Single Machine Scheduling Based on EDD-SDST-ACO Heuristic Algorithm

1 Introduction

Within the multi-item and small lot size production mode in the manufacturing industry, there are a large number of changeovers, such as changing moulds, assembling fixtures, and adjusting machines, which all lead to many setup time requirements. Thus, setup time is the changeover time between the end of processing the current job and the beginning of processing the next job. The time consists of the time required to change moulds, assemble fixtures, and adjust machines. Consequently, setup time affects completion time and production efficiency. For instance, the survey data of one manufacturing enterprise shows that the setup time is three to five times longer than the total processing time of 3000 jobs in a cold upsetting process. The setup time equals 40% of the total processing time of 200 jobs on a computer numerical control (CNC) turning machine. However, the traditional approach to addressing setup time is either to neglect or include it in the processing time. In fact, due to the different features of different kinds of jobs, the setup time usually depends on the processing sequences. For instance, in the plastic industry, França et al. [8] indicated that the setup time for different types and colours of products is sequence dependent.

The study of single machine scheduling (SMS) problem with a sequence-dependent setup time (SDST) is greatly significant in modern manufacturing. Wilbrecht and Prescott [22] found that the effect of SDST is significant when a job shop is operated at or near full capacity. Panwalkar et al. [19] reported that 70% of the managers should consider the SDST in production scheduling. Allahverdi et al. [1] presented an excellent survey of a scheduling problem with an SDST in many industries, including controlled manufacturing, electronics, and printed board manufacturing. Therefore, this paper investigates the issue of SMS-SDST by minimising the makespan.

The rest of this paper is organised as follows. In Section 2, the literature of the SMS-SDST problem is reviewed. Section 3 presents a model developed to optimise the SMS-SDST problem. In Section 4, the earliest due date (EDD)-SDST-ACO heuristic algorithm is presented. Section 5 presents a numerical example. Section 6 gives some conclusions and introduces some interesting directions for future studies.

2 Literature Review

The literature on scheduling problems involving setup time divides scheduling problems into the sequence-independent and sequence-dependent groups. Shop environment scheduling problems were further categorised as single machine, parallel machines, flow shops, and job shops involving setup time (costs) [1]. Mathematical programming methods have been used for solving the job shop scheduling problem with SDST [3, 4, 5, 10, 21, 23]. Naderi et al. [17] presented a genetic algorithm (GA) to solve job shop problems. The objective is to minimise the maximum completion time or the makespan based on SDST. The GA is hybridised with a diversification mechanism, namely the restart phase, and a simple form of localised searching to enrich the algorithm. Raman et al. [20] proposed a heuristic algorithm for the problem 1|S_ij|∑w_jT_j based on a static dispatching rule. An ACO algorithm was proposed for the SMS problem 1|S_ij|∑w_jT_j [14, 15]. Gagne et al. [9] compared the ACO algorithm with several heuristics for solving an SMS problem. Moreover, solution methods for the SMS problem with constant setup times are abundant in the literature.

Lageweg et al. [13] defined the minimisation of the total weighted tardiness (1||Pw_jT_j) on a single machine as strongly NP-hard. Das and Ooi [7] addressed the inverse prediction of unknown parameters in a rectangular fin geometry subjected to a variable surface heat transfer coefficient. A differential evolution algorithm [6] is shown and indicates that the present problem is inherently ill posed in terms of the retrieval of the value of fluid thermal diffusivity for which many possible solutions exist. The problem is expected to adapt the fin under different conditions.

The literature above studied job-machine scheduling with SDST. The objectives were to minimise the maximum completion time, makespan, total weighted tardiness time, etc. They presented some heuristic algorithms to solve the problems, such as ACO algorithms and iterated local search heuristic algorithm. They considered sequence-independent setup time and SDST because the setup time influences the completion and tardiness times directly.

3 Problem Description

We now give a formal description of the problem. We have n jobs, which are all available for processing at time zero on a continuously available single machine. The machine can process only one job at a time. Associated with each job j is the required processing time (p_j) and due date (d_j). In addition, there is a setup time (S_ij) incurred when job j follows job i immediately in the processing sequence. Let π be a sequence of the jobs, π=[π(0), π(1), …, π(n)], where π(k) is the index of the kth job in the sequence and π(0) is a dummy job representing the starting setup of the machine.

The objective of the problem is to find a sequence with minimised makespan of jobs. Using the three-field notation, the problem can be denoted by 1/S_ij/C_max.

3.1 Assumptions

In this section, the objective of SMS-SDST is to minimise makespan, as shown by 1/S_ij/C_max. Assuming there are five jobs needed for processing on a machine M, the problem is schematically depicted in Figure 1.

Figure 1:

Schematic Diagram of SMS with Setup Time.

Assume that the initial processing sequence is π(X)={1, 4, 2, 3, 5} as in Figure 1A. The matrix S_ij represents the setup time between the ith job and the adjacent jth job. It is sequence dependent and asymmetrical, so the element S_0i represents the setup time of the ith job, which is the first position in the sequence. Suppose that S_0i=0 and i=1, 2, 3, 4, 5. The optimal processing sequence is shown in Figure 1B: π(X′)={1, 3, 2, 4, 5}. It obviously reduces the completion time.

The characteristics of the problem considered in this paper are shown as follows:

1. All kinds of jobs are independent and available for their process at t₀ =0.

2. One kind of job includes many of the same jobs.

3. The processing time and setup time are non-negative integers. The processing time is sequence independent and is known in advance. The setup time depends on the sequence of the processing jobs.

4. The process of a kind of job on the machine cannot be interrupted.

4 EDD-SDST-ACO Heuristic Algorithm

SMS has been proven to be an NP-hard problem. The initial population of a traditional intelligent algorithm is randomly generated, and it increases the searching time. In order to improve the efficiency of searching, He et al. [11] proposed that shortened due date and on-time delivery were two important factors to improve the search efficiency and convergence of algorithm capabilities. This paper combines the EDD-SDST rule and an ACO algorithm to solve the problems. The initial population is part of a feasible solution that is generated using the heuristic algorithm. The ACO algorithm is a positive feedback mechanism and parallelism. Several solutions are built at the same time; they interchange information during the procedure, and they use information from previous iterations. Therefore, the computing capability and efficiency of the algorithm can be greatly increased. Furthermore, the heuristic probability method does not easily fall into the local optimum. The EDD-SDST-ACO algorithm relationship model is shown in Figure 2.

Figure 2:

EDD-SDST-ACO Algorithm Relational Models.

The specific steps of the EDD-SDST-ACO algorithm are described below:

1. n is the number of jobs waiting for processing.

2. All kinds of jobs are assumed to arrive simultaneously. According to the EDD rule, the kind of job is selected with the EDD in the first place of the sequence to process, and its setup time is 0.

3. Choose the successive neighbouring job. The successive neighbouring job is determined by the minimum setup time. If the same setup time occurs during the search, the minimum delivery time will be the sub-option. This procedure will continue until all of the work pieces are processed. Assuming there are m ants and n kinds of jobs, the setup time between the two neighbouring processing positions σ_i and σ_j is S (σ_i, σ_j) (i, j=1, 2, …, n), and the setup time of the first position job is 0. The makespan between the two neighbouring positions is C_σk−iσi(i=1, 2, …, n), and ant k (k=1, 2, …, m) can choose the successive neighbouring job through the density of the pheromones. For each ant k, the probability P^k (σ_iσ_j) (t) of moving from the sequence of the ith job to the jth job at time t is calculated as follows:

   (10)Pk(σi, σj)={[τ(σi, σj)(t)]α⋅[η(σi, σj)(t)]β∑s∈allowk[τ(σi, σs)(t)]α⋅[η(σi, σs)(t)]β,s∈allowk0,s∉allowk

   where τ (σ_i, σ_j) (t) is the pheromone trail associated with the assignment of the ith job and the jth job at time t. At the initial time, the pheromone trail is identical at each connection path between jobs, as τ (σ_i, σ_j) (0)=τ₀. α, β are parameters that are used to establish the relative influence correlativity of η versus τ. η (σ_i, σ_j) (t) is the heuristic function; η (σ_i, σ_j) (t)=1/C_σi) is the extent of expectation that the ant moves from the ith job to the jth job; and allow={0, 1, …, n−1} represents the possible successive neighbouring jobs except the beginning piece as time goes on. The element of this set continues to decrease until it reaches 0 (i.e. all of the jobs are processed).

In our algorithm, we use the dispatching rule of EDD-SDST-ACO as the heuristic η (σ_i, σ_j). As mentioned above, the pheromone levels will gradually disappear when the ants release their pheromones. Set ρ (0<ρ<1) as the volatilisation coefficient. After iteration t is completed, the pheromone concentrations are updated to

where Δτ^k(σ_i, σ_j) is the pheromone trail that the kth ant releases on the path from the ith job to the jth job, and Δτ(σ_i, σ_j) is the sum of the pheromone trail that all of the ants release on the path from the ith job to the jth job.

The ant cycle system model is used to calculate the release pheromone trail, as shown in Eq. (12) [2]:

where Q is a constant variable that represents the sum of the release pheromone trail of all ants in a full cycle and S_k means the length of the path, which is the kth ant moving. When the ant takes a shorter path, it releases a higher pheromone trail. The working diagram of ACO optimisation is shown in Figure 3.

Figure 3:

The Working Diagram of EDD-SDST-ACO Optimisation.

5 Experimental Evaluation

In this section, some experiments that are performed for the purpose of assessment of effectiveness and competitiveness of applied algorithms are presented. We test our proposed algorithm against recent successful reported meta-heuristic algorithms in the literature, including ACO [18] and GA. These algorithms belong to the most popular evolutionary algorithms. Also, the computational tests that are performed to evaluate the effectiveness and efficiency of algorithms are described. Experimental tests are divided into two groups including small- and large-scale problems. The procedure of data generation is described in the next subsection.

5.3 Computational Results

5.3.1 Parameters Setting, Taguchi Methods

The proposed EDD-SDST-ACO algorithm contains five factors that control the effectiveness and efficiency of the algorithm. The factors have to be adjusted to the problem in order to get the best solutions of problems. The value of the control factors including α, β, ρ, Nuts number, and Inter-Max are determined by a set of experiments that are designed by the Taguchi method. For this reason, at first, the efficient range of each parameter is determined using the experience by initial test problems relatively (Table 2), and then the exact values are tuned by analysing the interaction between factors.

The designed experiments of control factors are shown in Table 3, where the quality of solutions and computational time (CPU) are considered as responses. The analysis of variance is done for the SNR and mean of responses of experiments (Figure 4). Finally, the averages of responses and SNR for factors are summarised in Table 4.

Table 4:

Analysis of SNR (Smaller is Better).

Level	α	β	ρ	Nuts	Iter_max
1	−58.77	−58.67	−58.67	−58.59	−58.93
2	−58.78	−58.62	−58.61	−58.66	−58.60
3	−58.54	−58.70	−58.63	−58.74	−58.70
4	−58.61	−58.70	−58.80	−58.71	−58.47
Δ	0.25	0.08	0.19	0.16	0.46
Rank	2	5	3	4	1

Figure 4:

Mean of SNR.

Based on Tables 4–6, Iter_max and α have the first and second ranks. However, other factors have different ranks in these tables; the interactions of factors are depicted in Figures 5 and 6, and the optimal results are α=1, β=4, ρ=0.9, Nuts=30, and Iter_max=50.

Table 5:

Analysis of Means.

Level	α	β	ρ	Nuts	Iter_max
1	617.2	615.8	615.1	600.2	640.5
2	630.3	621.9	610.1	611.4	608.1
3	615.9	615.1	610.2	611.0	617.0
4	605.3	614.3	631.6	616.2	601.5
Δ	25.0	7.6	21.5	16	39.0
Rank	2	5	3	4	1

Table 6:

Analysis of Standard Deviations.

Level	α	β	ρ	Nuts	Iter_max
1	863.3	845.2	845.5	845.3	866.1
2	855.5	826.4	841.5	849.0	842.3
3	829.5	853.5	846.1	837.1	850.1
4	842.8	854.5	846.6	848.1	820.7
Δ	33.8	28.1	5.1	11.9	45.4
Rank	1	3	5	4	2

Figure 5:

Main Effects Plot for Means.

Figure 6:

Main Effects Plots for Standard Deviation.

5.3.2 Computational Experiments

In this section, the performances of the suggested EDD-SDST-ACO, ACO, and GA are evaluated. All the proposed algorithms are implemented on the CNC turning machine. All algorithms run in Matlab R2010a on a PC with CORE i3 M 380 CPU 2.53 GHz and 6.00 GB memory. All instance data are read from Excel 2010, and the results are written to Excel. Only the running time of the main program is counted, excluding data reading and writing. All test problems are solved by the EDD-SDST-ACO algorithm 20 times. A detailed survey is performed on the computational results regarding the four parameters included: Min.sol, Avg.sol, Max.sol, and CPU time.

As Tables 7 and 8 show, the performance of the EDD-SDST-ACO algorithm is obviously significant. Furthermore, in addition to increasing the number of job categories, the effect is more obvious than others, while the running time is greater than that of ACO and GA.

Table 7:

Comparison between the Results of EDD-SDST-ACO Algorithms and ACO.

n	EDD-SDST-ACO				ACO				Ave.sol improved (%)
	Min.sol	Max.sol	Ave.sol	CPU (s)	Min.sol	Max.sol	Ave.sol	CPU (s)
10	1068	1289	1178.5	14	1180	1339	1239.3	12	4.9
20	2160	2450	2305	21	2378	2590	2484	18.4	7.2
30	3456	3651	3553.5	27.3	3612	3832	3722	28	4.5

Table 8:

Comparison between the Results of EDD-SDST-ACO Algorithms and GA.

n	EDD-SDST-ACO				GA				Ave.sol improved (%)
	Min.sol	Max.sol	Ave.sol	CPU (s)	Min.sol	Max.sol	Ave.sol	CPU (s)
10	1068	1289	1178.5	14	1150	1301	1225.5	11.8	3.8
20	2160	2450	2305	21	2369	2610	2489.5	18.4	7.4
30	3456	3651	3553.5	27.3	3512	3792	3652	28	2.7

6 Conclusions

This study addresses SMS with an SDST. The studied objective is minimising C_max. For solving the above problem, we proposed a hybrid approach with hybridisation of EDD rule and ACO algorithm. This study developed a model of SMS-SDST to minimise the maximum makespan. Furthermore, an EDD-SDST-ACO heuristics algorithm was proposed to solve this model. Because of the sensitiveness of our proposed algorithm to parameter values, a specific tool named the Taguchi experimental method was employed. Finally, three cases were considered, and the numbers of job categories were 10, 20, and 30. The performance of the proposed EDD-SDST-ACO algorithm was examined by comparing with ACO and GA. The computational results showed that the proposed EDD-SDST-ACO statistically outperformed ACO and GA. In addition, the proposed EDD-SDST-ACO algorithm is also suitable for other machining processes and industries with multi-item and small lot size production mode. Future works should apply the EDD-SDST-ACO algorithm to solve other kinds of combinatorial optimisation problems and to develop some other hybrid algorithms for solving the SMS sequence dependent on the setup time.