | |
15 | 44955 | 609 | | 1184 | 45147 | | 47091 | 720 | 44755 | 1084 | |
20 | 44872 | 490 | | 1023 | 45138 | | 47382 | 588 | 44657 | 900 |
Solutions of the four algorithms for instances with problem sizes of
Solutions for an instance with
Solutions for an instance with
(a) The minimal makespan is often obtained by SMF-MPGA (Figure
To measure the algorithm efficiency, Table
CPU time (in seconds) of the four algorithms.
| | EFAM-MPGA | HPF-MPGA | SMF-MPGA | IFH-MPGA | NSGA-II |
---|---|---|---|---|---|---|
| 100 | 0.87 | 0.86 | | 0.89 | 0.88 |
| 100 | 2.15 | 2.08 | 1.99 | 2.96 | |
| 200 | 9.74 | 9.78 | 9.25 | 15.52 | |
| ||||||
Avg. | 3.92 | 3.73 | 3.63 | 5.04 | |
The above experimental results indicate that the performance of LS-MPGAs is much better than that of NSGA-II. According to these results, the application environments of the four LS-MPGA algorithms are suggested as follows.
(a) If a decision-maker prefers the objective of total net profit, HPF-MPGA is a good choice. However, HPF-MPGA has two significant drawbacks: its convergence is the worst and the obtained makespan is much larger than those of other algorithms.
(b) If a decision-maker prefers the objective of the makespan, they can adopt SMF-MPGA or EFAM-MPGA. In particular, SMF-MPGA can achieve the best convergence for the instances with small number of machines. The drawback of the two algorithms is the difficulty to produce a solution with high total net profit.
(c) If a decision-maker would like to find some trade-off solutions, especially for the instances with large number of machines, IFH-MPGA is the best. In terms of convergence, IFH-MPGA is able to produce the lowest value of
This paper studied the order acceptance and scheduling problem on unrelated parallel machines with machine eligibility constraints (OAS-ME), which is a NP-hard problem. Two objectives are considered as total net profit and the makespan. This paper analyses some properties of the objectives and presents four list scheduling rules named EFAM, HPF, SMF, and IFH. EFAM extends the classic first available machine (FAM) rule. HPF and SMF are the heuristics mainly designed for one objective as total net profit and the makespan, respectively. IFH presents an integrated function to balance the two objectives. Time complexities of these rules are all
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the Humanity and Social Science Youth Foundation of Ministry of Education of China (no. 17YJC630143), the National Natural Science Foundation of China (nos. 71701016 and 71471015), the Beijing Natural Science Foundation (no. 9174038), and the Fundamental Research Funds for the Central Universities (no. FRF-BD-16-006A).