The study of time-dependent cleaning policy within sequence-dependent scheduling problems

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The study of time-dependent cleaning

policy within sequence-dependent

scheduling problems

Master Thesis SCM & TOM

Qi Yuan (S3021203)

Supervisor: Dr O.A. (Onur) Kilic

Co-assessor: Dr J.A.C. (Jos) Bokhorst

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Abstract

Purpose: In many industries, the scheduling activities are often challenged by both sequence- and time-dependent setups. The problems regarding how to optimize the schedule while respecting sequence- and time-dependent constraint are often difficult to solve. Moreover, the company can come up with different policies to manage these setups and the outcome of these policies remains unclear. The purpose of this thesis is to help the company manage these setups such that the scheduling outcome has the best performance.

Method: In this study, we use mixed integer programming to formulate the problem under investigation. The experiment was designed to compare two different policies under the same scheduling case and we manipulate input variables such as processing time and setup time to see how different policy performs under different production environment.

Findings: We mainly use three different performance measures (makespan, weighted completion time, weighted tardiness) to evaluate the schedule outcome of respected cleaning policy.

In terms of makespan, we found that the fixed policy is the better choice. For weighted tardiness, the flexible policy performs much better in terms of tardiness reduction. Regarding the weighted completion time, both policies deliver similar results and we recommend fixed policy since it is easier to execute and takes less effort to compute.

Conclusion: When the company aims at minimizing makespan, we recommend company employ the fixed cleaning policy. When the company tries to minimize weighted tardiness, the flexible policy is the better choice. In case company prioritize weighted completion time, the fixed policy take fewer management resources and should be preferred over the flexible policy.

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Introduction

In this paper, we investigate a scheduling problem with sequence- and time-dependent setups. The problem is motivated by the fact that in many industries, both setups occur simultaneously. Take the dairy industry as an example:

Consider a dairy production line that involves a mixer machine for mixing sweet additives into yoghurt, and it can only process one job at a time. As a scheduler, the task is to decide the starting time of each job on this machine such that the mixer completes each job without violating their due date. While scheduling, we encounter two constraints: First, due to technical restrictions, If the sweetened yoghurt is produced before the pale one, the mixer needs to be cleaned entirely to ensure no sweet additives remains on the machine. On the other hand, if paled yoghurt is produced before sweetened one, then this cleaning action can be skipped (Kopanos et al. 2011). Second, to prevent contamination, the mixer is not allowed to run ten hours without sterilisation (Gellert et al. 2011). Under this context, The challenge is to find an optimal schedule that contains both the job starting time and the sterilisation window under highly restricted environment. The abovementioned example involves three challenges, and the first challenge is the job due date. In the literature, if the job completion time exceeds its due date, the amount of due date violation is called tardiness. When the scheduling problem has tardiness consideration, it is often difficult to solve. Du and Leung (1990) studied a single-machine sequence-independent scheduling problem with the objective to minimize total tardiness. They reported that the problem was hard to solve and describe it as the NP-hard problem. If the problem is NP-hard, then it falls into a class of complex problems that cannot be solved within polynomial time.

Apart from job due date, the second challenge in our case is sequence-dependent setups. In the literature, if the setup time of any paired jobs is dependent on their job sequencing, then it falls under the concept of sequence-dependent setups. In majority cases, if the sequence-dependency is violated, the setup time usually become much longer. The challenge of optimizing scheduling problem with sequence-dependent setups is also NP-hard: Pinedo (2002) studied a single-machine sequence-dependent scheduling problem with the objective to minimize total completion time. He translated the scheduling problem into travelling-salesman problem (TSP) and reported that the problem was NP-hard even without considering job release time and due date. For more examples regarding the hardness of sequence-dependent scheduling problem, see Allahverdi (2015).

Our third scheduling challenge is the sterilisation policy. Since it is exclusively triggered by time, the sterilisation action under investigation falls under the concept of time-dependent setups. While respecting the hygienic regulation, the company can come up with two different sterilisation policies: fixed and flexible

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2 Unlike sequence-dependent setups, we do not know much about time-dependent setups. Furthermore, the flexible cleaning policy is rarely studied in the field of scheduling. A recent study (Stefansdottir et al. 2017) reviewed cleaning action within the food processing industry and only found one paper (Kilic 2011) that is related to this setup. Apart from food-processing, it also occurs infrequently in chemical production and maintenance planning. Among these papers, authors use different terminologies such as “frequency dependent cleaning (Kondili et al. 1993)”, “time-dependent cleaning (Kilic 2011)” or “cleaning and sterilisation (Gellert et al. 2011)” to describe setups under investigation. One should suspect this setup frequently occur in industries that have high hygiene standards such as pharmaceutical, chemical or food processing, yet the knowledge towards flexible cleaning is rather limited.

For fixed cleaning policy, it can be defined as periodic downtime with the fixed time interval and has a more general appearance: management policies such as time-based maintenance or end-day machine cleaning all belong to this category. In the literature, scheduling problem with the fixed cleaning policy is also NP-hard even without sequence-dependent constraints: Graves and Lee (1999) investigated a single-machine sequence-independent scheduling problem where periodic maintenance has job preemption. Their objective is to minimize maximum tardiness, and they reported the problem was NP-hard. Cassady et al. (2003) investigated a single-machine sequence-independent scheduling problem with weighted jobs and periodic maintenance. They tried to minimize total tardiness and reported the problem was NP-hard. For more examples regarding the hardness of scheduling problem with fixed setup interval, see (Abdul-Razaq et al.1990, Allahverdi et al. 1999, 2008, Allahverdi 2015).

Although both fixed and flexible cleaning policies are feasible options, the company has to decide which one to choose. The objective of this paper is to compare fixed cleaning policy vs flexible cleaning policy regarding scheduling performance while respecting sequence-dependent setups. Since scheduling problem with either of these setups is NP-hard, the proposed problem which contains both of them will be NP-hard as least. Fortunately, early contributions have already covered some characteristics in our problem: Kelly and Zyngier (2007) have addressed the modelling approach for dependent setups, Sun et al. (1999) have covered sequence-dependent scheduling problem with release time and due date. Zammori et al. (2014) studied scheduling problem with sequence-dependent setups and maintenance considerations. There are even studies dedicated to optimising sequence-dependent scheduling problem with periodic downtime (Chen 2008a 2008b). Having said that, the research effort towards flexible cleaning policy is rather limited, and only a few studies (Kondili et al. 1993, Kilic 2011, Gellert et al. 2011) have considered it. Moreover, to our best knowledge, no research consider both types of time-dependent setup in one study, so the benchmark result of these two cleaning policies remains unknown.

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3 The rest paper is arranged as follows: Firstly, the definition of these setups will be provided and relevant papers regarding sequence- and time-dependent setups will be reviewed. Secondly, we present the details of the scheduling problem under consideration. Next, we develop a MIP model for the problem and solve it. After testing the model, the results will be presented and compared. Lastly, the research will end with the discussion and conclusion where we summarise research insights and discuss future research topics.

Theoretical background

This section is organized in the following order: First, we discuss the relevant characteristics of the setups under investigation; second, we review scheduling papers that are relevant to this study; last, we conclude the research insight from the time-dependent setups and conclude this research structure.

2.1 Sequence-dependent setups

In the literature, if setup time (or cost) is dependent on both the job itself and the predecessor or successor of the job, then it is called sequence-dependent setup (Allahverdi et al. 1999). To describe it in mathematical language, many papers (e.g., Ozgur and Brown 1995, Choi et al. 2003, Menezes et al. 2011, Shen and Yao 2015) use (𝑆𝑖𝑗𝑘) to denote setup time for job 𝑗 where job 𝑖 is its

predecessor and job 𝑘 is its successor. Figure 1 “Sequence-dependent setup illustration” shows two examples for this setup:

Figure 1 Sequence-dependent setup illustration

If a scheduling study with sequence-dependent setup does not consider job release time and due date, and the objective is to reduce the total completion time, then the problem is equivalent to travelling salesman problem (TSP) (Choi et al. 2003). In the TSP, jobs are translated into cities (nodes) and the setup time is translated into the distance between the cities (arc). The objective of TSP is to find the shortest route that connects all cities, which is equivalent to “find the job sequence that has the earliest completion time”. If arc distance does not differ between two nodes travelling from each other ( 𝑆𝑖𝑗= 𝑆𝑗𝑖), it belongs to symmetrical TSP problem otherwise it’s

asymmetrical TSP (𝑆𝑖𝑗 ≠ 𝑆𝑗𝑖) (Allahverdi et al 1999). For more examples regarding

sequence-dependent scheduling study that uses TSP as an approach, see Zhu and Wilhelm (2006).

From the modelling perspective, there are multiple approaches to formulate sequence-dependent setup. One straightforward approach is to employ a binary variable 𝑋𝑖𝑗 to indicate if job 𝑖 is

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4 Kelly and Zyngier (2007) investigated the abovementioned formulation and proposed a more efficient way to model the sequence-dependent setups. In their study, they defined the logical relationship between variables so that setup variable can be derived from other variables and is integral in solution. Since they do not have to declare setup variable as a binary search variable, substantial computational time was saved.

2.2 Time-dependent setups

Many studies consider setups are “time-dependent” and the setup of these studies are similar to the definition of ours: Naderi et al. (2009a 2009b) describe it as “periodic maintenance” where maintenance action occurs at a fixed time stamp. Kondili et al. (1993) described it as a cleaning action that the filter needs to be cleaned regularly to prevent impurity build up. A recent study (Stefansdottir et al. 2017) reviewed changeover actions observed in the food-processing industry and proposed their classification on time-dependent setups. Enlightened by their classification, this paper will follow their stream and classifies time-dependent setups into two planning policies: fixed and flexible.

For flexible policy, it is triggered by a timer since the last changeover. Kilic (2011) described the policy as “The maximum amount of time that machine can run without cleaning”. Note that in this definition, the cleaning allowance considers the machine idle time after cleaning: Suppose the machine stays idle for a long time since the last cleaning, the cleaning will be triggered once it reaches maximum allowance even no jobs are planned during this period. To avoid this, in this paper the time allowance does not consider the first idle period since the last cleaning. Figure 2 “Flexible cleaning” describes the timer for this setup.

Figure 2 Flexible cleaning

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Figure 3 Fixed cleaning

2.3 Relevant literature

In this section, we review the papers that are relevant to our research question. We choose scheduling papers that consider both sequence- and time-dependent setups because their questions are closer to ours. Table 1 “Relevant literature” provides an overview of these papers.

Author Year

Time-dependent policy

Objective Shop Setting

Kondili et al 1993 Flexible Max profit Multi-stage Papageorgiou & Pantelides 1996 Flexible Max Prod-value Multi-stage

Kilic 2011 Flexible Makespan 2-stages production

Gellert et al 2011 Flexible Makespan Single-machine

Entrup et al 2006 Fixed Min total cost Single-machine

Kopanos et al 2010 Fixed Min total cost Parallel

Doganis & Sarimveis 2007 Fixed Min total cost 2 Parallel machines

Chen et al 2008 Fixed Makespan Single-machine

Stefansdottir et al 2017 Fixed Makespan 2 stage flowshop

Chen et al 2009 Fixed Makespan Single-machine

Naderi et al 2009a Fixed Makespan Jobshop

Naderi et al 2009b Fixed Makespan Flex flow line

Ángel-Bello et al 2011a Fixed Makespan Single-machine Ángel-Bello et al 2011b Fixed Makespan Single-machine

Kaplanoğlu 2014 Fixed Makespan Single-machine

Zammori et al 2014 Fixed Min late job count Single-machine

Table 1 Relevant literature

2.3.1 Flexible time-dependent policy

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6 upper bound for this variable so that filter will be cleaned once the upper limit is reached. They developed the MILP model of their problem and solved with the branch-and-bound method. Papageorgiou and Pantelides (1996) studied a multi-stage scheduling problem for chemical production, and their objective was to maximize production value. In their research, they also used the “frequency dependent cleaning” to describe the cleaning of chemical impurities. They define frequency dependent cleaningas the max allowance for successive batches and count the number of batches since the last changeover. They also respect the sequence-dependent setup between jobs and developed the MILP model to their problem. They solved their model via branch-and-bound procedure and demonstrated it with case studies.

Kilic (2011) investigated a three-stage scheduling and lot-sizing problem for the evaporated milk production with the objective of minimizing total completion time. He respected the sequence-dependent setups as well as time-sequence-dependent setups (flexible). In his paper, time-sequence-dependent setup was defined as “maximum amount of time that production line can run without cleaning”. To model time-dependent setup, he employed the block planning method. The method consolidates several jobs into a planning block and makes the jobs within each block follow sequential order. Next, he added the upper bound to the block time length and planned the cleaning action between these blocks. Due to the complexity of the problem, it was formulated with constraint programming and solved with CPLEX solver.

Gellert et al. (2011) studied a single-machine scheduling problem for a dairy filling line, and their objective was to minimize total completion time. They respect the sequence-dependent setup and defined flexible setup as “The maximum cleaning and sterilising lag since the last cleaning”. In their paper, they assumed the cleaning and sterilisation can substitute job setup time and has job preemption. If flexible setup (cleaning) interrupts production, the job can be resumed immediately without additional setup time. Since their problem is NP-hard in the strong sense, they solved their problem by developing a genetic algorithm.

2.3.2 Fixed time-dependent policy

Entrup et al. (2006) investigated a single-machine problem with inventory and shelf-life considerations. Their objective was to minimize total cost, and they respect sequence- and time-dependent setups. In their paper, the packaging machine needs to be cleaned in a fixed time interval and the cleaning action will cause material loss. They employed a block-based planning approach similar to Kilic (2011) but with fixed time-dependent setups. In their model, the planning block has a fixed length of 2 days, and the jobs within each block follow sequential order. The cleaning action takes place at the end of each block, and the material loss caused by cleaning happens at the beginning of the next block. They considered a combination of discrete- and continuous- time representation and developed three different MILP models.

Kopanos et al. (2010) investigated a parallel packaging stage for dairy products with the objective to minimize total cost. In their paper, they consider two types of cleaning: a “cleaning and sterilization” as sequence-dependent setup, and a “shutdown cleaning” is performed at the end of each day. They considered the hybrid discrete- and continuous- time representation and developed MILP model for their problem.

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7 They respected the sequence-dependent changeovers and developed the MILP model with a hybrid discrete- and continuous-time representation.

While papers introduced above all refer fixed setup as “shutdown cleaning”, Chen (2008) suggest that calendar interruption such as weekends is important to consider. He addressed a single-machine scheduling problem with sequence- and time-dependent setups and the objective is to minimize total completion time. In his paper, he assumed that weekends have job preemption and may interrupt production. Moreover, if the job was interrupted during setup, it requires more time to resume. Because the problem is NP-hard in the strong sense, he used heuristics method to solve the problem and compared his heuristics with the branch-and-bound procedure.

Chen (2009) addressed a single-machine scheduling and lot-sizing problem for textile production. The objective of his study is to minimize total completion time. Since the fabric has different melting points, the job setup time obeys sequence-dependency. To prevent the machine from breaking down, the periodic maintenance was applied and has job preemption. He assumes that if the maintenance action interrupts production, the job will be resumed with additional setup time. The problem was solved via heuristics, and the results were compared with the branch-and-bound method.

Zammori et al. (2014) addressed “flexible periodic maintenance” with sequence-dependent scheduling problem for a single machine. The model objective was to minimize the total number of late jobs. In their paper, it is possible to perform maintenance earlier than its foreplanned time such that the idle time before the maintenance window is removed. Since they also consider the possibility of machine breakdown and repair, their problem is NP-hard in the strong sense, and it was solved via metaheuristics.

For the dynamic scheduling, Kaplanoğlu (2014) investigated a single-machine scheduling problem with periodic maintenance. In his model, job arrived in a dynamic manner and he proposed a “job inserting plan” with the objective to minimize total completion time. The model was solved via multi-agent-based approach (AI) where maintenance block and jobs are both considered as agents. The maintenance block will optimise itself first and then negotiate with job agents.

Ángel-Bello et al. (2011) addressed a single-machine scheduling problem with sequence-dependent setup and periodic maintenance, and the objective function is to minimize total completion time. In their paper, they translated maintenance action as “a job with setup time” which is similar to Doganis & Sarimveis (2007). The problem was solved by metaheuristics. Naderi et al. (2009a 2009b) studied the scheduling problem for jobshop and flexible flow line. In their research, both sequence-dependent setup and periodic maintenance take place. The objective was to minimize total completion time. Since the problem is complicated, they developed metaheuristics to solve their problem: In the first step, they translate maintenance into machine unavailability constraints; next, they plan the sequence-dependent jobs in between maintenance cycles.

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2.4 Discussion of time-dependent setups

After reviewing abovementioned papers, we conclude two characteristics for time-dependent setups, namely environment and substitutability.

The first characteristic of the time-dependent setup is its operating environment. For flexible policy, it mostly takes place in the food-processing industry and chemical industry. These production processes have a unique feature that product quality deteriorates over time and require regular cleaning to maintain high hygiene standards. We also noticed that the mechanics of flexible policy is similar to age-based maintenance. Qi et al. (1999) define age-based maintenance as “machine needs to be maintained after continuous working for a period of time.” In this sense, the maintenance is also exclusively triggered by time. Although they are similar, the flexible policy in this study does not consider the first idle time after cleaning whereas the time allowance in age-based maintenance usually includes all idle periods.

Unlike the flexible policy, the fixed policy does not occur in a specific industry and can be widely applied in general production cases. In the literature, these setups are usually referred to as the “shutdown cleaning” or “periodic maintenance” which is often chosen due to the ease of management. By using fixed policy, the company knows when will cleaning takes place and can arrange personnel and spare parts in advance. Moreover, in multi-machine case, the planner can disjoint fixed cleaning schedule for each machine, such that the maintenance (or cleaning) labour has a smooth workload.

The second characteristic of time-dependent setup is the substitutability: If the machine is totally cleaned by time-dependent setup, then the first job after cleaning does not have to follow the sequence dependency of the last job before cleaning. In this sense, the time-dependent setup has the function of “reset” or “substitute” sequence-dependent setups, and our plant production has been separated into chunks of cyclic pattern with cleaning action (time-dependent setup) indicating the end of each cycle. In fact, literature (Kondili 1993, Papageorgiou & Pantelides 1996) has already defined this production manner as “the campaign mode” and explains that “if the reliable long-term prediction is available, it is preferred to partition planning horizon into long periods of time”. Figure 4 the campaign mode demonstrates this concept.

Figure 4 the campaign mode

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9 We also notice that the modelling approach for our proposed problem can employ either continuous- or discrete- time representation. The continuous vs discrete modelling approach has been studied by Floudas and Lin (2004) and concludes that discrete time approach has the advantage of “provides a reference grid of time for all operation competing for shared resources” and also “formulates problem and constraint in a straightforward and simple manner.” However, the discrete time approach also has its own limitations: firstly, each time block in the discrete model is a decision variable, which makes the size of the model result extremely large. Secondly, for continuous production process such as dairy production process that consumes input from feeding pipe and produces products continuously, the discrete time grid only provides approximate estimation on the actual process. Due to the limitation of the discrete approach, This time we model the production process by using continuous time representation.

After reviewing existing literature, we only found four papers that are related to flexible setup, which proofs the setup is under-researched. Furthermore, among these papers, none of them covers both fixed and flexible policies. Motivated by empirical observations (e.g., the dairy example mentioned in the introduction), we believe there is a strong need to have a benchmark study for fixed vs flexible cleaning policy to support company decision making, and the implication of this study can be further extended to other time-based regulations such as maintenance planning. Furthermore, this study contributes literature by doing the first scheduling research that investigating both fixed and flexible time-dependent setup.

The core concept of this research can be summarized as:

Benchmark the schedule performance of fixed and flexible policy while also respecting the sequence-dependent setups.

Methodology

In this chapter, we provide details of scheduling problem under investigation. After the problem is formulated, we explain the modelling process step by step. Next, we provide a detailed testing method and explain how each variable is generated. After the model is fully explained, we will test the model in MIP solver, and the testing results will be presented in the next chapter.

Notation Table

Sets: Notation

1: List of jobs 𝑁

2: List of campaign K

Parameters:

1: Setup time of job j 𝐶𝑖𝑗

2: Job processing time 𝑃𝑗

3: Job due date 𝑑𝑗

4: Campaign time limit A

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Decision Variable: Notation

1: Starting time of job 𝑗 𝑆𝑗

2: Campaign allocation 𝐵𝑗𝑘

3: Sequence-dependent variable 𝑋𝑖𝑗

4: Objective functions 𝐶𝑚𝑎𝑥 , ∑ 𝐶𝑤 , ∑ 𝑇 and ∑ 𝑇𝑤

5: Tardiness variables 𝑇1𝑗 and 𝑇2𝑗

6: Tardiness of job 𝑗 𝑇𝑗

6: Starting time for campaign 𝑘 𝐶1𝑘

7: Ending time for campaign 𝑘 𝐶2𝑘

3.1 Problem Statement

Suppose there is a set of jobs waiting to be released on a single machine, and the machine can only process one job at a time. All jobs are available at time 0, and no preemption is allowed. For each job 𝑗 ∈ 𝑁, it has its own processing time 𝑃𝑗 and due date 𝑑𝑗.

All jobs have sequence-dependent setups. The setup duration is dependent on the predecessor of the job. Let job 𝑖 denote for the predecessor of job 𝑗 where both 𝑖 and 𝑗 are job index numbers. The planned jobs obey sequence dependency if 𝑖 < 𝑗 , and violates sequence dependency if 𝑖 > 𝑗. The machine also needs to be cleaned regularly, and there are two different cleaning policy, namely fixed cleaning policy and flexible cleaning policy. For fixed cleaning policy, the machine needs to be cleaned on fixed time interval every A hours and the cleaning duration is 𝐺. For flexible cleaning policy, the machine is allowed to run maximum A hours without cleaning, and the cleaning can be planned earlier than its designed time limit. The cleaning duration for flexible cleaning is 𝐺 as well. Note that cleaning is not allowed to interrupt production, and the first job after cleaning will never violate sequence dependency.

The task is to find a schedule that contains the release time of each job 𝑆𝑗 such that the schedule

has the best performance of respected performance measure while also respecting sequence- and time-dependent setups.

3.2 The Performance measures

In this research, we mainly consider three different scheduling performance measures namely: Total makespan, weighted completion time, total tardiness and weighted tardiness. We choose these because they are among the top most used performance measures in scheduling research. Apart from these three, we also consider one particular case where all jobs have the same weight and call it “summed tardiness”.

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Notation Name Description

𝐶𝑚𝑎𝑥 The maximum makespan The total schedule length

∑ 𝐶𝑤 Weighted completion time The weighted completion time ∑ 𝑇 The summed tardiness of schedule Total due date violation

∑ 𝑇𝑤 The weighted tardiness of schedule Summed weighted due-date violation

Table 2 Performance measures

3.3 The modelling approach

Although the proposed problem is at least NP-hard, it is not unsolvable. Existing methods such as MILP, heuristics, metaheuristics or genetic algorithm are all being used to solve NP-hard scheduling problems. Allahverdi et al. (2008) stated that the solution method largely depends on the context of the problem. For this research, we believe mixed-integer-programming (MIP) to be the best method of this research for three reasons:

Firstly, the majority scheduling papers that contain both sequence- and time-dependent setups using MIP model as the modelling approach. Allahverdi et al. (2015) found MIP to be one of the most popular methods among the exact solution methods. Secondly, the MIP approach is flexible with powerful solver support. Since this research emphasis on scheduling performance and not solving method, working with MIP solver is more efficient. Lastly, the MIP solver also provides additional insight into the problem itself. Suppose the solver cannot solve the problem within a reasonable time, it also provides the remaining MIP gap and current best solution, which gives additional information regarding the scale of the problem.

3.4 The mathematical model

3.4.1: The base model (without cleaning policy)

To get started, we first propose a base model which is the simple version of our proposed problem. In the base model, we do not consider time-dependent setups and will add time-dependent constraint later on during the modelling process. The base model is a single machine sequence-dependent scheduling problem. The complete model is presented below:

Scheduling constraint:

𝑆𝑗≥ 𝑆𝑖+ 𝑃𝑖+ 𝐶𝑖𝑗− 𝑀 ∗ (1 − 𝑋𝑖𝑗) ∀𝑖, 𝑗 ∈ 𝑁, 𝑖 ≠ 𝑗 (1.1)

𝑆𝑖 ≥ 𝑆𝑗+ 𝑃𝑗+ 𝐶𝑗𝑖− 𝑀 ∗ 𝑋𝑖𝑗 ∀𝑖, 𝑗 ∈ 𝑁, 𝑖 ≠ 𝑗 (1.2)

𝑋𝑖𝑗+ 𝑋𝑗𝑖= 1 ∀𝑖, 𝑗 ∈ 𝑁, 𝑖 ≠ 𝑗 (1.3)

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12 The building block for our base model contains a binary decision variable 𝑋𝑖𝑗 to indicate if job 𝑖 is

planned before job 𝑗 or not. In other words, 𝑋𝑖𝑗 = 1 if 𝑆𝑖 < 𝑆𝑗 and 𝑋𝑖𝑗 = 0 otherwise.

Eq. (1.1) describes the case where job 𝑗 starts later than completion time of job 𝑖 . Since the completion time equals the starting time plus setup time plus processing time, the r.h.s of (1.1) is the formulation of completion time of job 𝑖 and additional liner relaxation. If job 𝑖 starts before job 𝑗, then 𝑋𝑖𝑗= 1 and Eq. (1.1) holds; in case job 𝑖 starts after job 𝑗, r.h.s will become -∞ and Eq. (1.1)

will be relaxed.

Similar to Eq. (1.1), the Eq. (1.2) describes the case where job 𝑖 starts later than job 𝑗 gets completed. In this case, the r.h.s is the formulation of completion time of job 𝑗 and additional liner relaxation. In case 𝑆𝑗< 𝑆𝑖, then 𝑋𝑖𝑗= 0 which will relax Eq. (1.1) and tightens Eq. (1.2).

By combining Eq. (1.1) and (1.2), the formulation describes the scheduling constraint either job 𝑖 starts after job 𝑗 gets completed, or job 𝑖 ends before job 𝑗 started. The Eq. (1.3) and (1.4) are the non-overlap constraint and binary constraint, indicating the machine can only process one job at a time.

Objective function 1: Minimize 𝐶𝑚𝑎𝑥

𝐶𝑚𝑎𝑥 ≥ 𝑆𝑗+ 𝑃𝑗+ 𝐶𝑖𝑗∗ 𝑋𝑖𝑗 ∀𝑖, 𝑗 ∈ 𝑁, 𝑖 ≠ 𝑗 (1.5)

Objective function 2: Minimize ∑ 𝐶𝑤 ∑ 𝐶𝑤 = ∑(𝑆𝑗+ 𝑃𝑗+ 𝐶𝑖𝑗∗ 𝑋𝑖𝑗) ∗ 𝑊𝑗

𝑗

∀𝑖, 𝑗 ∈ 𝑁, 𝑖 ≠ 𝑗 (1.6)

Objective function 3: Minimize ∑ 𝑇

𝑆𝑗+ 𝑃𝑗+ 𝐶𝑖𝑗∗ 𝑋𝑖𝑗− 𝑑𝑗≤ 𝑇1𝑗− 𝑇2𝑗 ∀𝑖, 𝑗 ∈ 𝑁, 𝑖 ≠ 𝑗 (1.7)

𝑇𝑗≥ 𝑇1𝑗− 𝑇2𝑗 ∀ 𝑗 ∈ 𝑁 (1.8)

𝑇𝑗≥ 0 ∀ 𝑗 ∈ 𝑁 (1.9)

∑ 𝑇 = ∑ 𝑇𝑗 ∀ 𝑗 ∈ 𝑁 (1.10)

Objective function 4: Minimize ∑ 𝑇𝑤 ∑ 𝑇𝑤 = ∑ 𝑇𝑗∗ 𝑊𝑗

𝑗

∀ 𝑗 ∈ 𝑁 (1.11)

Eq. (1.5) is the formulation of makespan, indicating the timestamp of makespan is larger than any job completion time, and Eq. (1.6) is the formulation of weighted completion time. Regarding job tardiness, we introduce two different variable 𝑇1𝑗 and 𝑇2𝑗 such that 𝑇1𝑗− 𝑇2𝑗 is equal to job

completion time minus job due date. When the job is complete within its due date, the l.h.s will result in a negative value, in this case 𝑇1𝑗= 0 and 𝑇2𝑗 will become a positive integer. On the other

hand, if job completion time is longer than its due date, 𝑇2𝑗 will equal to 0 and 𝑇1𝑗 will be a

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13 tardiness of the job while ensure the entire formulation is liner. After introducing 𝑇1𝑗 and 𝑇2𝑗 ,

we further formulate the total tardiness in Eq. (1.10) and weighted tardiness in Eq. (1.11).

3.4.2: The fixed cleaning policy

The fixed cleaning policy contains both sequence- and time-dependent setup, which is equivalent to our base case plus the additional fixed time-dependent setup. By following the concept of campaign production, the continuous timeline is partitioned by cleaning action into blocks of the fixed planning horizon; here we call each planning block “campaign”. Figure 5 “the campaign model” illustrates this concept.

Figure 5 the campaign model

To model campaign production, we first index each campaign with 1, 2, 3...k and introduce another binary variable 𝐵𝑗𝑘 for allocating (inserting) job j into campaign k. This means, 𝐵𝑗𝑘 = 1 if job j is

allocated to campaign k and 𝐵𝑗𝑘= 0 otherwise. The complete formulation of fixed cleaning policy

is as follows:

Objective: Minimizing 𝐶𝑚𝑎𝑥 , ∑ 𝐶𝑤 , ∑ 𝑇 and ∑ 𝑇𝑤

Subject to: 𝑆𝑗≥ 𝑆𝑖+ 𝑃𝑖+ 𝐶𝑖𝑗− 𝑀 ∗ (3 − 𝑋𝑖𝑗− 𝐵𝑖𝑘− 𝐵𝑗𝑘) ∀𝑖, 𝑗 ∈ 𝑁, 𝑘 ∈ 𝐾, 𝑖 ≠ 𝑗 (2.1) 𝑆𝑖 ≥ 𝑆𝑗+ 𝑃𝑗+ 𝐶𝑗𝑖− 𝑀 ∗ (2 + 𝑋𝑖𝑗− 𝐵𝑖𝑘 − 𝐵𝑗𝑘) ∀𝑖, 𝑗 ∈ 𝑁, 𝑘 ∈ 𝐾, 𝑖 ≠ 𝑗 (2.2) 𝑆𝑗≥ 𝐶1𝑘− 𝑀 ∗ (1 − 𝐵𝑗𝑘) ∀𝑗 ∈ 𝑁, 𝑘 ∈ 𝐾 (2.3) 𝑆𝑗+ 𝑃𝑗+ 𝐶𝑖𝑗∗ 𝑋𝑖𝑗≤ 𝐶2𝑘+ 𝑀 ∗ (1 − 𝐵𝑗𝑘) ∀𝑖, 𝑗 ∈ 𝑁, 𝑘 ∈ 𝐾, 𝑖 ≠ 𝑗 (2.4) ∑ 𝐵𝑗𝑘= 1 𝑘 ∀𝑗 ∈ 𝑁, 𝑘 ∈ 𝐾 (2.5) 𝐵𝑗𝑘 ∈ [0,1] ∀𝑗 ∈ 𝑁, 𝑘 ∈ 𝐾 (2.6) Plus (1.3) and (1.4)

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14 the same campaign. To formulate sequence substitution, the liner relaxation on the r.h.s indicating when 𝑋𝑖𝑗= 1 , 𝐵𝑖𝑘 = 1 and 𝐵𝑗𝑘= 1, the big M relaxation will equal to zero and tightens the

constraint. If any of these variable equals 0, the r.h.s will be close to -∞ and relaxes the Eq. (2.1) The Eq. (2.2) describes the case when job 𝑖 is planned after job 𝑗 similar to Eq. (1.2) This time, the constraint also needs to ensure both job 𝑖 and 𝑗 are allocated in the same campaign. The liner relaxation shows that if 𝑋𝑖𝑗 = 0 , 𝐵𝑖𝑘 = 1 and 𝐵𝑗𝑘 = 1, then the big M will be multiplied by zero

and tightens the constraint.

The Eq. (2.3) shows that if job 𝑗 is assigned into campaign 𝑘, then its starting time cannot be earlier than the starting time of the allocated campaign. In this case, we use liner relaxation to ensure this constraint holds only if job j is assigned to campaign 𝑘.

Regarding the job completion time, (2.4) ensures that the completion time of job 𝑗 is not allowed to exceed the ending time of campaign 𝑘. here we use the same liner relaxation function as (2.5). The Eq. (2.6) is the binary constraint and Eq. (2.5) ensures that each job is assigned once

Since the cleaning duration (G) and fixed cleaning interval (A) is known, the starting and ending time of each campaign can be derived from the following equations:

𝐶2𝑘− 𝐶1𝑘 = 𝐴 ∀𝑘 = 1,2,3. . 𝑘

𝐶1_(𝑘+1)− 𝐶2𝑘 = 𝐺 ∀ 𝑘 = 1,2,3. . 𝑘 − 1

𝐶11= 0

The objective function of the fixed cleaning policy is the same as aforementioned base case (1.5) -(1.11), and we minimize 𝐶𝑚𝑎𝑥 , ∑ 𝐶𝑤 , ∑ 𝑇 and ∑ 𝑇𝑤 respectively.

3.4.3: The flexible cleaning policy

Similar to fixed cleaning policy, the flexible cleaning policy also follows the concept of campaign production and the continuous time is partitioned into different planning blocks. However, this time, the starting and ending time of each campaign is unknown, and each campaign has a maximum time length. In this case, the model has to decide when does each campaign start and end, thus 𝐶1𝑘 and 𝐶2𝑘 are the decision variables. The complete formulation is as follows:

Objective: Minimizing 𝐶𝑚𝑎𝑥 , ∑ 𝐶𝑤 , ∑ 𝑇 and ∑ 𝑇𝑤 separately

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15 Since there is no change on job relationship and job allocation, Eq. (2.1) - (2.6) stays the same for the flexible policy. However, this time we do not know the starting and ending time of each campaign, therefore we add additional constraints (3.1) - (3.4) to optimize the campaign length. Eq. (3.1) formulates the constraint that the time gap between each campaign should always be more than the minimum cleaning duration G. Eq. (3.2) describe the constraint that the time length of each campaign should not exceed the maximum time allowance A.

Eq. (3.3) set the lower bound of campaign length, and it ensures that for each campaign, the minimal length should be at least as long as one job. Eq. (3.4) forces the first campaign starts from time 0.

The objective function from the base case (1.5) – (1.11) can still be applied here as the MIP formulation stays the same.

3.5 Computational design

We proceed our experiments by generating different sets of data to test both fixed cleaning policy and flexible cleaning policy. During data generation, we follow the stream of Abdul-Razaq, et al. (1990) and Keha et al. (2009) that use parameter selection as our data generation strategy. The key parameters of our experiment including job processing time, due date, setup time matrix, cleaning interval and cleaning duration.

3.5.1 The job processing time and weight

In this experiment, the job processing time contains two different dimensions: the processing time distribution and value selection. For processing time distribution, we generalized two different distribution types: uniform distribution and ABC distribution.

In the uniform distribution, the probability to select processing time from the pool is equally likely, whereas, in ABC distribution, we designed 10% time-consuming jobs, 30% Normal jobs and 60% easy jobs. The ABC distribution simulates the case when the distribution of customer order is not always uniform; Company may have one or few real big customers with large demand, and many small customers with small orders.

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16

Processing time types Description

# Value Pool Distribution

1 Arbitrary Uniform Uniform distribution from [1, 100] 2 Cyclic Uniform Uniform distribution from 5*[1,20]

3 Arbitrary ABC 10% [90,100], 30% [40, 90], 60% [1, 40] 4 Cyclic ABC 10% 90+5*[0,2], 30% 40+5*[1,10] 60% 5*[1,8]

Table 3 Processing time generation method

The weight of each job is generated with the following rule: If the processing time is uniformly distributed, the weight will be uniform [1, 10]. In case the processing time is ABC distributed, the weight for category A is uniform [9, 10], category B is uniform [4, 9] and category C is uniform [1, 4].

3.5.2 The job due date

The job due date is generalized by using location and range selection method inspired by Keha, khowala and Fowler (2009). For this experiment, 𝑑𝑗 has the uniform distribution [P*(L-R/2),

P*(L+R/2)]. Where P is the summed total processing time of all jobs, L is the location parameter and R is the range. In this experiment, L = 0.5 and R = 0.4 respectively.

3.5.3 The setup time matrix

The setup time matrix is another important parameter in this experiment. For studies that consider sequence-dependent setup, triangular inequality is a common assumption (Menezes et al. 2011). It is defined as “for any given three jobs, the setup time required from one to another is always shorter than summed time for three jobs” (𝑆𝑖𝑗+ 𝑆𝑗𝑘≥ 𝑆𝑖𝑘) (Clark et al 2010). Under this

assumption, for any paired jobs, there is no third job that can be inserted in between as “shortcut” to reduce makespan. In this study, we follow this assumption and generate setup matrix under this rule. Figure 6 “Example of triangular inequality” illustrates this concept.

Figure 6 Example of triangular inequality

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17

Figure 7 The setup matrix

The symmetrical and asymmetrical setup time matrix is originated from the sequence-dependent scheduling problem that if the author does not consider release time and due date and the model objective is the maximum makespan, then it is equivalent to travelling salesman problem. Under this context, the setup time matrix is representing the distance between cities. Symmetrical setup time matrix is representing the case when the distance for travelling from one city to another is always the same as its return route (𝑆𝑖𝑗 = 𝑆𝑗𝑖). If this is not the case, then the setup time matrix is

asymmetrical (𝑆𝑖𝑗 ≠ 𝑆𝑗𝑖). For these two matrix, all setup time has uniform distribution [1, 10].

Under the case of the asymmetrical matrix, we further investigate one special case: lower triangular matrix. In this particular case, for any two jobs 𝑖 𝑗 in the set 𝑁 where job 𝑖 is the predecessor of job 𝑗, if 𝑖<𝑗. then setup time of job j is always small. If 𝑖>𝑗, then setup time for job j is always large. Take ink production as an example, if dark coloured ink is produced after light coloured ink, the setup time for dark ink production will be small, and vice versa. For this particular case, we prepared two different setup time matrixes namely free changeover and fixed changeover.

In the free changeover case, the setup time for following sequence-dependency will always be 0, and violating sequence-dependency takes Uniform [1, 10] of setup time. This matrix simulates production case where if two jobs follow sequence-dependency, the setup can be prepared offline and no additional changeover is needed. Recall the diary mixer case mentioned in introduction chapter, if sweetened yoghurt is produced after paled one, the company can prepare sweet additives during the production and add additives into the next batch via a feeding pipe. This principle falls into the concept of “single minute exchange of dice” (SMED) and is an important management tool in setup time reduction.

For the Fixed changeover case, the setup time for following sequence-dependency takes uniform [1, 3], whereas violating sequence-dependency takes a fixed time of 10 hours. This matrix aims at simulating the production case where sequence violation will always result in the same amount of change over time. One example of fixed changeover matrix can be found in ink production: If dark coloured ink is produced before the light coloured one, the machine has to be totally cleaned, and this total cleaning will always take the same amount of time regardless the volume or the colour of the previous job.

3.5.4 Schedule size and cleaning duration

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18 Apart from the size of each experiment, we also found the number of campaigns heavily impacts the computational time. if the number of campaigns is more than 3, the solver takes much more time to solve. On the other hand, if we reduce the number of campaigns to 2, then there is only one time-dependent cleaning takes place for each run, which greatly reduces the impact of cleaning times. To ensure the solver will be able to solve the case within a reasonable time while also maintaining the relevance of time-dependent setup, we fixed the number of campaigns to 3. The cleaning duration (G) and the campaign length (A) will be set as 40 hours and 200 hours for both policies. To ensure the generated processing time always produce three campaigns under this setting, we will keep rerolling the job processing time until it hits appropriate value. Since we have four different setup time matrix, four different processing time and due date, and four different objective functions, two policies and two testing for each dataset, we did in total 256 runs of experiments for policy comparison.

Results

We translated the mathematical formulation into Mosel language and implemented the model on Xpress-Mosel 8.1 (64-bit version) MIP solver. The experiments are run on a machine with Intel i3 3.2 GHz CPU platform with 1GB RAM. All computational experiments have a time limit of 10 minutes. If the solver cannot solve the problem within the time limit, the model will be terminated and the “current best solution” will be exported as the model output. For each “current best solution”, we recorded the remaining MIP gap as additional references.

4.1 Computational time

Before jump into the comparison result, it is essential to mention computational time first as we notice the solver does not always solve the case within the time limit. During the experiment, we obtained many suboptimal results, thus great attention needs to be paid when comparing two policy results together.

This experiment contains 128 different runs for both fixed and flexible cleaning policy. For each scheduling task that containing 10 jobs, the fixed cleaning policy solved 44 runs within 10 minutes and the average time for each run takes 287 seconds. In the flexible cleaning policy, the solver cannot solve majority cases (104) with 10 minutes, and the average time for each run takes 543 seconds. Table 4 “computational time” summarize the time spent on this experiment.

Computational time Cleaning Policy

Fixed cleaning policy Flexible cleaning policy

Average time to solve the case 287 543

Average MIP gap remaining 28.8 42.23

Case solved within the time limit 66% 19%

Remaining gap within 0-20 9% 13%

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19 We also got three important insight from computational time result:

Firstly, the cleaning policy is the primary influence factor on computational time, and it takes way more effort to solve for flexible policy than fixed policy. This is because, in the flexible policy, we have more decision variables and the starting and ending time of each campaign needs to be decided.

Secondly, the processing time and due date is the second most significant factor that influences computational time. We notice the uniform distribution of processing time takes less time to solve than cyclic processing time. After further checking with MIP solver, we notice the cyclic processing time often finds many similar solutions on the same depth of BB (branch and bound) tree but cannot close the MIP gap quick enough.

Lastly, the setup time matrix also has some impact on computational time, That the lower triangular matrix takes less time to solve than standard setup time matrix. However, the impact of setup time is not as significant as abovementioned two factors.

4.2 Comparison result

All experiments are divided into four different groups distinguished by their setup time matrix. Each case contains four different processing time, and for each processing time, we run the model four times such that each model objective gets optimized once. The full computational runs are summarized in Table 5 “The primary result”.

# Setup time matrix Processing time Performance differences (Fixed – Flexible)

𝐶𝑚𝑎𝑥 ∑ 𝐶𝑤 _{∑ 𝑇} _{∑ 𝑇𝑤}

1 Asymmetrical Arbitrary 1% -1% 5% -15%

2 Asymmetrical Cyclic 1% 0% -1% -3%

3 Asymmetrical ABC 2% 4% 16% 7%

4 Asymmetrical Cyclic ABC 1% 0% 11% -1%

5 Symmetrical Arbitrary 2% 0% 8% 9%

6 Symmetrical Cyclic 1% -3% 3% 3%

7 Symmetrical ABC 1% 3% 3% 5%

8 Symmetrical Cyclic ABC 3% 2% 14% 22%

9 Free changeover Arbitrary 0% 4% 23% 12%

10 Free changeover Cyclic 0% 0% 17% 12%

11 Free changeover ABC 1% 0% 12% 1%

12 Free changeover Cyclic ABC 0% 0% 19% 3%

13 Fixed changeover Arbitrary 0% 5% 6% 3%

14 Fixed changeover Cyclic 3% 0% 1% 5%

15 Fixed changeover ABC 3% 1% 16% 3%

16 Fixed changeover Cyclic ABC 1% 1% 10% 5%

Table 5 The primary result

4.2.1: Policy comparison on the primary result

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20 of flexible cleaning that it eliminates machine idle time before cleaning action. However, the amount of time saved is not significant (1%-3%).

In terms of the weighted completion time, the flexible cleaning does not always outperform fixed cleaning. The computational comparison shows that if the setup time matrix is lower triangular, then the flexible policy has 0%-5% minor advantage on ∑ 𝐶𝑤 compare with fixed policy. However, for asymmetrical and symmetrical setup matrix, the performance difference range from -3% to 4%.

The results obtained from total tardiness objective (∑ 𝑇) suggests that the flexible cleaning policy has a major advantage compare with fixed cleaning policy. Apart from one result (-1%), the average performance difference shows that shifting from fixed cleaning towards flexible cleaning deliver more than 10% tardiness reduction. In one extreme case, the flexible cleaning policy even achieved 23% tardiness reduction.

For weighted tardiness objective (∑ 𝑇𝑤), shifting from fixed cleaning to flexible cleaning does not always deliver a positive result. Although the average figure shows 4.5% positive improvement, the result ranges from -15% to 22%. Recall that the solver does not always solve the scheduling case within the time limit, the results of negative value here can be the case that we are comparing optimal solution with suboptimal one.

Several insights can be concluded from this comparison table:

When the primary performance measure is makespan (or weighted completion time), there is no significant advantage by switching from fixed cleaning to flexible cleaning. On top of that, the fixed cleaning has two additional advantages: firstly, the fixed policy is easier to execute. Since we know when does cleaning takes place, the planner can arrange resources, equipment and personnel in advance. Secondly, it takes less computational time to solve and allows the planner to work more efficiently.

When the primary performance measure is total tardiness, then the flexible cleaning out-performs fixed cleaning thus it is worth to switch. If the primary objective is weighted tardiness, then the flexible policy does not always out-perform fixed policy within the time limit, and we advise the company to do data testing before switching.

4.2.2: Impact of setup time matrix

Since the improvement on makespan or weighted completion time is minor, the impact of setup time matrix on this objective is not significant. However, when the model objective is to minimize total tardiness, the flexible cleaning policy has the best performance on the free changeover matrix (average 18% tardiness reduction) whereas other three matrices only obtained 7%, 8% and 8% improvement on average. This interesting finding suggests that when changeover time becomes 0 by following sequence-dependency, the flexible cleaning policy will greatly reduce the schedule tardiness.

4.2.3: Impact of processing time

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21 which is why the flexible cleaning policy performs best under cyclic ABC distribution. Figure 8 “policy selection” shows the overall result of policy comparison.

Figure 8 Policy selection

4.3 The secondary result compares with optimal result

Aside from the performance comparison between two different policies, additional insight can be obtained by analyzing the performance difference within each policy. This time, we are looking at the gap between the secondary result and theoretical optimal result. The table 6 “gap between the secondary result and theoretical result” shows the average gap between the secondary result and optimized result.

Fixed cleaning policy Flexible cleaning policy Model objective 𝐶𝑚𝑎𝑥 ∑ 𝐶𝑤 _{∑ 𝑇} _{∑ 𝑇𝑤} 𝐶𝑚𝑎𝑥 ∑ 𝐶𝑤 _{∑ 𝑇 ∑ 𝑇𝑤}

𝐶𝑚𝑎𝑥 -- 8% 19% 22% -- 10% 23% 30%

∑ 𝐶𝑤 _1% _-- _11% _4% _2% _-- _23% _5%

∑ 𝑇 _1% _7% _-- _16% _1% _7% _-- _16%

∑ 𝑇𝑤 _1% _2% _10% _-- _2% _2% _18% _--

Table 6 Gap between the secondary result and optimal result

The result shows that even when solver is optimizing total tardiness or weighted tardiness, the schedule makespan of these solution does not differ much with theoretical optimal makespan. However, when solver is minimizing makespan, the obtained scheduling solution often has much worse tardiness performance than theoretical optimal value.

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22

Discussion

The experiment shows several interesting findings that need to be further discussed here: Firstly, we found there is no significant improvement in makespan or weighted completion time for flexible policy. This is in line with the concept of flexible cleaning that it only removes the “idle window” at the end of each campaign. Since this idle window is too small to insert the shortest job in current remaining list, plus we only have three campaigns in our experiment, the overall improvement is not significant.

Secondly, the flexible policy has excellent improvement in tardiness reduction. After taking an in-depth look at schedule solution, we found there is a major difference in job-campaign allocation when minimizing schedule tardiness.

Figure 9 Gantt chart example

Figure 9 “Gantt chart example” shows one schedule solution when minimizing weighted tardiness. In this example, we can see the fixed cleaning starts at 200, and solver inserted job 7 in the first campaign to maximize the utilization of fixed campaign window. For flexible cleaning, it inserts job 9 for the first campaign and starts cleaning at 146 instead of 200. We notice in this particular example, job 2 has high weight and medium due date, the outcome of flexible policy positioned job 2 much earlier and significantly reduced overall tardiness.

Thirdly, regarding the computational time, we notice cyclic processing time takes much longer to solve than arbitrary processing time. We suspect the cyclic processing time has more probability to have duplicated value, which makes solver take extra time to search.

We also find the flexible cleaning performs much better on free changeover matrix when the model objective is to minimize weighted (or total) tardiness. We suspect the following reasons for this observation:

When compare free changeover with asymmetrical and symmetrical, we notice the sequence penalty for these three are similar, However the zero changeover time when obeying sequence dependency gives free changeover additional advantage than these two standard matrix.

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23

Conclusion

In this study, we investigate a single-machine scheduling problem that contains sequence- and time-dependent setups. We tried to compare the scheduling performance under two different time-dependent cleaning policies, and the problem is challenging to solve. We contribute the literature by presenting the first benchmark study between two different time-dependent setups via modelling approach, and the proposed comparison also consider schedule prioritization and production configurations.

The aim of this study is to help the company decide which policy to choose, and we suggest company choose fixed policy when makespan or weighted completion time is the priority measure and pick flexible policy if tardiness is more important. In case company already using fixed policy and want to switch, we also suggest company pay additional attention on management effort since flexible policy takes way more time to compute and is more challenging to execute.

We also realize the proposed study contains the following limitations:

Firstly, this study only considers single objective function. When makespan and tardiness are both important, the model should incorporate with multi-objective function by giving weight on these performance measures and combine them together.

Secondly, we only tested four different objective measures in this study, and each experiment has fixed two cleaning times. In real production case (Kilic 2011), the maximum amount of time between each cleaning range from 16 hours to 72 hours, which may result in more campaigns in our model. Since the current model already has difficulty in solving flexible cleaning policy within a reasonable time, additional improvement can be made by using different modelling approach such as arc-time index model proposed by Nesello et al. (2018), or by tightening the MIP constraint such as employing “linear ordering formulation” introduced by Keha et al. (2009) Thirdly, we assume all jobs are available from time 0 and does not consider the order release date. In some business environment such as online shop; the planner is operating in a dynamic environment and performs order inserting and schedule fixing constantly. The impact of time-dependent cleaning under the dynamic environment remains unknown and shall be studied in the future.

Lastly, this research only studied the cleaning policy for a single machine. This is partly due to the technical limitation that solver can only solve 10 jobs within 10 minutes. For more complex configurations such as jobshop or flowshop problem, different modelling approach should be used to reduce the computational time during model solving.

There are several future research directions for this study:

Firstly, the time-dependent setup time can result in a different cost. By employing the flexible policy, the planner must pay extra efforts to arrange resources and personnel. Therefore, it is safe to assume the flexible policy is much more expensive. Under this assumption, the trade-offs regarding cost and benefit of flexible policy remain to be studied.

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24 Lastly, since the solver can only solve limited 10 jobs within 10 minutes, the planner can use the decomposition method to separate one large list of jobs into several small lists and optimize them separately. By using this approach, the global performance difference between fixed and flexible policy for a large number of jobs remains to be discovered.

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