Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 6024631, 12 pages

https://doi.org/10.1155/2018/6024631

## Multiobjective Order Acceptance and Scheduling on Unrelated Parallel Machines with Machine Eligibility Constraints

Donlinks School of Economics and Management, University of Science and Technology Beijing, Beijing 100083, China

Correspondence should be addressed to Bailin Wang

Received 27 October 2017; Revised 19 January 2018; Accepted 6 February 2018; Published 7 March 2018

Academic Editor: Eusebio Valero

Copyright © 2018 Bailin Wang and Haifeng Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 times of that of traditional scheduling problem, where is the number of orders.

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 [1–3]). Moreover, in the assemble line with machine eligibility constraints, processing technologies used in machines are different; thus production costs of one order on different machines are often different and* do not depend on the processing times*. That is, an order on an eligible machine may have short processing time but high production cost, due to the skilled but not advanced processing technology. Therefore, order assignments on machines affect not only the productivity but also the profit.

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* weighted* tardiness). Moreover, because production cost is independent of the processing time, the minimum makespan does not imply minimal total production cost. Therefore, minimizing the makespan does not mean minimizing the total net profit, and vice versa.

As shown in Section 3.1, OAS-ME is NP-hard; thus this paper presents four list scheduling (LS) rules and then develops a list-scheduling-based multiobjective parthenogenetic algorithm (LS-MPGA). The contributions of this paper are in the following areas: (a) for the OAS-ME, analyse some properties with respect to the objective of total net profit; (b) extend a classic LS rule with the consideration of the net profit, and present three new LS rules according to problem characteristics; (c) propose a LS-based algorithm (named LS-MPGA) with parthenogenetic operators and Pareto-ranking and selection method; (d) suggest application environments of the four LS rules under the framework of LS-MPGA through computational studies. The rest of this paper is organized as follows. Section 2 reviews the related works. Section 3 models the problem OAS-ME and analyses some properties. Section 4 proposes four LS rules and Section 5 presents the LS-MPGA. Experimental research in Section 6 inspects performance of these LS rules under the framework of LS-MPGA. Finally, Section 7 concludes the paper.

#### 2. Literature Review

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 [4] provided a comprehensive literature review on it. For parallel machine environment, there are some new studies presented after the review [4]. Wang et al. [5] studied the problem with two identical parallel machines and developed two heuristics and an exact algorithm based on optimal properties and the Lagrangian relaxation technique. Emami et al. [6] considered nonidentical parallel machine environment in which the revenue from an accepted order and processing times are uncertain and developed a Lagrangian relaxation algorithm. Moreover, a series of related papers treat OAS from the perspective of order rejection, and a comprehensive survey was made by Shabtay et al. [7]. For the unrelated parallel machine scheduling with rejection, the new studies after the survey [7] are reviewed as follows. Hsu and Chang [8] studied the problem with deteriorating jobs to minimize of the sum of total rejection cost and a scheduling criterion and proved that if the scheduling criterion is either total load or total completion time, the problem is solvable in polynomial time. Lin et al. [9] presented a deterministic 3-approximation algorithm and a randomized 3-approximation algorithm for two unrelated parallel machine scheduling problem with rejection. Jiang and Tan [10] presented a heuristic with worst-case ratio of 2 for an unrelated parallel machine scheduling with rejection and nonsimultaneous machine available time to minimize the sum of the makespan and total rejection cost. Above studies, including literature on unrelated parallel machines in the two survey papers, mostly employed the a priori optimization approach to sum objectives as an aggregated function, which cannot provide Pareto-optimal solutions for trade-off. Moreover, to the best of our knowledge, the machine eligibility constraint and time-independent production cost in our problem have not been considered in OAS previously in the literature.

Our problem is an extension of multiobjective unrelated parallel machine scheduling problem. Pareto-based metaheuristic is an effective approach for the multiobjective scheduling problem [11]. For the unrelated parallel machine environment, Lin et al. [12] proposed two heuristics and a Pareto-based genetic algorithm for unrelated parallel machine scheduling problem to minimize the makespan, total weighted completion time, and total weighted tardiness. Lin and Ying [13] presented a multiobjective multipoint simulated annealing (MOMSA) algorithm and Lin et al. [14] proposed a Tabu-enhanced iterated Pareto greedy algorithm for the same problem. Afzalirad and Rezaeian [15] studied the unrelated parallel machine scheduling problem with sequence-dependent setup times, release times, machine eligibility, and precedence constraints to minimize mean weighted flow time and mean weighted tardiness and improved two classic Pareto-based algorithms, NSGA-II and MOACO. Above literature indicates that Pareto-based metaheuristic is an effect approach for multiobjective scheduling problem on unrelated parallel machines.

#### 3. Problem Modelling and Analysis

##### 3.1. Problem Model

OAS-ME can be formally described as follows. Given a set of nonpreemptive orders and a set of unrelated parallel machines, this problem is to decide which orders should be accepted and how to schedule them. Each accepted order must be processed on one machine, and rejected orders are not allowed to be processed. Each machine can process at most one order at a time. Moreover, each order can only be processed on specific machines (machine eligibility constraints), and processing times and production costs of an order on the eligible machines are different from each other. Two objectives are considered. One is to maximize the total net profit, which is equal to revenues minus the sum of production costs and tardiness costs of accepted orders. The other is to minimize the makespan.

For convenience, following notations are introduced.

*(a) Indexes and Sets* : order index; : the number of orders; : the order with index . : machine index; : the number of machines; : the machine with index . : the set of machines which are eligible to process . .

*(b) Problem Parameters* , : production cost and processing time of on , respectively. , , : revenue, due date, and unit tardiness cost of , respectively. : a very large positive number.

*(c) Decision Variables* : binary variable for order acceptance decision. If , is accepted, otherwise, rejected. : binary variable representing whether is the th processed order on () or not (). : nonnegative continuous variable representing the completion time of .

*(d) Objective Parameters* , : the net profit and tardiness of , respectively. , : total net profit and the makespan, respectively.

Mathematical model for the OAS-ME can be formulated as a multiobjective mixed integer linear programming (MILP) as follows.

Objective (1) is to maximize the sum of net profits (revenue minus tardiness cost and production cost) of all accepted orders. Objective (2) is to minimize the makespan.

Constraints (3) restrict the relationship between variables and . If is rejected , it cannot be scheduled on any machine; otherwise, namely, is accepted , it must be processed on one and only one position of a machine. Constraints (4) present a position arrangement rule that if one position of a machine is free (), its succeeding positions are no longer assigned to any orders as well (). Constraints (5) indicate that each machine can process at most one order at a time. Constraints (3), (4), and (5) work together to restrict a feasible order sequence on a machine. Constraints (6) make sure that the completion time of any accepted order is no less than its processing time. Constraints (7) and (8) define the tardiness, and Constraints (9) define the makespan. Constraints (10) are machine eligibility constraints where an order is not allowed to be processed on an ineligible machine. Constraints (11) are binary constraints for and .

Scheduling decision in OAS-ME has a special case. While , OAS-ME is equivalent to the identical parallel machine scheduling with machine eligibility constraints. Liao and Sheen [16] indicate that the latter problem to minimize the makespan is NP-hard; thus OAS-ME is NP-hard as well.

##### 3.2. Objective Analysis

For the objective of total net profit, Theorem 1 holds.

Theorem 1. *If there is at least one accepted order with a negative net profit, the solution is certainly not optimal.*

*Proof. *Denote this solution by and its total net profit and the makespan by and , respectively. Let the set include all the orders with negative net profits in . By rejecting the orders in , we can get a solution named with two objectives and .

First, consider the total net profits of and . For the orders in , their net profits in (which are equal to zero) are greater than those in (which are negative). For the accepted orders which are not in , their tardiness in must not be greater than those in ; thus their net profits in are not smaller than those in . Therefore, .

Second, consider the makespan of and . Rejecting an order will not increase the completion time of other accepted orders; thus .

Since and , the solution is dominated by ; thus is certainly not optimal.

Based on Theorem 1, we can get following corollaries.

Corollary 2. *The solution in which all orders are rejected is an extreme optimal solution with and .*

Corollary 3. *If an order satisfies , then in any optimal solution, it must be rejected; otherwise, if , and is accepted in an optimal solution, then there must exist an optimal solution in which is rejected.*

For those orders that will surely be rejected in optimal solutions, they can be rejected in advance; therefore, according to Corollary 3, the following rule is set for preliminary rejection decision.

*Rule 1. *If , then reject .

Moreover, from Theorem 1 and Corollary 3, we can get the following theorem that gives the upper and lower bounds of the total net profits in optimal solutions. This theorem provides the basis for the design of the scheduling rule named IFH in Section 4.5 (see Section 4.5.1).

Theorem 4. *Denote the total net profit in an optimal solution as ; then*

*Proof. *For an order satisfying , ideal condition of its net profit is to schedule to the machine with the minimal production cost, and this assignment will not cause tardiness, namely, . According to Theorem 1, the worst condition is to reject ; then the net profit is 0, namely, . Therefore, if , then ; otherwise, according to Corollary 3, ; thus ; then Theorem 4 holds.

#### 4. List Scheduling Rules

##### 4.1. Motivation and Algorithm Overview

As aforementioned, OAS-ME is a joint decision of order acceptance and order scheduling. In the problem model in Section 3.1, the variables correspond to the order acceptance decision, and the variables and correspond to the order assignment subdecision and completion time subdecision of scheduling decision, respectively. According to Constraints (5) and (6), while and are determined, completion times can be calculated by (13). Therefore, the key to solve OAS-ME is how to find the optimal values of variables and .

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 [17]. In this section, we introduce the order acceptance decision to a classic LS rule in Section 4.2 and present three new LS rules in Sections 4.3–4.5, which can make the order acceptance decision and order assignment decision simultaneously.

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 :

Reject ; then the variables and are all assigned as 0.

Accept , assign it to a machine , and insert it at the end of the order sequence on . Suppose is the th order scheduled to ; then , , and where , and where .

Hereinafter, the above two decisions are briefly described as “reject ” and “schedule to ” in the LS rules. The calculation of the variables , ,* and * corresponding to each decision has been described in detail herein, therefore not further described in the subsections.

For ease of algorithm description, some notations are defined. : available time of , which is equal to the total processing time of scheduled orders on ; , : the net profit and completion time of while it is scheduled to , respectively; : the makespan of scheduled jobs ; : the makespan of partial schedule in which is scheduled to .

##### 4.2. Extended FAM (EFAM)

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 [17]. We develop an extended FAM (EFAM) for OAS-ME, in which order rejection decision is complemented and machine eligibility constraint is considered.

According to Theorem 1, while scheduling , EFAM first creates a candidate set of machines (). contains the eligible machines which produce positive net profit for (14). EFAM then defines Rule 2 to reject or select a machine for it.

*Rule 2. *If , then reject ; otherwise, schedule to where .

In EFAM, for one order, creating the candidate set and selecting the first available machine both take . There are orders; thus the complexity of EFAM is .

##### 4.3. Highest Profit First (HPF)

This subsection presents a LS rule named highest profit first (HPF) rule, which prefers solutions with high net profit. HPF first finds a machine satisfying (15) and then calls Rule 3.

*Rule 3. *If , then reject ; otherwise, schedule to .

HPF finds for an order in ; thus its time complexity is .

##### 4.4. Smallest Makespan First (SMF)

Besides total net profit, OAS-ME has the other objective, the makespan. This subsection presents a smallest makespan first (SMF) rule. SMF schedule to the machine with the minimal completion time of if is accepted, and the difficulty to develop SMF is how to determine whether to accept . Here the criterion of EFAM is adopted, namely, if none of the machines can make a positive net profit for the considered order; then reject this order. Therefore, In SMF, the candidate set is first created by (14); then Rule 6 is employed. Time complexity of SMF is .

*Rule 4. *If , then reject ; otherwise, schedule to , where

##### 4.5. Integrated-Function-Based Rule (IFH)

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 5.

*Rule 5. *If , then reject ; otherwise, schedule to , whereIn (17), and are the functions reflecting the influence of assigning an order on the net profit and the makespan, and and are the weights of and , respectively. and should satisfy two requirements: (a) Minimizing and represents the same effect trend of optimization; (b) and should be on the same order of magnitude. The following gives the ideas to design these functions.

###### 4.5.1. Function with respect to the Net Profit

For the net profit of (), according to Theorem 4, Proposition 5 holds.

Proposition 5. *, where .*

Therefore, is defined by (18), in which , and a small corresponds to a high net profit of .

###### 4.5.2. Function with respect to the Makespan

For the makespan of partial schedule (), Proposition 6 holds.

Proposition 6. *.*

*Proof. *If is rejected, then . If is accepted and scheduled to , then , and the worst condition is to schedule to the machine with the latest available time and the longest processing time, that is, . Therefore, Proposition 6 holds.

Therefore, (19) defines which belongs to . Clearly, small represents a small makespan of .

###### 4.5.3. Time Complexity of IFH

In IFH, for each order , to calculate and , and should be found at first, which takes . Therefore, finding for takes , and time complexity of IFH is .

#### 5. Multiobjective Parthenogenetic Algorithm

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 [18]; thus this paper employs PGA with LS rules to produce order lists and then generate solutions for OAS-ME. This algorithm is named LS-based multiobjective PGA (LS-MPGA), and its key points are elaborated in the following subsections.

##### 5.1. Chromosome Encoding and Initialization

A chromosome in LS-MPGA corresponds to an order list; that is, it contains genes which are all different, and otherwise this chromosome is invalid.

In population initialization process, LS-MPGA randomly generates chromosomes (order lists) to maintain diversity, where is the population size.

##### 5.2. Parthenogenetic Operators

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 [19], in which each chromosome has only one parent [20]. In PGA, gene recombination operators are presented as a parthenogenesis approach instead of crossover, and it can guarantee the validity of the offspring for the chromosomes encoded as permutation. There are three types of gene recombination operators: gene shift operator, gene exchange operator, and gene inverse operators, which are employed in LS-MPGA.

(a)* Gene shift operator* evolves chromosomes by an* insertion* process with probability . It inserts the gene in position into position , where .

(b)* Gene exchange operator* implements a* swap* process with probability . This operator swaps the genes in positions and , where .

(c)* Gene inverse operator* is a* reversion* process with probability . This operator reverses sequence of genes between position and , where .

Figure 1 illustrates examples of these operators, in which a parent evolves offspring chromosomes by these operators with and .