SIMULTANEOUS CONSIDERATION OF SEQUENCE-DEPENDENT AND POSITION-DEPENDENT SETUP TIMES IN SINGLE MACHINE SCHEDULING PROBLEMS

In this work we address a single machine scheduling problems with sequence-dependent setup times, in which the setup time and the processing time may depend on the job position in the processing order. We consider two manufacturing environments. In the first one, jobs are processed automatically, then the job positions affect only on the setup times. In the second environment, the operators modify the machine settings between different types of jobs and operate it to process the jobs, then the job positions affect both setup time and processing time. In this work, we will investigate the validity of the assumption: scheduling problems render high-quality solutions that reckon with job positions just as the those reckoned without job positions. Minimising the maximum completion time and the sum of completion time of the jobs are the objective functions. To tackle this scenario, we introduce four mathematical formulations: one formulation for each combination of objective function and manufacturing environment. The validity of the models is established by the results of extensive computational experiments on the proposed models.


INTRODUCTION
There is an immense time-saving when setup times (costs) have been explicitly included in scheduling decisions in various real world industrial and service realms. It is crucial, in some cases, that the setup times must be explicitly considered. Food processing, chemical, printing or metal processing industries just to name a few [2]. Whenever the time required to prepare a machine for processing a given job is also depends on the last scheduled job, the setup time is called as sequence-dependent setup time (ST sd ).
The explicit consideration of ST sd in scheduling problems substantially increases the complexity and hence the problems will be much harder to solve or have to approximate from a computational point of view. For instance, the problem of minimising the Total Completion Time (TCT ) on a machine with independent setup times can be easily solved by the simple shortest processing time rule, whereas if ST sd are considered, the problem becomes an asymmetric minimum latency problem, which is NP-hard in the strong sense [9].
The dependency can be position-dependent, which means that the time required to perform the operations vary depending on the job positions in the processing sequence. When the time variations are a consequence of operation repetitions, they are known as learning effect or deterioration effect in the scheduling literature. If it decreases as the job position grows then it is possible to consider that there is a learning effect, while if it increases it is possible to consider that there is a deterioration effect. Hence, the variation in the operation times due to job positions is regarded as either of these effects.
Many researchers have been studied extensively the deterioration and learning effects. In the literature have found only the problems in which it is assumed that there are no setup times or that they depend only on the job that is going to be processed that can be included in the job processing times. De facto, the studies have been carried out only on the processing times of the jobs. Having said that, we address single machine scheduling problems with ST sd , where both, setup times and processing times, may be affected by position-dependent learning effects.
We consider two manufacturing environments. In the first one, the jobs are processed automatically, while the machine settings between different types of jobs are executed manually (the position affects only the setup times). In the second one, the operators modify the machine settings between jobs and operate the machine to process the jobs (the position affects both setup and processing times). Two performance measures, maximum completion time (makespan, C max ) and total completion time (TCT ), are used. The former is focused on the efficient use of machines and the latter focused on the maximisation of the production flow and the minimisation of the work-in-process inventories.
In this work, we will validate the assumption that prime solutions of the problems that reckoned without job position effect on ST sd are also prime solutions for the problems reckon with the effect. It is less obscure to obtain quality solutions when the job position effect is ignored.
Then, if the assumption were true we could take those solutions and simply evaluate them using the corresponding job position factor to obtain the true values of the objective functions.
Otherwise, it would be necessary to consider job position effect in the production programming process to obtain high-quality solutions.
This study will get through the decision makers to decide whether or not to include this kind of learning effect in the production planning to make more efficient the production process. For, we propose four mathematical formulations, one formulation for each combination of manufacturing environment and objective function. The effectiveness of the models are verified by carrying out extensive computational experiments on the models.
To the best of our knowledge, this is the first time that: -mathematical formulations have been proposed for scheduling problems where setup times simultaneously depend on the sequence and on the positions of the jobs in the processing order.
-investigate how the quality of the solutions is affected by the positions of the jobs in the processing order when setup times are sequence-dependent.
-both ST sd and processing time are affected by the positions of the jobs in the processing order.
In the next section, we will discuss the related literature. In section 3 the mathematical formulations of the problem are presented, then the computational tests followed by the conclusions.

LITERATURE REVIEW
The seminal work of learning effect in scheduling problems was presented by [6], hereinafter a myriad of works have been published about the learning effects on the processing times in scheduling problems. The approaches to tackle the problem have been distinguished [20] as: time-dependent [10], position-dependent [7] and cumulative [14].
In the time-dependent approach, the time needed to produce a unit decrease depends on the starting time of the job. [10] presents a comprehensive description of scheduling models with this effect. While, in the position-dependent approach, the time required to produce a unit decrease rely on the number of repetitions of jobs [6]. Here, processing time p jr of job j in position r is calculated by: p jr = p j · f (a, r), where p j is the processing time without learning effect (normal processing time), a is a constant learning factor and f is a decreasing function with respect to r.
In the cumulative approach, the time required to produce a unit decrease is depending on the normal processing time p j and on an accumulated value of a parameter. Typically, it decreases depending on the sum of the processing times of the all jobs already scheduled [14]. Here, processing time p jr is calculated by: p jr = p j · f a, is the normal processing time of job in position k, and f is inversely proportional to . This effect is also known as the time-dependent learning effect or sum-of-processing-times-based learning effect.
Numerous studies use these approaches to investigate the effect of learning on various scheduling problems. Some recent works addressing single machine scheduling problems are [18,8,15,13,11,23]. For more details about scheduling problems with learning effects see the surveys by [7,19,5].
A particular case of learning effect on processing times so-called past-sequence-dependent (p-s-d) setup time approach, is introduced by [12]. In p-s-d setup time approach the processing time p jr is obtained as the normal processing time plus a value that depends on the sum of the processing times of the all already scheduled jobs; that is, p jr = s [r] + p j , where s [1] = 0, , for r = 1, 2, 3, · · · , n and b is a constant. The s [r] is interpreted as a setup time that depends upon the sum of the processing times of the all already scheduled jobs. [24,21,22] used p-s-d approach and [1] presented a comprehensive survey on scheduling problems with ps-d setup times.
All the works does not consider setup times for the machines or they depend only on the job to be processed and therefore they can be included in the job processing times. However, in practical industrial scenario, it is indispensable to consider the setup time explicitly. Despite the growing interest in scheduling problems involving sequence-dependent setup times, we only found one paper studying learning effect with this kind of setup times in single-machine scheduling problems. Recently, [16] addressed a scheduling problem on a single machine with ST sd .
They considered a position-dependent learning effect only on processing times and their goal was to minimise total tardiness.
In this paper, we address the minimisation of the C max and the TCT in single machine scheduling problems with sequence-dependent setup times. We analyse position-dependent learning effects in two contexts: when the job positions affect only the setup times, and when they affect both setup times and processing times. The main objective is to investigate when it is necessary to consider the job positions in the production programming process to acquire finest solutions.

FORMULATION OF THE PROBLEMS
We consider a set of n independent jobs to be processed on a single machine. Each job j has a processing time p j and there is a machine setup time s i j , which is incurred when job j immediately follows job i. In general, s i j = s ji . At the beginning, the machine is at an initial state 0 (or dummy job 0), and there are setup time s 0 j , prior to process the first job in the machine. All the jobs are available initially, there is an ST sd between jobs, and the preemption is not allowed. The objective function is the minimisation of the C max or the TCT for each manufacturing environment, considering position-dependent learning effects.
A learning effect is represented, in general, by the function y = f (b, r), which depends on the number of r (setups) that have already been done in the machine and on a parameter b (learning factor) associated with the learning rate. In the first manufacturing environment, we assume that the position affects only the setup times. Then the total time, t r i j , required for processing the job j in position r just after job i in position r − 1 is defined as: where s r i j denotes the machine setup time from job i in position r − 1 to job j in position r. In the second manufacturing environment, we assume that both the setup times and the processing times are affected. Then the total time, t r i j , required to process the job j in position r just after job i in position r − 1 is defined as: where p r j denotes the processing time of job j in position r. The function f (b, r) = b r−1 , b ∈ (0, 1], represents the learning rate that is inversely proportional to the learning factor b. In the four formulations, the objective is to find a sequence of jobs P that minimises the corresponding objective function (C max or TCT ). In order to evaluate the objective functions, the solutions are represented by a sequence of n jobs; where, [r] denotes the job in the position r in the sequence P.
For a given sequence P, the makespan considering learning effect can be calculated by: However, the TCT for a sequence P without learning effect can be calculated by [3]: . Further, when the learning effect is considered, it can be calculated by: In expressions (4) and (6) the contribution of each job to the objective function depends on its position in the sequence. Consequently, we should define decision variables that consider the position of the jobs in the sequence. Moreover, the time-dependent formulations for the minimum latency problem [3] can be adapted to this problem.
To formulate the mathematical models, we define the decision variables x r i j as: 1, if and only if jobs i and j occupy positions r and (r + 1) in the sequence 0, otherwise Note that x r i j = 1 means that there are (n − r) jobs after job i in the sequence. Using these variables, it is possible to define the objective functions with learning effect as follows. For the makespan: and for the total completion time: The values of t r i j are calculated by (1) or (2) according to the manufacturing environment under consideration.
Note that, variables x 1 i j appear in these two expressions, (8) and (9), of the objective functions. In the first expression, the variables x 1 i j are used to calculate the contribution of the job in the first position of the processing order. While in (9) they are used to calculate the contribution of the job in the second position.
The set of constraints is defined as follows: Constraints (14) establish the binary nature of the variables x r i j . The studied problems can be seen as particular cases of time-dependent Asymmetric Travelling Salesman Problems (ATSPs). In addition, for b = 1, the problems for minimising the C max can be transformed into ATSPs and the problems for minimising the TCT can be transformed into Asymmetric Minimum Latency Problems (AMLPs). Both the ATSP and the AMLP are NP-hard problems in a strong sense [17,9].
Considering the objective functions and the manufacturing environments, we obtain 4 integer linear formulations that share the set of constraints (10)- (14). We will refer to these models according to the nomenclature presented in Table 1.
The proposed models will be used with different set of instances to study the impact of the learning effects on solution quality when the setup times are sequence-dependent.

COMPUTATIONAL EXPERIMENTS
To

Results for the makespan.
In this section we analyse the impact of the learning effect on the job processing order when the objective is to minimise the makespan.
In Table 2 we show the optimal sequences given by CPLEX using Model I and Model II on the instances with number of jobs 15 and for different levels of b . We can observe that different processing orders have been obtained for different learning levels. In the last column we show the quality of the optimal solution (obtained without learning effect) when it is implemented for the different levels of b. The gap for each value of b is calculated by: where OptimalVal(b) is the optimal solution value for a given value of b, and SolVal withoutLE (b) is the value of the solution without learning effect that is evaluated for the same b value.  for the first manufacturing environment with C max as objective function has little influence over the makespan, whereas in the second manufacturing environment the opposite happens. In other words, using solutions obtained without considering learning effects lead us to implement mediocre sequences in the second manufacturing environment.
The behaviour observed in Table 2 is common for all tested instances as shown in Table 3.
The first column in Table 3   A graphical representation of behaviour of the gaps (on average) for each level of learning factor b and each setup range R is presented in Figure 1. It can be observed that in the first manufacturing environment the learning has very little effect on the production process duration, since the solution obtained without learning effect generally provides job sequences of good quality for all levels of the learning factor. Having said that, in the second manufacturing environment, we note that the sequence obtained without learning has low quality for the different levels of the learning factor. Therefore, in this case if we look for quality sequences it is quintessential to consider the learning effect in the production programming.
For the Model I (learning effect only on the setup times), the gaps grow along with the setup range, for each b. That is, the quality of the solution without learning effect is inferior when the range of variation of the setup times is greater than the range of variation of the processing times (R 3 ) for all levels of the learning factor. However, for Model II (learning effect on both the setup times and the processing times), the opposite happens. That is, the solutions without learning effect are degraded when the range of variation of the processing times is greater than the range of variation of the setup times (R 1 ). In addition, it can be observed that the gaps grow as the b values decrease, i.e., the gaps grow with the learning rate.

Results for the total completion time.
Here we analyse the impact of the learning effect on the job order when the objective is to minimise the total completion time. Firstly, it is presented the results given by CPLEX solver for Model III and Model IV using instances with number of jobs 15.    Table 5 has the same structure as of Table 3. The gap values are calculated using the expression (15). Each class is a set of 20 instances, based on the number of jobs (n), setup range (R), and level of learning factor (b), and then the average of each class is given. The gaps are shown in columns R i corresponding to both models and that are averaged over each class. From these results we can conclude that the learning effect affects in a different way to each manufacturing environment. While it is true that, in both environments, the gap is inversely proportional to b they are much larger for Model IV. This can be better observed when n = 25 and 30.
A graphical representation of the gaps behaviour (on average) for each level of b and each setup range R for the TCT objective function is presented in Figure 2. Then, the gaps are  Thus, from the experimental study, on the one hand, we can conclude that for the makespan the job position effect on setup times could be ignored and the effect on the quality of the solution would be minute, while for the TCT it is vital to consider the job position effect on setup times in the production programming process. On the other hand, when the job position effect affects both setup times and processing times, the solutions obtained without taking into account this effect, for both the C max and the TCT, have an inferior solution. The effects are more pronounced for the makespan. In the manufacturing environment our study suggests, for C max and TCT, that the job positions should be considered in the production programming process. Additionally, it was observed that, the solutions worsen as the range of setup times decreases.
The results of this study allow decision makers to decide whether or not to include the position-dependent learning effect in the production planning to make more efficient the production process. If inclusion is necessary, the proposed models can be used to obtain prime solutions when the number of batches to be scheduled is 30 or less.

CONCLUSIONS
Single machine scheduling problem with sequence-dependent setup times is studied in this article. The setup times and the processing times are affected by the job positions in the processing order. We studied the cases of the position-dependent learning effect affects only the setup times, and affects both setup times and processing times. Two performance measures, the makespan and the total completion time, are employed. To tackle the problem, four mathematical formulations are proposed and validated the efficiency of the models over a set of instances adapted from the literature.
In future work, we will extend design of a heuristic algorithms to tackle for larger instances considering position-dependent learning effect in the production programming process. The generalisation of this study to other scheduling problems with sequence-dependent setup times is also another realm of research.