A metric-based framework for sustainable production scheduling

A metric-based framework for sustainable production scheduling

Journal of Manufacturing Systems 54 (2020) 174–185 Contents lists available at ScienceDirect Journal of Manufacturing Systems journal homepage: www...

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Journal of Manufacturing Systems 54 (2020) 174–185

Contents lists available at ScienceDirect

Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys

Technical Paper

A metric-based framework for sustainable production scheduling Amin Abedini*, Wei Li, Fazleena Badurdeen, I.S. Jawahir

T

Institute for Sustainable Manufacturing, Department of Mechanical Engineering, University of Kentucky, Lexington, KY, USA

ARTICLE INFO

ABSTRACT

Keywords: Sustainability Production scheduling Trade-off balancing

Production scheduling involves operational level decision making at the shop floor that covers not only the manufacturing stage of the product life-cycle, but also the use stage of the processes. Triple bottom line (TBL) including economic, environmental, and social pillars has been introduced to holistically evaluate the performance of a production firm. Despite the substantial research in sustainable manufacturing, a holistic model that considers all three pillars of the TBL for sustainable production scheduling is virtually absent. This paper presents a metric-based model to systematically and holistically evaluate the sustainability of the production schedules. To this aim, we first perform an extensive literature review to identify the fundamental performance metrics in production scheduling. Second, we assess those metrics with respect to the TBL . Third, we show the inconsistencies among the fundamental performance metrics, and consequently among the objectives defined in the TBL . Finally, we propose a generic model for production scheduling for sustainability based on balancing the trade-offs among the inconsistent objectives. The efficiency and effectiveness of the proposed model is demonstrated using a comprehensive numerical study. The proposed model not only provides a sustainable schedule, but also results in better control over the fundamental performance metrics of the production scheduling.

1. Introduction Trade-off balancing in production scheduling is important for sustainable manufacturing because sustainable manufacturing can be regarded as an optimization problem with many inconsistent objectives. Regardless of the studied system, an optimization model can be expressed by a linear programming (LP) model in the canonical form. For the purpose of illustration, consider a system with only two objectives of min(z1 = cT1 x) and min(z2 = cT2 x) as shown by Eqs. (1) and (4), where x = [x1, x2 , …, xK ] is the vector of decision variables, c = [c1, c2, …, cK ] is the coefficients of the objective function (also known as weights), A is a P × K matrix, b = [b1, b2, …, bP ] is the vector of nonnegative constants, and LB1 and LB2 are the minimum values of z1 and z2 , respectively:

LB1 = min

z1 = cT1 x

(1)

Ax < b

(2)

x

(3)

0

LB2 = min z2 = cT2 x s. t . (2), (3)

(4)

If z1 and z2 are consistent objectives, we can rewrite Eqs. (1) and (4) into Eq. (5): ⁎

min s. t .

(c 1 + c 2)T x = LB1 + LB2 (2), (3)

(5)

The system of linear equations expressed by Eq. (5) is called inconsistent if it has no solutions [1]. Elkington (1998) introduced the triple bottom line (TBL) including economic, environmental, and social pillars to holistically evaluate the performance of a production firm [2]. Considering all three pillars of the TBL, an obvious observation is that the objectives defined in the pillars are not consistent with each other. For example, a production plant may demonstrate excellent monetary profit (i.e., economic pillar) but at the cost of water/air pollution (i.e., environmental pillar). Therefore, sustainable production scheduling can be seen as an approach to balancing the trade-offs among the inconsistent objectives defined in the TBL. United State Environmental Protection Agency [3] defines sustainable manufacturing as follows: “Sustainable manufacturing is the creation of manufactured products through economically-sound processes that minimize negative environmental impacts while conserving energy and natural resources. Sustainable manufacturing also enhances employee, community and product safety”. We can conclude that sustainable manufacturing is able to address all the three pillars of the TBL. Jawahir et al. (2015) point out that sustainable manufacturing must

Corresponding author. E-mail address: [email protected] (A. Abedini).

https://doi.org/10.1016/j.jmsy.2019.12.003 Received 27 May 2019; Received in revised form 16 December 2019; Accepted 18 December 2019 0278-6125/ Published by Elsevier Ltd on behalf of The Society of Manufacturing Engineers.

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address the TBL at product, process, and system levels [4]. Sustainable manufacturing must also cover all four stages of the product life-cycle including: pre-manufacturing, manufacturing, use, and post-use [5]. Therefore, sustainable manufacturing can serve as a technological tool for the transition from the Linear Economy to the Circular Economy [6]. Production scheduling is an operational level decision made at the shop floor that falls into the manufacturing stage of the product lifecycle, and also covers the use stage of the processes [7]. Production scheduling sequences a set of jobs on one or multiple machines in order to optimize a given objective [8,9]. The common objectives for production scheduling are minimization of maximum completion time (MCT), total completion time (TCT), flow time mean and variance, and tardiness/earliness [10]. Although production scheduling with regard to single pillars of TBL such as greenhouse gas emission [11], energy consumption [12–14], and labor cost [15] have been reported, there is no report on production scheduling with regard to all three pillars of the TBL [16]. Therefore, developing an approach to holistically evaluate production scheduling sustainability is necessary. Currently, there is no holistic evaluation scheme available which is capable of balancing the trade-offs not only among the inconsistent objectives at the lower level (i.e., shop floor), but also among different pillars of the TBL at the higher level (i.e., organization level). The closest work to the sustainable production scheduling evaluation is the work proposed by Badurdeen et al. [17] for evaluating process sustainability. They proposed a four-level hierarchical structure called ProcSI (Process Sustainability Index) that segregates the overall process sustainability into individual process-level quantifiable metrics. ProcSI includes four levels: ProcSI, clusters, sub-clusters, and individual metrics. Their work provides detailed structures for each level. Since different metrics are measured with different units (e.g., cost ($), energy (Kw), etc.), a normalization scheme is adopted to normalize individual metrics into a 0–10 scale. After normalization, a weighting scheme is developed to balance the normalized values according to their relative importance (managerial preference) or their levels of impact. Once the weights are assigned, the normalized values are aggregated to calculate the scores for sub-clusters, clusters, and ProcSI . The aggregation follows a bottom-up approach. However, ProcSI is merely focused on evaluating the sustainability of the process and it does not have any consideration of the production schedule. For example, given a machining process, ProcSI evaluates the sustainability of the machining process without considering the effects of production schedules on the sustainability of the overall production. In this paper, we propose a generic model for balancing trade-offs among inconsistent objectives defined in the TBL for production scheduling problems. Analogous to inconsistent objectives of the TBL, inconsistencies exist among objectives in production scheduling [18]. We hypothesize that balancing trade-offs among inconsistent objectives of production scheduling can result in a sustainable schedule. We model trade-off balancing as a function of z = f (·) , which is generally applicable to min(·) or max(·) problems at the low (metrics) level, but also consistent at the high (sub-cluster, cluster) levels. We show that balancing trade-offs among production scheduling objectives (e.g., MCT, TCT, etc.) can indirectly balance the trade-offs among the inconsistent objectives defined in the TBL , which results in a sustainable schedule. Contributions of this paper are as follows:



The remainder of this paper is organized as follows: in Section 2, we provide a comprehensive literature review of practices in production scheduling for sustainability; in Section 3, we develop the proposed mathematical model of production scheduling for sustainability; in Section 4, we present extensive numerical studies, followed by results and discussions. Finally, in Section 5, we draw conclusions and give future research directions. 2. Literature review In this section, we provide a comprehensive literature review on sustainability practices in production scheduling. For detailed literature on production scheduling and sustainable manufacturing, readers are referred to [10,16,19], respectively. The objective of this section is to systematically review the production scheduling objectives, and establish a bridge between them and the objectives defined in the TBL . To this aim, at the end of this section, we categorize the production scheduling objectives into three clusters of (i) economic-oriented scheduling, (ii) environmental-oriented scheduling, and (iii) social-oriented scheduling. Each cluster is then divided into several sub-clusters, and subsequently, each sub-cluster is divided into multiple measurable performance metrics. 2.1. Economic-oriented scheduling In this subsection, we review recent papers with economic-oriented objectives. We identify four types of economic performance indicators for production scheduling, including: (i) production cost, (ii) energy cost, (iii) labor cost, and (iv) inventory cost. 2.1.1. Production cost Consider a production system with N jobs. Let Cn be the completion time of job n {1, 2, …, N } . Maximum completion time, MCT = max(C1, C2, …, CN ) , is the time elapsed from the start-time of the first job to the finish-time of the last job. Thus, minimizing MCT, i.e., min(MCT) , is equivalent to maximizing the production system utilization, i.e., max(Util) [10]. Utilization is the ratio between the actual output of a given resource and its potential output if that resource were fully utilized. A production system with a utilization less than 100% theoretically has the potential to increase its production output without any recurring and/or capital cost. Production scheduling with the objective of min(MCT) has been extensively studied, and numerous methods have been proposed for different production environments such as flow shop [20,21], job shop [22], and continuous manufacturing [23]. Fang et al. (2011) studied a flow shop scheduling problem where machines could be operated at different speeds that affect the required power peak. A mixed integer optimization model was proposed to simultaneously minimize MCT , power peak, and the carbon footprint [24]. Wang and Li (2017) studied a flow shop scheduling problem with the objectives of maximizing production profit by implicitly integrating the objectives of min(MCT) , electricity cost, and labor cost [25]. Liu et al. (2015) studied a seru production scheduling problem with the objectives of min(MCT) and carbon dioxide emission. A non-dominated sorting algorithm was proposed to solve the NP -complete problem [26]. Ye et al. [27,28] proposed two constructive heuristics to sub-optimally minimize MCT in a no-wait flow shop scheduling problem, based on a case studies performed on the Taillard's benchmark, their proposed heuristics outperformed the best knows heuristics to minimize MCT in the no-wait flow shop scheduling problem. Shahsavari-Pour and Ghasemishabankareh (2013) proposed a hybrid metha-heuristic to simultaneously optimize MCT , workload of

• We provide a comprehensive list of fundamental production sche•

making, and (iv) better control over the production at the lower level. Through extensive case studies, we show that decision making must be performed with regard to both sustainability and process control.

duling performance metrics that drive multiple areas of system performance. We propose a generic model that attributes the fundamental production scheduling performance metrics to the three pillars of the TBL . Production scheduling sustainability index (PSSI ) is proposed to systematically evaluate the sustainability of the production schedules. PSSI offers a flexible optimization model capable of addressing: (i) the decision maker preferences, (ii) inconsistencies among objectives, (iii) sustainability at the higher level of decision 175

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bottleneck machines and the total workload [29] in a flexible job shop scheduling problem, Pareto optimality conditions were integrated in the proposed metha-heuristic to efficiently solve the NP -complete problem in a shorter computation time. Tardiness and earliness costs also affect the production cost [30]. Earliness costs may occur if job n is completed earlier than its due date (dn ). The earliness of a job is defined as En = max(dn Cn, 0) [10]. Earliness costs could result from deterioration of the final products or from the need for an extended time/capacity for holding the inventory [31]. Tardiness costs occur if job n is finished after its due date, and includes expedited delivery costs, lost customer, contract penalties, etc. [32,30,33]. Tardiness is defined as Tn = max(Cn dn, 0) [10]. If both tardiness and earliness are considered, the objective function can be defined by Eq. (6): N

min(z ) =

objectives of minimizing the total energy consumption and the maximum tardiness with a power-down mechanism. A mixed integer programming model was proposed and -constraint method was developed to solve large scale problems [49]. The power consumed for inventory handling is directly related to the inventory level incurred during the period of the schedule. Total completion time TCT = Cn gives an indication of the inventory level in the system during the period of schedule [10]. Therefore, min(TCT) can minimize the inventory level, and, consequently, the power required for inventory handling. Sun et al. (2014) proposed an inventory management model to support load peak management strategies with fewer units in the inventory. A multi-objective optimization model was proposed to minimize the summation of the inventory holding cost, power consumption cost, and lost production cost [50]. Ding and Song (2015) studied a parallel machines scheduling problem with the aim of balancing the trade-offs between total energy consumption and MCT [51]. Ye et al. (2017) proposed a heuristic to sub-optimally solve the problem of minimizing TCT in a no-wait flow shop system [52], Based on case studies performed on the Taillard's benchmark, their heuristic was superior compared to the best known heuristics to minimize TCT . Another factor affecting the energy costs is the energy-efficiency of the machines. Therefore, developing energy-efficient machines has been the subject of multiple studies [53,54]. However, machine selection and layout design falls into a higher level of production management known as production planning, and requires a significant capital [11]. Thus, energy-efficient production scheduling is more practical and fits the purpose of this paper.

N

(En + Tn) = n=1

|Cn

dn |

(6)

n=1

Liu et al. (2017) studied a flow shop scheduling problem with the objective of minimizing energy consumption and tardiness penalty. A fuzzy set theory was developed to determine the optimal sequence of jobs and the state transition of each machine [32]. Artigues et al. (2013) proposed a two-step approach for a parallel machine scheduling problem with the objective of minimizing tardiness at the first step and minimizing energy and overrun cost at the second step [34]. In many production systems, products are produced in batches with a common due date; d [35–37]. Therefore, the objective of minimizing earliness and tardiness can be rewritten by Eq. (7). It is worth mentioning that in this way, earliness and tardiness are equally penalized:

2.1.3. Labor cost Labor wages are higher in the overtime hours [15]; therefore, finishing the production in the regular hours results in a lower labor cost. min(MCT) can finish the production in a shorter time period that may yield a lower labor overtime cost. Inventory handling requires staffed hours (e.g., for transportation, security, etc.), therefore, reducing the inventory level can result in a lower inventory handling cost. As mentioned earlier, min(TCT) can reduce the inventory level, thus resulting in a lower inventory handling labor cost.

N

dn = d n

min(z ) =

|Cn

d|

(7)

n=1

For a special case where the common due date d is equal to the N average completion time, i.e., d = ACT = TCT/ N = n = 1 Cn/N , it has been shown that min(z ) =

N n=1

|Cn

ACT| is equivalent to minimizing N

completion time variance (CTV ) where CTV = n= 1 (Cn ACT) 2/ N . Therefore, min(CTV) can serve as an alternative objective function to minimize tardiness and earliness around a common due date [38–40].

2.1.4. Inventory cost Inventory cost refers to all the costs of holding an inventory, including opportunity cost of the inventory monetary value, infrastructure cost, insurance, depreciation cost, taxes, etc. [55]. As mentioned earlier, min(TCT) is directly associated with minimizing the inventory level, and consequently the inventory cost. Yavari and Isvandi [56] proposed a mixed integer programming model to minimize the sum of total weighted completion time, holding cost, and part ordering cost in a two stage flow shop problem, a genetic algorithm (GA) was also proposed to solve the NP -hard mixed integer programming model. Al-Anzi and Allahverdi proposed an artificial immune system heuristic to minimize TCT in a two-stage multi-machine assembly flow shop scheduling problem [57], computational examples showed that the propose heuristic can reduce the error by 60% compared to the best known heuristics.

2.1.2. Energy cost Manufacturing consumes 31% of primary energy [32], and utility companies have step-wise pricing policies in which the power costs are significantly higher if the demand exceeds a base load. Therefore, considering a system-wide approach for energy efficiency in production scheduling is necessary to attain sustainable manufacturing [41]. Production power must be managed with respect to three main components including, (i) idle energy consumption [42,43], (ii) power peak [44], and (iii) inventory handling power consumption. Minimizing idle energy consumption is equivalent to minimizing a machine's idle time, which is addressed by min(MCT) . Some papers also propose an “ON–OFF” strategy for shutting the non-bottleneck machines down while idle [45–47]; however, the latter seems to be impractical in many production systems [48]. Power peak management has been addressed by varying the machines speeds in order to shift production load to the off-peak periods [24]; however, varying the machines speed is not practical in many situations where the quality of products is a function of the machine speed (e.g., roughness of machined parts). Bruzzone et al. (2012) studied a flexible flow shop problem where the reference production schedule is obtained with the N primary objective of minimizing tardiness (i.e., min( n= 1 Tn) and min(MCT) as the secondary objective). Given the reference production schedule (i.e., job to machine assignment and job sequence), an energy aware scheduling approach was proposed to minimize the required power peak by modifying the reference production schedule [41]. Che et al. (2017) studied a single-machine scheduling problem with the

2.2. Environment-oriented scheduling Energy consumption and greenhouse gas emissions are the most studied environmental indicators in production scheduling [16]. As discussed in Section 2.1.2, energy consumption is a function of machines’ state, processing speed, and power peak [49]. More specifically, energy consumption with regard to the machine state and processing speed has appeared in [58] with optimizing MCT and energy efficiency as objectives in a permutation flow shop scheduling problem, and [59] with the objectives of optimizing MCT and energy consumption in a flexible flow shop scheduling problem. Peak power management has 176

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appeared in [24,60,41,50,49,32] with either MCT or tardiness in the objective or constraints forms. Greenhouse gas emission has been studied by multiple works [61,62,11]. Since fossil fuels are the primary source of energy generation, managing energy consumption significantly reduces greenhouse gas emissions [11]. Most studies calculate the greenhouse gas emissions as a function of energy consumption [16].

Sustainability Index (PSSI ), we propose a top-to-bottom decomposition followed by a bottom-to-top aggregation scheme. At the decomposition phase, we divide PSSI into three clusters covering the three pillars of the TBL , including economic, environmental, and social. Each cluster is then divided into sub-clusters. Each sub-cluster covers a specific area of impact of its cluster. Accordingly, each sub-cluster is then divided into individual metrics that specifically measure a single performance indicator. Once the top-bottom structure is developed and all the individual metrics are measured, a bottom-up aggregation approach including normalization and weighting is utilized to calculate PSSI .

2.3. Social-oriented scheduling Average completion time, ACT = TCT/ N , also known as flow time is the average time that a job spends in the system. For a fixed number of jobs (N ), minimizing average completion time, min(ACT) , is equivalent to min(TCT) , and as discussed earlier, it has been extensively studied [63]. The attractiveness of min(ACT) is that it is equivalent to minimizing waiting time, which is of importance for customers [64]. Minimizing waiting time can help the society to have access to products/services with the minimum waiting time. Therefore, we conclude that min(TCT) can also serve social-oriented scheduling objectives. N Completion time variance, CTV = n= 1 (Cn ACT) 2/ N , is another performance metric which has been the subject of numerous studies in production scheduling [65,66,38,67]. In service-oriented production systems such as hospitals, call centers, etc., it is important to provide customers a uniform service (in terms of waiting time or completion time) [64,68]. Therefore, min(CTV) can also serve social-oriented scheduling objectives. min(CTV) and min(TCT) are possibly inconsistent objectives, and balancing trade-off between them is required.

Notations

i Index of clusters, i {1, 2, 3} Ji Number of sub-clusters in cluster i ji Index of sub-clusters in cluster i , ji {1, 2, …, Ji}k Index of the individual metrics, k {1, 2, 3} with k = 1 for TCT , k = 2 for MCT , and k = 3 for CTV . Index of production scheduling alternatives, = {1, 2, …, } x [k]The value of metric k for production schedule D[k]Normalized deviation of metric k for production schedule M [k, i], j Sustainability score of for metric k in sub-cluster j of cluster i . Let i {1, 2, 3} denote the index of clusters, with 1 for economic, 2 for environmental, and 3 for social pillars of the TBL . Ji is the number of sub-clusters in each cluster, therefore, ji {1, 2, …, Ji} denotes subcluster j in cluster i . Let k {1, 2, 3} denote the index of the individual metrics, with k = 1 for TCT , k = 2 for MCT , and k = 3 for CTV . = {1, 2, …, } , where is a set of Given production schedule alternative production schedules, x [k] is the value of metric k for production schedule . Because individual metrics could be in different scales/units, we define the normalized deviation from the best possible

2.3.1. Summary The metrics used for production scheduling to achieve different goals are summarized in Table 1 based on their impacts on different pillars of the TBL . We deliberately kept these metrics generic, so that practitioners may use/modify them according to different production environments such as flow shop, job shop, single/multiple machines, etc. MCT , TCT , and CTV were identified as the most fundamental metrics for production scheduling. As shown by Table 1, some of these metrics are simultaneously evaluated in multiple TBL pillars. As mentioned earlier, min(MCT) , min(TCT), and min(CTV) are inconsistent objectives, which means optimizing one may result in worsening the others. Therefore, a production schedule that simultaneously optimizes all objectives is infeasible.

value as D[k] =

This section presents the mathematical formulations of the proposed models. In order to develop a comprehensive Production Scheduling Table 1 Metric-based hierarchical decomposition for sustainable production scheduling. Sub-cluster

Metric

Economic-oriented scheduling

Production cost

MCT CTV MCT TCT MCT TCT TCT MCT TCT MCT TCT TCT CTV

Energy cost Labor cost Environmental-oriented scheduling

Inventory cost Energy consumption Greenhouse gas emissions

Social-oriented scheduling

Waiting time Waiting time variance

LB(x[k ] )

LB(x[k] )

, where UB(x [k] ) = max

(x [k] ) , and

= min (x ) . In the minimization sense, D is the degree of closeness between x [k] and its best value LB(x [k] ) . At the metric level, we use M [k] = 10(1 D[k] ) to calculate the sustainability score of production schedule for metric k . M [k] [0, 10] normalizes each metric to a scale of 0 to 10, where 0 is the worst performance and 10 is the best performance in terms of sustainability. M [k] attributes the sustainability of an individual metric to its normalized deviation. For example, as shown in Table 1, waiting time variance is directly affected by CTV ; therefore, production schedule with x [3] = LB(x[3] ) generates the highest sustainability score for the waiting time variance, i.e., M [3] = 10 . Once the top-bottom structure is developed and all the individual metrics are measured, a bottom-up aggregation approach by Eqs. (8)–(13) is utilized to calculate PSSI . Eq. (8) is the aggregation of the individual metrics sustainability scores into the Sub-cluster Sustainability Score (SSS ), where k, j, i [0, 1] is the weight assigned to the metric k of sub-cluster j in cluster i . Eq. (9) imposes that the sum of all weights must be equal to 1. Eq. (10) is the aggregation of sub-cluster sustainability scores to the Cluster Sustainability Score (CSS ), where j, i is the weight of sub-cluster j in cluster i . Eq. (11) indicates that the sum of sub-cluster weights must be 1. Eq. (12) aggregates cluster sustainability scores into the Production Scheduling Sustainability Index (PSSI ), where, i is the weight of cluster i . Eq. (13) indicates that the sum of cluster weights is equal to 1:

LB(x [k] )

3. Problem formulation

Cluster

x [k ]

UB(x[k ] ) [k ]

[k ]

K

SSS

, i, j

=

i, j , k M

[k ]

k=1

(8)

K i, j , k

=1

(9)

k= 1 Ji

CSS

TCT : total completion time, MCT : maximum completion time, CTV : completion time variance.

,i

=

i, j SSS , i, j j=1

177

(10)

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A. Abedini, et al. Ji

L

i, j

=1

s. t .

(11)

j=1

,i

C

(12)

C

=1

(13)

i=1

(l),1

(17)

kD

k

=1

k

C

]+p

(L), m 1

(L), m

(19) (20)

(N ), M

C

(l), M

(21) (22)

1 N

N

(C

(l), M

ACT )2

l =1

(23)

=1

(24)

LB(x [k] ) = min(x [k] )

(25)

UB(x [k] ) = max(x [k] )

(26)

Eq. (16) is the objective function that minimizes the weighted sum of the deviations of performance metrics from their optimum values (i.e., D[k]), where k is the weight of min(x [k] ) in the objective function. Eqs. (17)–(19) are used to recursively compute the completion time of the job in position L of sequence on machine m . Eq. (20) calculates MCT for sequence , Eq. (21) is the TCT of sequence , and Eq. (23) computes CTV of sequence . Eq. (24) imposes that the summation of weights must be equal to 1. Considering the NP -completeness of the optimization model and the quadratic nature of constraint (23), it is extremely difficult to optimally solve this model for a large number of jobs. The most viable practice to solve such models is the use of metaheuristics. For the purpose of demonstration, we carry out a series of case studies. The number of jobs N changes from N = 5, …, 10 , resulting in six choices. Number of machines M changes from 3 to 19 (M = 2l + 1, l = 1, 2, …, 9), yielding nine choices. This configuration results in 54 combinations of number of jobs and number of machines. For each combination, 100 instances are randomly generated. The processing times are randomly drawn from a uniform distribution in [1,99]. Therefore, in total we have 5400 instances. Given k changing from [0.0: 0.1: 1.0], we have 66 combinations of three weights with 3 = 1, i.e., { 1, 2 , …, 66} . The 66 minimization functions k=1 k of min(z [ ]) cover the three single-objective minimizations of min (TCT ), min(MCT ) and min(CTV ) with weights equal [1, 0, 0], [0, 1, 0], and [0, 0, 1], respectively. Since the number of jobs is relatively small (N 10 ), we are able to use enumeration/optimization packages to find UB(x [k] ) and LB(x [k] ) for all k as well as z [ ] for each weight.

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Notations

N Number of jobsn Job index, n =1, 2,…, N M Number of machinesm Machine index, m =1, 2,…, M Decision space, | | = N ! , a permutation of N jobs, (l) = n if job n occupies position l in l Index of portions in , l {1, 2, …, N }P Matrix of processing times, P(n, m) = pn, m Given an instance with N jobs, there are N! different possible se{ (1), (2), …, (N )} denote a sequences (i.e., | | = N !). Let quence of N jobs, and (l) = n if job n occupies position l in sequence . We modify the optimization model presented by Eqs. (14) and (15) with regards to the constraints of an N -job, M -machine permutation flow shop as shown by Eqs. (16)–(26):

k=1

(L 1), m,

k= 1

The PSSI model is demonstrated here by evaluating the sustainability performance of different scheduling methods in a permutation flow shop. The flow shop scheduling problem arises where a set of jobs on one or multiple machines must be sequenced in order to optimize a given objective function. Permutation flow shop is a special type of flow shop in which the processing order of jobs is identical on all machines. Permutation flow shop has been the subject of a massive body of literature [10].

x [k] LB(x[k] ) UB(x [k] ) LB(x [k] )

[

= max C

3

4. Numerical study

k

(L), m

x [3] =

It is worth mentioning that the proposed CP model is a generic model that needs to be worked out with regards to the production system constraints, i.e., flow shop, job shop, continuous manufacturing, etc. An example of such modifications is given in Section 4 for an N -job M -machine permutation flow shop problem.

3

(18)

ACT = x [2]/N

(15)

z ( , , P) =

(1), r

l=1

3 k=

p

N

[k ]

k=1

=

x [2] =

3

z ( , , P) =

(1), m

x [1] = C

PSSI provides the ability to evaluate the sustainability of production scheduling alternatives, which then allows for a systematic comparison among them across different levels of the production system, i.e., organizational vs. shop floor levels. We also propose a compromise programming (CP ) model [69] to balance the trade-offs among the inconsistent objectives of min(TCT), min(MCT) and min(CTV) as shown by Eqs. (14)–(15), where is a sequence of jobs that can be written in the form of an assignment matrix, P is the matrix of processing times, k is the weight of objective k in the objective function, and = { 1, 2 , 3} is the set of objectives weights. Eq. (14) explicitly integrates the decision maker's preference into the trade-off function by assigning a weight ( k ) to each metric. With dynamics in production, decision makers’ preferences might change as the process reveals its performance over time. min(z ) is precisely equivalent to minimizing the deviations from an ideal but infeasible point at which all objectives are at their optimum values:

min

p

r=1

3

s. t .

=

m

i CSS i=1

min

(L ),1

l=1

3

PSSI =

i

C

4.1. Inconsistencies among objectives Fig. 1 clearly shows that single optimization of min(TCT), min(MCT) , and min(CTV) are inconsistent with each other, since there is no single point that yields the best value for all three objectives. In order to statistically confirm the inconsistency among min(TCT), min(MCT) , and min(CTV) , we perform Spearman's rank correlation analyses for the sequences generating min(TCT), min(MCT) , and min(CTV) . Table 2 shows the Spearman's rank correlation coefficient ( ) between sequences. Small values of ( ) confirm that the sequences generating minimum values for single objectives are not significantly correlated.

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Fig. 1. Scatter plots of normalized deviations of 66 weights, Pareto-dominant solutions are shown by red markers. Table 2 Spearman's

4.3. Production Scheduling Sustainability

among sequence for scheduling objectives.

Spearman's

TCT vs. MCT

TCT vs. CTV

MCT vs. CTV

0.0823

0.2336

0.0528

In order to demonstrate the efficiency and effectiveness of trade-off balancing in achieving a sustainable production schedule, we perform the scheduling with respect to 66 trade-off functions as described earlier. As shown by Table 3, without loss of generality, we consider a special case for calculating PSSI in which equal weights are assigned to all metrics in each sub-cluster, and equal weights for all clusters as well.

4.2. Pareto dominance In the case of multi-objective optimization with inconsistent objectives, Pareto dominance is useful for decision making [70,71]. For K minimization problems, if xA[k] and xB[k] are two vectors that measure a positive attribute k , such as the utility of decision A and B , respectively, decision A dominates decision B if the following conditions are satisfied:

xB[k],

k

{1, 2, …, K }

(27)

xA[k] < xB[k],

k

{1, 2, …, K }

(28)

xA[k]

Table 3 PSSI weight assignment. i

PSSI

Eq. (27) states that decision A is not worse than decision B in any dimension, while Eq. (28) states that decision A is better than decision B in at least one dimension. Pareto optimal outcomes cannot be improved without sacrificing at least one objective. Pareto dominant solutions are shown in Fig. 1 by red markers; each cross shows the data point obtained from one weight. It is observed that there is no Pareto-optimal solution when all three objectives are taken into consideration.

0.333

0.333

0.333

179

Cluster Economic-oriented scheduling

Environmentaloriented scheduling

Social-oriented scheduling

i, j

Sub-cluster

0.25

Production cost

0.25

Energy cost

0.25

Labor cost

0.25 0.5

Inventory cost Energy consumption

0.5

Greenhouse gas emission

0.5

Waiting time

0.5

Waiting time variance

i, j , k

Metric

0.5

MCT

0.5 0.5 0.5 0.5 0.5 1 0.5

CTV MCT TCT MCT TCT TCT MCT

0.5 0.5

TCT MCT

0.5 1

TCT TCT

1

CTV

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The equal weights assignment means that all aspects of TBL have the same importance for the decision makers.

Table 4 shows the statistics of sustainability index for the 66 studied (z [ ] ) ). It is observed that single-obtrade-off functions (i.e., min jective objective functions of min(TCT), min(MCT) , and min(CTV) (i.e., min (z [1,0,0] ) , min (z [0,1,0] ) , and min (z [0,0,1] ) , respectively) are (z [0.5,0.3,0.2] ) in terms of sustainability index. outperformed by min min (z [0.5,0.3,0.2] ) has not only the highest average sustainability index of 9.06 but also the minimum standard deviation of 0.48. In terms of the fundamental performance metrics as shown by Table 5, single-objective functions generate no deviation on their intended metric but very large deviations on the others. On the other hand, min (z [0.5,0.3,0.2] ) generate small and uniform deviations on all three metrics, which provides a better control over the system performance, meaning that a stable compromise among the inconsistent objectives is achieved. In order to further discuss the performance of production scheduling objective functions, we narrow our discussion to four production min (z [1,0,0] ) , min (z [0,1,0] ) , scheduling alternatives of min (z [0,0,1] ) , and min (z [0.5,0.3,0.2] ) which we hereafter name min(TCT), min(MCT) , min(CTV) , and min(TO) , respectively. These weights are shown in bold fonts in Table 4.

4.4. Results and discussion In this subsection, we statistically present the results of our case study in terms of production sustainability indices, sustainability scores at sub-cluster level, and fundamental performance metrics of TCT , MCT , and CTV . Using equal weights presented by Table 3, we can rewrite Eqs. (8)–(13) as follows:

PSSI = 10[0.5(1 PSSI = 10[1

D[1] ) + 0.292(1

(0.5D

[1]

+ 0.292D

[2]

D[2] ) + 0.208(1

D[3] )]

[3]

+ 0.208D ) ] (29)

= z [0.5,0.292,0.208] z [0.5,0.3,0.2]

As shown by Eq. (29), production scheduling to maximize PSSI (i.e., max (PSSI) ) as the objective function is equivalent to min (z [0.5,0.3,0.2] ) . Therefore, depending on the weight assignment for PSSI , scheduling for sustainable production always leads to a trade-off function among the fundamental performance metrics of TCT , MCT , and CTV . It is worth mentioning that the weight assignments may change over time as the production system reveals its nature or due to production/market needs. Therefore, different weights for the trade-off function may be used. The contribution of the PSSI model is twofold: (i) at a higher level, PSSI evaluates the sustainability of the production schedules in terms of a quantifiable value, (ii) at the lower level, PSSI provides insight into the system performance in terms of the fundamental performance metrics. These two contributions together provide a comprehensive control over the production performance. Table 4 PSSI statistics of trade-off functions (min

Ave Std Max Min

Ave Std Max Min

Ave Std Max Min

Ave Std Max Min

Ave Std Max Min

Ave Std Max Min

4.4.1. Production scheduling sustainability In this subsection, we compare the performance of the aforementioned production scheduling alternatives including min(TCT), min(MCT) , min(CTV) , and min(TO) in terms of sustainability index (PSSI ) and the fundamental performance metrics of TCT , MCT , and CTV . Table 6 shows the PSSI for the studied production scheduling alternatives. min(TO) has the greatest average and also the smallest

(z[ ] ) ).

(1,0,0)

(0,1,0)

(0,0,1)

(0.9,0.1,0)

(0.8,0.2,0)

(0.7,0.3,0)

(0.6,0.4,0)

(0.5,0.5,0)

(0.4,0.6,0)

(0.3,0.7,0)

(0.2,0.8,0)

8.52 0.74 10.00 5.36

8.69 0.67 10.00 5.44

7.04 1.26 10.00 2.61

8.72 0.68 10.00 5.58

8.87 0.62 10.00 5.67

8.96 0.57 10.00 5.86

9.00 0.52 10.00 6.21

9.01 0.51 10.00 6.48

9.00 0.52 10.00 6.23

8.97 0.54 10.00 5.70

8.93 0.57 10.00 5.64

(0.1,0.9,0)

(0.9,0,0.1)

(0.8,0,0.2)

(0,7,0,0.3)

(0.6,0,0.4)

(0.5,0,0.5)

(0.4,0,0.6)

(0.3,0,0.7)

(0.2,0,0.8)

(0.1,0,0.9)

(0,0.1,0.9)

8.87 0.61 10.00 5.64

8.63 0.71 10.00 5.57

8.74 0.68 10.00 5.57

8.83 0.64 10.00 5.70

8.87 0.62 10.00 5.50

8.80 0.68 10.00 5.03

8.61 0.81 10.00 4.78

8.29 1.00 10.00 3.86

7.88 1.18 10.00 3.10

7.46 1.26 10.00 2.67

7.30 1.24 10.00 2.79

(0,0.2,0.8)

(0,0.3,0.7)

(0,0.4,0.6)

(0,0.5,0.5)

(0,0.6,0.4)

(0,0.7,0.3)

(0,0.8,0.2)

(0,0.9,0.1)

(0.1,0.1,0.8)

(0.1,0.2,0.7)

(0.1,0.3,0.6)

7.55 1.21 10.00 3.10

7.79 1.15 10.00 3.57

8.00 1.07 10.00 3.86

8.18 0.98 10.00 4.46

8.33 0.91 10.00 4.66

8.46 0.83 10.00 4.84

8.57 0.76 10.00 5.21

8.64 0.71 10.00 5.25

7.73 1.21 10.00 3.10

7.97 1.12 10.00 3.86

8.18 1.03 10.00 3.86

(0.1,0.4,0.5)

(0.1,0.5,0.4)

(0.1,0.6,0.3)

(0.1,0.7,0.2)

(0.1,0.8,0.1)

(0.2,0.1,0.7)

(0.2,0.2,0.6)

(0.2,0.3,0.5)

(0.2,0.4,0.4)

(0.2,0.5,0.3)

(0.2,0.6,0.2)

8.36 0.93 10.00 4.66

8.51 0.85 10.00 4.69

8.63 0.76 10.00 5.27

8.73 0.70 10.00 5.49

8.82 0.64 10.00 5.64

8.15 1.07 10.00 3.86

8.35 0.96 10.00 3.86

8.53 0.87 10.00 4.66

8.67 0.78 10.00 4.86

8.78 0.69 10.00 5.46

8.78 0.63 10.00 5.49

(0.2,0.7,0.1)

(0.3,0.1,0.6)

(0.3,0.2,0.5)

(0.3,0.3,0.4)

(0.3,0.4,0.3)

(0.4,0.3,0.3)

(0.3,0.5,0.2)

(0.3,0.6,0.1)

(0.4,0.1,0.5)

(0.4,0.2,0.4)

(0.4,0.4,0.2)

8.92 0.59 10.00 5.64

8.52 0.88 10.00 4.58

8.67 0.79 10.00 4.69

8.81 0.69 10.00 5.32

8.91 0.62 10.00 5.49

9.00 0.54 10.00 5.87

8.97 0.56 10.00 5.64

8.98 0.54 10.00 5.64

8.80 0.70 10.00 4.91

8.92 0.61 10.00 5.55

9.03 0.51 10.00 6.04

(0.4,0.5,0.1)

(0.5,0.1,0,4)

(0.5,0.2,0.3)

(0.5,0.3,0.2)

(0.5,0.4,0.1)

(0.6,0.1,0.3)

(0.6,0.2,0.2)

(0.6,0.3,0.1)

(0.7,0.1,0.2)

(0.7,0.2,0.1)

(0.8,0.1,0.1)

9.02 0.51 10.00 6.23

8.96 0.56 10.00 5.95

9.04 0.50 10.00 6.46

9.06 0.48 10.00 6.65

9.04 0.49 10.00 6.52

8.98 0.55 10.00 5.70

9.02 0.51 10.00 6.04

9.02 0.51 10.00 5.86

8.92 0.60 10.00 5.67

8.95 0.57 10.00 5.67

8.83 0.65 10.00 5.67

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Table 5 Normalized deviation statistics of selected trade-off functions (min (1,0,0)

(z[ ] ) ).

(0,1,0)

(0,0,1)

(0.5,0.3,0.2)

D[k]

D[1]

D[2]

D[3]

D[1]

D[2]

D[3]

D[1]

D[2]

D[3]

D[1]

D[2]

D[3]

Ave Std Max Min

0.000 0.000 0.000 0.000

0.245 0.164 1.000 0.000

0.368 0.165 0.932 0.000

0.187 0.133 0.870 0.000

0.000 0.000 0.000 0.000

0.180 0.134 0.751 0.000

0.489 0.194 0.985 0.000

0.176 0.135 1.000 0.000

0.000 0.000 0.000 0.000

0.065 0.070 0.654 0.000

0.045 0.064 0.653 0.000

0.236 0.149 0.938 0.000

Table 6 PSSI for alternative production schedules.

min(TCT ) min(MCT ) min(CTV ) min(TO)

Table 7 PSSI capabilities indices for alternative production schedules.

PSSI (Ave)

PSSI (Std)

p -value, t -test against min(TO)

8.52 8.69 7.04 9.06

0.74 0.67 1.27 0.48

0.000 0.000 0.000 –

min(TCT ) min(MCT ) min(CTV ) min(TO)

standard deviation. t -tests revealed that the performance of min(TO) is significantly different compared to the other production scheduling alternatives. Fig. 2 shows the Pareto frontiers of the sustainability index mean and standard deviation for the studied production scheduling alternatives where min(TO) dominates the others. (USL LSL) Cp Process capability indices, and 6 USL

µ

µ

Cp

Cpk

% < LSL

0.35 0.38 0.20 0.53

0.01 0.10 0.39 0.40

49.00 38.65 87.76 11.76

11.76% of min(TO) outputs fall below 8.5. As it is graphically shown by Fig. 3, the distributions of sustainability index of min(TO) has a negative skewness, which means that the mass of distribution is concentrated close to the maximum value of 10. min(CTV) has the worst sustainability status with PSSI = 7.04 , Cp = 0.20 , and Cpk = 0.39, which mean that the outputs of the production schedule are highly scattered away from the average value. Single-objective production scheduling alternatives perform poorly with a high percentage of observations below LSL .

LSL

Cpk = min[ 3 , 3 ], are used to further compare the sustainability of different production scheduling alternatives. Given LSL and USL , greater values of Cp and Cpk imply that a process generate outputs which are more centered with smaller variations [72]. To perform process capability analyses, we first need to define the specification limits of PSSI . Generally, a sustainability score of 8-10 indicates an excellent sustainability status [17]; therefore, in this case study we consider (LSL, USL) = (8.50, 10) . Table 7 shows Cp , Cpk , and the percentage of results less than LSL for the alternative production schedules. It is observed that min(TO) outperforms all other production scheduling alternatives. The outputs of min(TO) not only center around the average value, but also provide greater values of Cpk , which means the process is better under control in terms of sustainability index. In order to further evaluate the capability of the production scheduling alternatives, we have provided the percentage of observations that fall below the lower specification limits (i.e., % < LSL ) in Table 7. Lower values of % < LSL demonstrate that a process has greater capability relative to the lower specification limit. We use % < USL to evaluate the capability of solutions, because the greater PSSI the more sustainable production schedule. Therefore, those observations that fall below the LSL show large deviations from the excellent sustainability status. Only

4.4.2. Sustainability at sub-cluster level A summary of observations and a comparison of the results for the application of PSSI for the studied production scheduling alternatives is presented here. Table 8 shows the statistics of sustainability scores at sub-cluster level. It is observed that even though min(TO) generates the highest value of PSSI , it does not necessarily generate the highest sustainability score for each sub-cluster. That is because some of subclusters are driven by a single metric (e.g., waiting time is driven only by TCT ) and the weighting approach plays a big role on the contribution of each metric on the final PSSI value. As is shown graphically in Fig. 4, min(TO) has the largest area compared to the other alternative schedules. min(CTV) has the smallest area in Fig. 4 with the lowest score in all sub-clusters except for the waiting time variance. Therefore, min(CTV) can be seen as the least sustainable scheduling alternative with a PSSI of 7.04. Fig. 5 shows the Pareto frontiers of sustainability scores average and standard deviation at the sub-cluster level. It is observed that although min(TO) dominates all the other alternatives at the PSSI level, it does not necessarily need the dominance at the sub-cluster level. In fact, the min(TO) model provides the best compromise among the inconsistent objectives that leads to the best outcome at the higher level, and also provides the decision maker with a desirable flexibility at the lower levels. These two together are effective tools for the decision maker to ensure the stability of the overall performance while being able to hedge against the inconsistencies at the lower levels. 4.5. Fundamental performance metrics In order to evaluate the performance of production schedule alternatives in terms of the fundamental performance metrics of TCT , MCT , and CTV , we perform a series of statistical analyses, including statistical process control (SPC) and process capabilities analysis. Table 9 presents the summary of SPC charts (i.e., x¯ –R charts) for the studied production schedule alternatives. For the sake of brevity, the SPC charts are not presented.

Fig. 2. PSSI Pareto frontier. 181

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Fig. 3. PSSI process capability charts for alternative production schedules.

As shown by Table 9, single-objective production scheduling alternatives of min(TCT), min(MCT) , and min(CTV) generate no deviations on their intended objective but very large deviations on the others. Furthermore, single objective production scheduling alternatives generate lower PSSI compared to min(TO). On the other hand, min(TO) not only provides the two best PSSI , respectively, but also small and uniform deviations on the fundamental performance metrics. As discussed earlier, the first step in trade-off balancing is to establish an ‘ideal point’. The coordinates of the ideal point are given by the optimum values of all objectives. It is obvious that with inconsistent objectives, the ideal point is not feasible to achieve; therefore, the ideal point is only a point of reference. The second step is to establish an “anti-ideal” point. The coordinates of the anti-ideal point are given by the worst values of all objectives. The objective of trade-off balancing is to find the closest efficient solution to the ideal-point. To measure the distances between a solution and the ideal point, a family of distance 3 functions are introduced as follows: Lp ( , k ) = [ k = 1 (wk D[k] ) p]1/ p , where wk is the preference of the decision maker for metric k [73]. When p = 1, L1 measures the longest distance (geometrically speaking) between thr solution and the ideal point. Let us assume that all three fundamental metrics have the same importance for the decision maker, 3

Fig. 4. Sustainability scores at sub-cluster level.

analyses for the fundamental performance metrics. In order to perform capability analyses, we first need to define the specification limits of each performance indicator. Eqs. (30) and (31) represent the LSL and USL of performance indictor k using the performance of min(To) (because it has min(L1, h ). This definition not only provides tight specification limits with only one standard deviation, but also drives the specification limits towards 0, which is desirable for minimizing the deviation from the best value for each performance indicator. Table 11 shows the specification limits used in this study:

D[k ]

that is, L1 = k =31 . Table 10 shows the SPC statistics of L1 for the studied alternatives. It is observed that min(TO) generates the minimum average L1 of 0.115 with the tightest bounds of [0.099, 0.131]. Therefore, it can be concluded that minimizing the trade-offs among the fundamental production scheduling performance metrics of TCT , MCT , and CTV not only results in a sustainable production schedule, but also provides a better control over the system performance at the metric level. In addition to SPC analyses, we perform process capabilities Table 8 Statistics of sustainability scores at sub-cluster level.

min(TCT ) min(MCT ) min(CTV ) min(TO)

Production cost

Energy cost

Labor cost

Inventory cost

Energy consumption

Greenhouse gas emissions

Waiting time

Waiting time variance

µ

µ

µ

µ

µ

µ

µ

µ

8.16 8.16 7.55 8.50

0.82 0.84 0.97 0.77

8.78 9.06 6.67 9.45

0.82 0.67 1.51 0.43

8.78 9.06 6.67 9.45

0.82 0.67 1.51 0.43

10.00 8.13 5.11 9.35

0.00 1.33 1.94 0.70

8.78 9.06 6.67 9.45

µ : average, : standard deviation. 182

0.82 0.67 1.51 0.43

8.78 9.06 6.67 9.45

0.82 0.67 1.51 0.43

10.00 8.13 5.11 9.35

0.00 1.33 1.94 0.70

6.32 8.20 10.00 7.64

1.65 1.34 0.00 1.49

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Table 11 Specification limits of performance indicators. min(TO)

Specification limits

µ

LSL

USL

0.000 0.000 0.087

0.135 0.109 0.385

0.065 0.045 0.236

TCT MCT CTV

0.070 0.064 0.149

min(TO ) = min(TO[0.5,0.3,0.2] ) . Table 12 Capability results for the scheduling alternatives. min(TCT )

min(MCT )

min(CTV )

min(TO)

Cp

TCT

inf

0.17

0.12

0.32

Cpk

MCT CTV TCT

0.28 0.04 0.00 79.72 45.81

inf 0.23 65.33 0.00 6.09

0.17 inf 96.83 69.13 100.00

0.23 0.34 15.49 15.56 15.24

Fig. 5. Sub-cluster sustainability score Pareto frontier. Table 9 Average performance and control limits of fundamental performance metrics. min(TCT )

min(MCT )

min(CTV )

min(TO)

TCT MCT CTV PSSI

0.000 0.203 0.327 8.32

0.151 0.000 0.147 8.50

0.442 0.142 0.000 6.72

0.045 0.027 0.199 8.92

TCT MCT CTV PSSI

0.000 0.245 0.368 8.52

0.187 0.000 0.180 8.69

0.490 0.176 0.000 7.04

0.065 0.045 0.236 9.06

x -UCL

TCT MCT CTV PSSI

0.000 0.287 0.410 8.71

0.223 0.000 0.213 8.87

0.537 0.211 0.000 7.36

0.084 0.062 0.272 9.19

R -LCL

TCT MCT CTV PSSI

0.000 0.446 0.441 2.07

0.383 0.000 0.348 1.95

0.504 0.366 0.000 3.43

0.209 0.190 0.385 1.41

R

TCT MCT CTV PSSI

0.000 0.702 0.695 3.27

0.603 0.000 0.548 3.08

0.796 0.576 0.000 5.40

0.329 0.299 0.606 2.22

R -UCL

TCT MCT CTV PSSI

0.000 0.958 0.949 4.46

0.823 0.000 0.748 4.20

1.085 0.786 0.000 7.36

0.449 0.408 0.828 3.03

x -LCL



% > USL

0.11 0.32 inf

MCT CTV TCT MCT CTV

[k ] LSL[k] = max[0, µmin(TO)

[k ] USL[k] = µ min(TO) +

[k ] min(TO)

[k ] min(TO) ]

inf 0.37 0.13

0.14 inf 0.62

0.29 0.34 0.31

(30) (31)

Given the specification limits shown in Table 11, we calculate Cp and Cpk of the fundamental performance metrics for each scheduling alternative. The results of the capability analyses are shown in Table 12. It is observed that the single objective optimizations perform poorly in terms of Cp and Cpk throughout all three fundamental metrics. min(TO) exhibits a uniform and acceptable performance. The outputs of min(TO) not only center around the average value, but also provide greater values of Cpk , which means the process is better under control. In order to further evaluate the capability of different solutions, we have provided the percentage of observations that fall above the upper specification limits (i.e., % > USL ) in Table 12. Lower values of % > USL demonstrate that a process has greater capability relative to the upper specification limit. We use % > USL to evaluate the capability of solutions, because in minimization problems the objective is to minimize the deviations from the minimum possible value. Therefore, those observations that fall above the USL show large deviations from the best solution and are of extreme importance in decision making. However, single-objective functions show a very heterogeneous performance on different metrics, i.e., no deviation on their intended objectives but a very large percentage of outputs above USL for the others. On the other hand, min(TO) evenly performs with roughly 15.50% above USL for all three metrics.

LCL : lower control limit, UCL : upper control limit, x¯ : AVERAGE, R : VARIATION range.

Table 10 L1 average performance and control limits.

5. Conclusion

min(TCT )

min(MCT )

min(CTV )

min(TO)

x -LCL x¯ x -UCL

0.178 0.204 0.231

0.107 0.122 0.138

0.196 0.222 0.248

0.099 0.115 0.131

R -LCL R R -USL

0.276 0.435 0.593

0.162 0.256 0.349

0.276 0.435 0.594

0.171 0.270 0.368

std

0.098

0.056

0.100

0.060

Production scheduling is an operational level of decision making that covers not only the manufacturing stage of the product life-cycle, but also the use stage of the processes life-cycle. Our comprehensive literature review shows that a systematic model for evaluating the sustainability of a production schedule with respect to all aspects of the triple bottom line (TBL ) is virtually absent. To fill this gap, in this paper, we propose a quantitative yet generic sustainability evaluation scheme for production scheduling. Our proposed Production Scheduling Sustainability Index (PSSI ) is a hierarchical procedure that decomposes the objectives defined in the TBL into three clusters of (i) economic-oriented scheduling, (ii) environment-oriented scheduling, and (iii) social-oriented scheduling. Each cluster is then decomposed

std : standard deviation.

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into multiple sub-clusters, each covering a specific area of impact in the production. Finally, each sub-cluster is divided into multiple measurable fundamental metrics. Once the fundamental metrics are measured, an aggregation procedure including normalization and weighting is adopted to calculate the PSSI . The weighting scheme can reflect the decision maker's preferences according to policies, regulations, and/or market necessities. Through an extensive literature review, maximum completion time (MCT ), total completion time (TCT ), and completion time variance (CTV ) were identified as the most fundamental performance metrics that drive many other aspects of the production. We statistically show that the objectives of min(TCT), min(MCT) , and min(CTV) are inconsistent- that is, a schedule optimizing all metrics is infeasible. We propose a compromise programming model in order to minimize the distance between the production scheduling performance and an ideal yet infeasible point at which all the fundamental performance metrics are at their optimum values. We then utilize the normalized deviation of each metric from its best possible value to construct the sustainability scores at the metric-level of the PSSI . We show that balancing trade-offs not only supports the scheduling for sustainability at the organizational level, but also results in a better control over the processes at the shop floor level. In order to evaluate the efficiency and effectiveness of the proposed models, a comprehensive case study on a permutation flow shop scheduling problem was performed. Multiple production scheduling alternatives were compared in terms of sustainability index as well as the fundamental performance metrics. The results show that scheduling for sustainability outperforms all the studied production scheduling alternatives with maximum average and minimum standard deviation of PSSI . It was observed that although production scheduling for sustainability results in the best and tightest bounds for PSSI , it allows larger variation ranges for the fundamental metrics which provide a less sensitive schedule. The limitations of the proposed PSSI model include (i) the weight assignments can be very subjective and is driven by the market and the system constraints, thus, it must be worked out before implementation of the PSSI . However, any weight assignment will result in a trade-off function among the fundamental performance metrics, thus, it does not affect the validity of the proposed model. (ii) Considering the NP -completeness of most of the production scheduling problems, the solvability of the proposed compromise programming model depends on the problem size and the applied operations research tools. Our future studies will be in two directions. (i) Since the weighting scheme can dramatically affect the skewness of the PSSI distribution, further studies are required to find the most suited weighting scheme for production scheduling sustainability. (ii) Considering the NP -completeness of most production scheduling problems, it is extremely time consuming to determine optimum schedules for a large number of jobs. Therefore, developing heuristics/metaheuristics is necessary for large scale problems. Our next step will be developing a heuristic for balancing trade-offs among min(TCT), min(MCT) , and min(CTV) .

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Conflict of interest None declared. Acknowledgement We appreciate the supports from the Department of Mechanical Engineering and the Institute for Sustainable Manufacturing (ISM) at the University of Kentucky. The authors would like to appreciate the anonymous reviewers for their constructive comments and suggestions. We also appreciate Ms. Brooke Lawson for her diligent proofreading of this paper. 184

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