Inventory Based Bi-Objective Flow Shop Scheduling Model and Its Hybrid Genetic Algorithm

Abstract

Flow shop scheduling problem is a typical NP-hard problem, and the researchers have established many different multi-objective models for this problem, but none of these models have taken the inventory capacity into account. In this paper, an inventory based bi-objective flow shop scheduling model was proposed, in which both the total completion time and the inventory capacity were as objectives to be optimized simultaneously. To solve the proposed model more effectively, we used a tailor-made crossover operator, and mutation operator, and designed a new local search operator, which can improve the local search ability of GA greatly. Based on all these, a hybrid genetic algorithm was proposed. The computer simulations were made on a set of benchmark problems, and the results indicated the effectiveness of the proposed algorithm.

1. Introduction

Scheduling is a major issue faced everyday in manufacturing systems and industrial settings, so it is significant to develop effective and efficient advanced manufacturing and scheduling technologies and approaches. The first research on the flow shop scheduling problem (FSSP) was completed by Johnson [1]. Johnson developed an exact algorithm to minimize makespan for the n-jobs and 2-machines FSSP. With the increase in the number of jobs and machines, FSSP becomes a strong NP-complete combinatorial optimization problem [2]. Most researches for single objective FSSPs result in a schedule to minimize the time required to complete all jobs, that is, to minimize the makespan. Historically, FSSP has been primarily treated by exact methods [3], such as linear programming, branch and bound, and Lagrangian relaxation. Recently, meta-heuristics have gained a wide attention [4], including the topics such as genetic algorithms (GAs), simulated annealing (SA), ant colony optimization (ACO), particle swarm optimization (PSO), tabu search (TS), and differential evolution (DE).

However, in many real world situations, several objectives must be considered. The purely single objective FSSP cannot totally reflect the needs of the modern practical applications, that is to say, multiple objectives must be optimized simultaneously. It is important to find out the trade-off solutions among these objectives, particularly in conflicting situations. So far, few researchers have attempted to tackle the multi-objective FSSPs (MOFSSPs). The most commonly used method in the current study of MOFSSPs is constant weight weighting method. It assigns constant weights to objectives according to the importance of each objective. By this way, an MOFSSP is transformed into a single objective FSSP through a linear combination of objectives and weights. The following works are examples of the representative works of this approach: Ravindran et al. [5] proposed some multi-criterion approaches to FSSP by considering makespan time and total flow time. Eren and Güner [6] considered a bicriteria scheduling problem with sequence-dependent setup times on a single machine. The objective function of the problem is the minimization of the weighted sum of total completion time and total tardiness. Lemesre et al. [7] proposed a parallel exact method to solve bi-objective permutation flow shop problem. Tseng and Liao [8] considered an n-job, m-machine lot-streaming problem in a flow shop with equal-size sublots where the objective is to minimize the total weighted sum of earliness and tardiness. Naderi et al. [9] applied a novel simulated annealing to hybrid FSSP to minimize both total completion time and total tardiness. The major disadvantage of this transformation is that the weight of each objective must be given subjectively in advance. It results in great gaps between the real and expected solutions if the decision maker is not experienced enough. An alternative method used to solve the MOFSSP does not specify the weights in advance. It provides various solutions to the decision makers for reference. These solutions are called Pareto optimal solutions. The number of Pareto optimal solutions is often not sole, and decision makers can select one of the solutions to meet their demands. It is a more flexible way than the constant weight weighting method. The representative works of this approach include the following: Qian et al. [10] proposed an effective differential evolution based hybrid algorithm for MOFSSP. Pasupathy et al. [11] proposed a multi-objective genetic algorithm named Pareto GA with an archive of nondominated solutions subjected to a local search. Melab et al. [12] proposed a grid-based parallel GA with the aim of obtaining an accurate Pareto front. They focused on FSSP with the minimization of the makespan and the total tardiness criteria. Tavakkoli-Moghaddam et al. [13] investigated a novel multi-objective model for a no-wait FSSP that minimizes both the weighted mean completion time and weighted mean tardiness. Li and Wang [14] solved the MOFSSP using a hybrid quantum-inspired GA. Chang et al. [15] presented a mining gene structures on subpopulation genetic algorithm which is combined with mining gene structure approach and subpopulation genetic algorithm. Dugardin et al. [16] focused on the multi-objective resolution of a reentrant hybrid FSSP. The two objectives are the maximization of the utilization rate of the bottleneck and the minimization of the maximum completion time. Karimi et al. [17] presented a multiphase approach to tackle hybrid flexible FSSP considering the minimization of makespan and total weighted tardiness simultaneously. Geiger [18] described the proposition and application of a local search metaheuristic for MOFSSP. Chiang et al. [19] considered the makespan and the total flow time as objectives and proposed a memetic algorithm to solve the FSSP. Dubois-Lacoste et al. [20] presented a new, carefully designed algorithm for five bi-objective permutation FSSPs that arise from the pair wise combinations of the objectives: makespan, the sum of the completion times of the jobs, and the weighted and nonweighted total tardiness of all jobs. Cho et al. [21] dealt with a scheduling problem for reentrant hybrid flow shop with serial stages where each stage consists of identical parallel machines.

The previous literature shows that the researchers have established many different multi-objective models for FSSP by considering different factors, but none of these models have taken the inventory capacity into account. The completion time, tardiness, flow time, lateness, earliness, and the number of tardy jobs are often used as the objectives in MOFSSP. Therefore, we considered both the inventory capacity and the completion situation of the jobs, established an inventory based MOFSSP model in this paper. To solve the proposed MOFSSP model more effectively, we used some tailor-made genetic operators and designed a new local search operator. Based on these genetic operators, we proposed a hybrid genetic algorithm (HGA). Finally, the efficiency of the proposed algorithm was verified by computer simulations on some typical FSSPs.

The remainder of the paper is organized as follows. In Section 2, we introduce the concepts of multi-objective problems. Then, we describe MOFSSP and its model in Section 3. In Section 4, a hybrid genetic algorithm to the MOFSSP is presented. Section 5 presents the experimental results. The conclusions are made in Section 6.

2. Multi-Objective Optimization and Pareto Optimality

Usually, a 𝑘 objective multi-objective optimization problem can be described as follows [22]: minimize𝑦=𝑓(𝑋)=𝑓₁(𝑋),𝑓₂(𝑋),…,𝑓_𝑘(𝑋),subjectto𝑔_𝑖(𝑋)≥0𝑖=1,2,…𝐷, (1) where 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 is the decision vector, 𝑋∈Θ⊂𝑅^𝑛, Θ is the search space. 𝑌???is the objective space, and 𝑦∈𝑌 is the objective vector. 𝑔_𝑖,𝑖=1,2,…𝐷 is a constraint function.

The following concepts are often used in multi-objective optimization problems.

Definition 1. Let a decision vector 𝑋₁∈Θ.(1)𝑋₁ is said to dominate a decision vector 𝑋₂∈Θ(𝑋₁≺𝑋₂) if and only if 𝑓_𝑖(𝑋₁)≤𝑓_𝑖(𝑋₂)𝑖=1,2,…,𝑘, and ∃𝑖∈{1,2,…,𝑘}satisfying𝑓_𝑖(𝑋₁)<𝑓_𝑖(𝑋₂).(2)𝑋₁ is said to be Pareto optimal if and only if ¬∃𝑋₂∈Θsatisfying𝑋₂≺𝑋₁.(3)𝑃_𝑆={𝑋₁∈Θ∣¬∃𝑋₂∈Θsatisfying𝑋₂≺𝑋₁} is said to be Pareto optimal set of all Pareto optimal decision vectors.(4)𝑃_𝐹={𝑓(𝑋)=(𝑓₁(𝑋),𝑓₂(𝑋),…,𝑓_𝑘(𝑋))∣𝑋∈𝑃_𝑆} is said to be Pareto optimal front of all objective function values corresponding to the decision vectors in 𝑃_𝑆.(5)𝑋₁ is said to be non-dominated regarding a given set if 𝑋₁ is not dominated by any decision vectors in the set.

Pareto optimal decision vector cannot be improved in any objectives without causing degradation in at least one other objective. When a decision vector is non-dominated on the whole search space, it is Pareto optimal.

3. Problem Definition and Mathematical Modeling

In classical FSSP [23], we have a set of 𝑛 jobs and a set of 𝑚 machines. Each job consists of 𝑚 operations that have to be processed in a specified sequence. All jobs visit machines in the same processing route, starting from machine 1 until finishing on machine 𝑚. The operation of job 𝑖 on machine 𝑗 lasts for a fixed and predetermined amount of time, denoted by 𝑡_𝑖,𝑗. Other assumptions commonly characterized for FSSP are as follows. (i)Each machine can only process one operation at a time, and each job can be processed by one machine at a time.(ii)A job can visit a machine once and only once, and preemption of operations is not allowed.(iii)There are no precedence constraints among the operations of different jobs.(iv)Neither release times nor due dates are specified, and setup times are included in processing times.(v)There is no machine failure; therefore, machines are always available.

The objective of FSSP is to find the optimal schedule, that is, the schedule of the operation sequences and starting time on each machine so that one or more given criteria are optimized.

In the most of previous MOFSSPs, the completion time, flow time, lateness, earliness, tardiness, and the number of tardy jobs are often used as the optimization criteria. These objectives are established based on the completion situation of the jobs and are almost not taking the inventory capacity into account. As we all know, the completion situation of each job is the most concerned problem in FSSP, but the inventory capacity is also an important issue which should not be ignored. Especially for those factories with high inventory costs but very low transportation costs, if the volume (or size) of each operation is large, then we need a very large depot to put all the operations, and this will increase inventory cost. In order to reduce the inventory cost, we can try to use as little space of depot as possible by delivering the operations required in batches to the depot and transported them away after process. Of course, the operations are different for different schedules at a fixed time, and what we need to do is finding a reasonable scheduling scheme with the inventory capacity needed as small as possible. Thus, we can construct an inventory based MOFSSP model to minimize the makespan and the inventory capacity simultaneously.

Given a feasible schedule, we can know the operation sequences and the completion time on each machine. We introduce several notations at first:(i)𝑛: the total number of jobs to be scheduled;(ii)𝑚: the total number of machines in the process;(iii)𝑡_𝑖,𝑗: the processing time for job 𝑖 on machine 𝑗;(iv)𝐶_𝑖,𝑗: the completion time for job 𝑖 on machine 𝑗;(v)𝑆=(𝑠_𝑖,𝑗)_𝑚×𝑛: the processing sequence matrix of all jobs, where (𝑠_𝑖,𝑗) means the 𝑗th processed job number on machine 𝑖.(vi)𝑊_𝑖𝑗: the volume of the operation for job 𝑖 on machine 𝑗.

We use 𝐶_max=𝐶_{𝑠𝑚,𝑛,𝑚} to represent the maximum makespan and 𝑆_max=max_{0≤𝑝≤𝐶max}∑^𝑛_𝑖=1∑^𝑚_𝑗=1(𝑁_𝑖𝑗𝑝×𝑊_𝑖𝑗) to represent the maximum inventory capacity needed, where 𝑁_𝑖𝑗𝑝=1,𝐶_𝑖𝑗−𝑡_𝑖𝑗=𝑝0,𝐶_𝑖𝑗−𝑡_𝑖𝑗≠𝑝,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚,𝑝=0,1,…𝐶_max. (2)

Then, a two objective optimization model for FSSP can be formulated as follows: minimize𝑓𝑋=𝐶_max,𝑆_max,subjectto𝐶_𝑠1,1,1=𝑡_𝑠1,1,1𝐶_𝑠1,𝑖,1=𝐶_{𝑠1,𝑖−1,1}+𝑡_𝑠1,𝑖,1𝐶,𝑖=2,…,𝑛_{𝑠𝑗,1,𝑗}=𝐶_{𝑠𝑗,1,𝑗−1}+𝑡_{𝑠𝑗,1,𝑗}𝐶,𝑗=2,…,𝑚_{𝑠𝑗,𝑖,𝑗}=max𝐶_{𝑠𝑗,𝑖−1,𝑗},𝐶_{𝑠𝑗,𝑖,𝑗−1}+𝑡_{𝑠𝑗,𝑖,𝑗},𝑖=2,…,𝑛;𝑗=2,…,𝑚, (3) where (3) is to minimize a vector of objective functions, in which 𝑋 is a feasible schedule.

4. A Hybrid Genetic Algorithm for MOFSSP

Genetic algorithms were proposed by Holland [24] and have been successfully used in a variety of combinatorial optimization problems. In general, GA works by keeping a fixed number of candidate individuals (population). Each corresponds to a solution of the problem. To describe or formulate a solution, an appropriate representation of individual must be defined, which encodes a set of genes and joins them together to form a string of values. In each generation of GA, a fitness function is required to evaluate the fitness value of each individual. Then, a scheme which favors fitter individuals is used to randomly select parents from the population to form the mating pool. Next, crossover and mutation operators are used to generate better offspring. Furthermore, the offspring are inserted into the current population to form new population for next generation. The process is iterated until a specified stopping criterion is reached.

Our proposed hybrid genetic algorithm adopts the general structure described earlier. We will explain the details of our implementation of the genetic algorithm for the MOFSSP in the following.

4.1. Encoding and Decoding

Defining an appropriate encoding scheme is one of the most important and critical issues for constructing an efficient GA. The following encoding scheme proposed by [25] is adopted in this paper: a chromosome is composed of 𝑚 substrings, corresponding to 𝑚 different machines. Each substring represents the sequence of the operations processed on each machine, denoted by a permutation of integers from 1 to 𝑛. The total length of a chromosome is 𝑛×𝑚. Consider a 4-job and 3-machine FSSP as an example. Suppose a chromosome is: [(2 1 4 3) (3 4 1 2) (2 1 3 4)], then the substrings (2 1 4 3), (3 4 1 2), (2 1 3 4) represent, respectively, the sequences of the operations processed on machine 1, 2, and 3.

4.2. Fitness Function and Selection Operation

The fitness function is defined as follows in this study.

Suppose the current population set is 𝑃(𝑡). Each individual 𝑋_𝑖∈𝑃(𝑡) is compared with all other individuals in 𝑃(𝑡) first and then we get a set 𝐶(𝑋_𝑖)={𝑋_𝑗∣𝑋_𝑗∈𝑃(𝑡)(𝑋_𝑗≻𝑋_𝑖)}. Let 𝑓(𝑋_𝑖)=|𝐶(𝑋_𝑖)|, where 𝑓(𝑋_𝑖) is the fitness value of individual 𝑋_𝑖∈𝑃(𝑡), and |𝐶(𝑋_𝑖)| represents the number of the individuals in 𝐶(𝑋_𝑖).

Suppose 𝑁 is the population size and 𝑓(𝑋_𝑖) means the fitness value of individual 𝑋_𝑖∈𝑃(𝑡), then the selection probability of 𝑋_𝑖 is defined by 𝑝(𝑋_𝑖)=𝑓(𝑋_𝑖)/∑^𝑁_𝑘=1𝑓(𝑋_𝑘).

4.3. The Crossover Operator

Information is swapped among the chromosomes presenting in the mating pool to create new chromosomes in crossover operator. A crossover operator proposed by [26] is adopted in this paper. Suppose that 𝑥, 𝑦 are two parent individuals, the crossover operator can be described as follows.

Algorithm 2 (crossover operator).

Step??1. Divide the machines 1,2,…,𝑚 into two complementary sets 𝐴 and 𝐵.

Step??2. For two parent individuals 𝑥 and 𝑦 exchange the genes in the substrings that belong to set 𝐴 with probability of 𝑝_𝑐 and get two children 𝑥^′ and 𝑦^′.

Exchanging the genes in the substrings that belong to set 𝐴 is equivalent to exchanging the sequences of the operations processed on machines which belong to set 𝐴. In this way, the children can effectively inherit the sequences of the operations processed on machines.

From the proposed crossover operator, we can see that this operator is easy to implement, and the children can inherit the sequences of the operations processed on some machines of their parents.

4.4. The Mutation Operator

The mutation operator avoids convergence to local optima solution and diversifies the search directions.

We use a mutation operator designed in [27]. It can be described as follows. Given chromosome 𝑥, mutation generates 𝑦 by the following procedure:

Algorithm 3 (a mutation operator based on the critical path).

Step??1. Calculate the critical path of individual 𝑥.

Step??2. Swapping two successive operations 𝑣 and 𝑤 processed on the same machine with probability 𝑝_𝑚, where 𝑣 and 𝑤 are two successive operations on a critical path of parent 1. Then, we can get the offspring, denoted by 𝑦.

As can be seen from [28], for any feasible solution 𝑥, it is possible to construct a finite sequence of mutation operations, which will lead 𝑥 to a globally minimal solution. Thus, the mutation operator used in this paper improved the possibility of production of the excellent individuals and ensured the convergence of the algorithm.

4.5. A New Local Search Operator

To improve the convergence speed of the algorithm, a new local search operator is designed in this paper. For individual 𝑥, suppose 𝑥^′ is the final offspring through our local search operator, and 𝑘 is the number of the new solutions to be generated by our local search operator.

Algorithm 4 (a new local search operator).

Step??1. Let 𝑖=1, 𝑁_𝜀=𝜙 (𝜙 represents an empty set).

Step??2. ?If 𝑖≤𝜅, go to Step 3, else, go to Step 5.

Step??3. Randomly select a substring of 𝑥, randomly generate a sequence of the selected substring and get a ?new solution 𝑁_𝑖.

Step??4. Put solution 𝑁_𝑖into set 𝑁_𝜀, 𝑖=𝑖+1, and then turn to Step 2.

Step??5. ?Choose the best individual from 𝑁_𝜀 as offspring 𝑥^′.

4.6. The Framework of the HGA

We proposed a hybrid genetic algorithm (briefly HGA) to solve the MOFSSP based on aforementioned genetic operators. The framework of the HGA can be described as follows.

Algorithm 5 (HGA).

Step??1. Generate initial population 𝑃(0), and let 𝑡=0. Calculate the fitness value of each individual in 𝑃(0).

Step??2. Randomly select two individuals from 𝑃(𝑡) and use Algorithm 2 to get two offsprings. Repeat the ?above procedure 𝑁/2 times and get 𝑁 offspring, denoted the set of these offspring by 𝑃^′(𝑡).

Step??3. For each individuals in 𝑃^′(𝑡), use Algorithm 3 to get a set of offspring, denoted by 𝑃^′′(𝑡).

Step??4. Calculate the fitness value of each individual in 𝑃^′′(𝑡).

Step??5. For each individual in 𝑃^′′(𝑡), use Algorithm 4 with probability 𝑝_𝑙 and get a set of offspring, denoted by ?𝑃^′′′(𝑡).

Step??6. Select 𝑁 individuals from 𝑃^′′′(𝑡) to get a tentative population, still denoted as 𝑃^′′′(𝑡), then use the ?elitist strategy to the union of 𝑃^′′′(𝑡) and 𝑃(𝑡) to get the next generation population 𝑃(𝑡+1).

Step??7. If the stop criterion is satisfied, then stop. Otherwise, let 𝑡=𝑡+1, and turn to Step 2.

5. Simulation Results

5.1. Algorithm Description and Test Problems

We selected several benchmarks with different scales to test the performances of HGA in this Section. Benchmarks Rec01-Rec41 are selected from [29]. The proposed HGA was compared with a non-dominated sorting multi-objective evolutionary algorithm, called NSGA-II reported in [30], which is recently used by many researchers to illustrate the superiority of their algorithms.

5.2. Measures for Comparing Nondominated Sets

Multiple and uniformly distributed solutions are required to form a Pareto front in multi-objective problems. It is not easy to compare the non-dominated solution sets between two algorithms. The following three comparison metrics are considered in this paper [31]. Suppose algorithm 𝑗,𝑗=1,2 obtains a non-dominated solution set 𝑆_𝑗. The union of two non-dominated solution sets (i.e., 𝑆=𝑆₁∪𝑆₂) is denoted by 𝑆.

(1) The number of the obtained non-dominated solutions can be defined as NDSN𝑆_𝑗=𝑆_𝑗−𝑥∈𝑆_𝑗∣∃𝑦∈𝑆∶𝑦≺𝑥. (4) The larger the value NDSN(𝑆_𝑗) is, the better the solution set 𝑆_𝑗 is.

(2) The distance of the obtained non-dominated front to Pareto front (if known) can be defined as NDSAD𝑆_𝑗=∑^|𝑆𝑗|_𝑖=1𝐷𝑥_𝑖,𝑥^∗𝑆_𝑗. (5)NDSAD(𝑆_𝑗) denotes the average distance from each solution 𝑥_𝑖 (𝑥_𝑖∈𝑆_𝑗) to an ideal solution 𝑥^∗, where 𝐷(𝑥_𝑖,𝑥^∗) is the distance between 𝑥_𝑖 and an ideal solution 𝑥^∗in the objective space, 𝐷𝑥_𝑖,𝑥^∗=𝑓₁𝑥_𝑖−𝑓₁(𝑥^∗)²+𝑓₂𝑥_𝑖−𝑓₂(𝑥^∗)². (6) Since 𝑥^∗is unknown, 𝐷(𝑥_𝑖,𝑥^∗) cannot be directly measured. We use an approximated value 𝑓₁(𝑥^∗)=min_𝑗𝑓₁(𝑥_𝑗) and 𝑓₂(𝑥^∗)=min_𝑗𝑓₂(𝑥_𝑗) to replace 𝑓₁(𝑥^∗) and𝑓₂(𝑥^∗). Obviously, the smaller the value NDSAD(𝑆_𝑗) is, the better the solution set 𝑆_𝑗 is.

(3) The spread and distribution of the obtained non-dominated solutions can be defined as SM𝑆_𝑗=∑^𝑛_𝑖=1𝑑_𝑖−𝑑(),𝑑𝑛−1_𝑖=min_𝑗𝑓^𝑖₁(𝑋)−𝑓^𝑗₁(𝑋)+𝑓^𝑖₂(𝑋)−𝑓^𝑗₂(𝑋), (7) where 𝑗=1,2,…𝑛 and 𝑖≠𝑗. 𝑑 are the mean value of all 𝑑_𝑖, and 𝑛 is the number of solutions in set 𝑆_𝑗.

5.3. Test Results and Discussions

The algorithms are implemented in Visual C++ on a computer with a 3.20?GHz Pentium(R) Dual-Core CPU. The parameters used in simulations are as follows: for both the NSGA-II and HGA, the population size 𝑁=100, the crossover probability 𝑝_𝑐=0.7, and the mutation probability 𝑝_𝑚=0.1. For HGA, the local search probability 𝑝_𝑙=0.5 and 𝜎=10. For all benchmarks, suppose the volume of operation 𝑂_𝑖𝑘 is 𝑊_𝑖𝑘=𝑖, where 𝑂_𝑖𝑗 represents job 𝑖 when processed on machine 𝑗, 𝑖=1,2,…𝑛, 𝑗=1,2,…𝑚. The algorithms were run 10 independent runs on each benchmark. The proposed HGA is applied to a number of benchmarks, and its performance is compared with NSGA-II.

Table 1 lists the values of metrics NDSN(𝑆_𝑗), NDSAD(𝑆_𝑗), and SM(𝑆_𝑗) obtained by HGA and NSGA-II, respectively. From the values of NDSN(𝑆_𝑗), we can know that the set of the non-dominated solutions obtained by NSGA-II are dominated by that obtained by HGA for all test problems. It indicates that HGA tends to find non-dominated solutions with higher quality than NSGA-II. From the values of NDSAD(𝑆_𝑗), we can know that the values of NDSAD(𝑆_𝑗) obtained by HGA are smaller than those of NSGA-II for all the test problems. It shows that the distance between the Pareto front and the non-dominated solutions obtained by HGA is smaller than those by NSGA-II. From Table 1 we also can know that the values of SM(𝑆_𝑗) obtained by HGA are smaller than those of NSGA-II for all the test problems. It shows that uniformity of the distribution of the non-dominated solutions obtained by HGA is better than that of NSGA-II.

Table 1 

Comparisons of metrics NDSN(𝑆_𝑗), NDSAD(𝑆_𝑗), and SM𝑆_𝑗.

By the above comparison, we can know that both the quality and the distribution of non-dominated solutions obtained by our algorithm are better than those by NSGA-II. This shows that the genetic operator designed in this paper is effective, and HGA performs better than NSGA-II.

6. Conclusions

We considered both the completion time of the jobs and the inventory capacity as objectives and constructed an inventory based MOFSSP model in this paper. To solve the proposed model, a local search operator was designed in order to improve the quality of the solutions. Based on these, a hybrid genetic algorithm was proposed. The experimental results show that the proposed algorithm is effective and performs better than the compared algorithm.

Acknowledgment

The work is supported by the National Natural Science Foundation of China (61272119).