Energy efficient scheduling problems under Time-Of-Use tariffs with different energy consumption of the jobs⁎

Proceedings,16th IFAC Symposium on Proceedings,16th IFAC Symposium on Information Control Problems in Manufacturing Proceedings,16th IFAC Symposium on Proceedings,16th IFAC Symposium on Available online at www.sciencedirect.com Information Control Problems in Manufacturing Proceedings,16th IFAC Symposium on Bergamo, Italy, June 11-13, 2018 Information Control Problems in Information Control in Manufacturing Manufacturing Bergamo, Italy, JuneProblems 11-13, 2018 Information Control Problems in Manufacturing Bergamo, Italy, June 11-13, 2018 Bergamo, Italy, June 11-13, 2018 Bergamo, Italy, June 11-13, 2018

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IFAC PapersOnLine 51-11 (2018) 1053–1058

Energy Energy efficient efficient scheduling scheduling problems problems under under Energy efficient scheduling problems under Time-Of-Use with different energy Energy efficienttariffs scheduling problems under Time-Of-Use tariffs with different energy Time-Of-Use tariffs with different energy   energy consumption of the jobs Time-Of-Use tariffs with different consumption of the jobs consumption of the jobs consumption of the jobs  MohammadMohsen Aghelinejad, Yassine Ouazene, Alice Yalaoui

MohammadMohsen Aghelinejad, Yassine Ouazene, Alice Yalaoui MohammadMohsen Aghelinejad, Aghelinejad, Yassine Yassine Ouazene, Ouazene, Alice Alice Yalaoui Yalaoui MohammadMohsen MohammadMohsen Aghelinejad, Yassine Ouazene, Alice Yalaoui Industrial Systems Optimization Laboratory (ICD, UMR 6281, CNRS) Industrial Systems Optimization Laboratory (ICD, UMR 6281, CNRS) Industrial Optimization Laboratory (ICD, UMR Universit´ ee de de Industrial Systems Systems Optimization Laboratory (ICD, France UMR 6281, 6281, CNRS) CNRS) Universit´ de Technologie Technologie de Troyes, Troyes, France Industrial Systems Optimization Laboratory (ICD, UMR 6281, CNRS) Universit´ ee de Technologie de Troyes, France E-mail: (mohsen.aghelinejad, yassine.ouazene, alice.yalaoui)@utt.fr Universit´ de Technologie de Troyes, France E-mail: (mohsen.aghelinejad, yassine.ouazene, alice.yalaoui)@utt.fr Universit´e de Technologie de Troyes, France E-mail: yassine.ouazene, alice.yalaoui)@utt.fr E-mail: (mohsen.aghelinejad, (mohsen.aghelinejad, yassine.ouazene, alice.yalaoui)@utt.fr E-mail: (mohsen.aghelinejad, yassine.ouazene, alice.yalaoui)@utt.fr Abstract: This This paper paper deals deals with with single single machine machine scheduling scheduling problems, problems, where where the the machine machine may may Abstract: Abstract: This paper with single machine scheduling problems, where the may be in in processing, processing, idle or deals off state. state. The machine consumes a different different amount of energy energy in function function Abstract: This idle paper deals withThe single machine scheduling problems, where the machine machine may be or off machine consumes a amount of in Abstract: paper deals with single machine scheduling problems, where the machine be in idle or off The machine aa different amount of in of its its state. This Moreover, during processing (ONconsumes state), the energy consumption depends onmay the be in processing, processing, idle or during off state. state. The machine consumes different amount of energy energy in function function of state. Moreover, processing (ON state), the energy consumption depends on the be in processing, idle or off state. The machine consumes a different amount of energy in function of its state. Moreover, during processing (ON state), the energy consumption depends on the job. The complexity of these problems, when the jobs’ sequence is fix, for the uniform-speed of its state. Moreover, during processing (ON state), the energy consumption depends on the job. complexity ofduring these problems, when the jobs’ is fix, for thedepends uniform-speed of itsThe state. Moreover, processing (ON state), thesequence energy consumption on the job. The complexity of when the jobs’ sequence is for problem and the speed-scalable speed-scalable problem are are analyzed. For this purpose, a dynamic dynamic programming job. The complexity of these these problems, problems, when the jobs’ sequence is fix, fix, for the the uniform-speed uniform-speed problem and the problem analyzed. For this purpose, a programming job. The complexity of these problems, when the jobs’ sequence is fix, for the uniform-speed problem and the speed-scalable problem are analyzed. For this purpose, a dynamic programming approach is proposed to solve these problems by using a finite graph. The results demonstrate problem and the speed-scalable problem are analyzed. For this purpose, a dynamic programming approach is proposed to solve these problems by usingFor a finite graph. The resultsprogramming demonstrate problem the speed-scalable problem are analyzed. this a dynamic approach is to solve problems by using graph. The results demonstrate that the the and uniform-speed problem is polynomial polynomial of degree degree 3,finite andpurpose, its speed-scalable case is pseudo pseudo approach is proposed proposed toproblem solve these these problems by using aa3, finite graph. The resultscase demonstrate that uniform-speed is of and its speed-scalable is approach is proposed toproblem solve these problems by using a3, finite graph. The resultscase demonstrate that the uniform-speed is polynomial of degree and its speed-scalable is pseudo polynomial. that the uniform-speed problem is polynomial of degree 3, and its speed-scalable case is pseudo polynomial. that the uniform-speed problem is polynomial of degree 3, and its speed-scalable case is pseudo polynomial. polynomial. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. polynomial. Keywords: Energy Energy efficiency; efficiency; Time Time of of use use electricity electricity tariffs; tariffs; Non-preemption Non-preemption Single Single machine machine Keywords: Keywords: Energy efficiency; Time of of use electricity electricity tariffs; Non-preemption Single machine machine scheduling; Dynamic programming approach; Speed scalable problem Keywords: Energy efficiency; Time use tariffs; Non-preemption Single scheduling; Dynamic programming approach; Speed tariffs; scalableNon-preemption problem Keywords: Energy efficiency; Time of use electricity Single machine scheduling; Dynamic programming approach; Speed scalable problem scheduling; Dynamic programming approach; Speed scalable problem scheduling; Dynamic programming approach; Speed scalable problem 1. INTRODUCTION INTRODUCTION the total energy costs by just changing the order of the 1. the total energy costs by just changing the order of the 1. INTRODUCTION INTRODUCTION the total energy costs by just changing the order of the jobs or machine’s state, based on the on-peak and off1. the total energy costs by just changing the order of the jobs or machine’s state, based on the on-peak and off1. INTRODUCTION the total energy costs by just changing the order of the jobs or machine’s state, based on the on-peak and offpeek periods. In other words, it is possible to minimize In the most industrial countries, a significant portion of jobs or machine’s state, based on the on-peak and offpeek periods. In other words, it is possible to minimize In the most industrial countries, a significant portion of jobs or machine’s state, based on the on-peak and offpeek periods. In other words, it is possible to minimize In the most industrial countries, a significant portion of the energy consumption costs without minimizing the all the energy consumption is associated with industrial peek periods. In other words, it is possible to minimize In the most industrial countries, a significant portion of consumption costs without minimizing the all the the most energyindustrial consumption is associated with portion industrial peek periods. In other words, is possible to minimize In countries, a significant of the the energy energy consumption costsit without minimizing the all the energy energy consumption associated with industrial amount of energy consumption of the system. The energy activities, where electricity is used as the main energy the energy consumption costs without minimizing the all the consumption associated with industrial amount of energy consumption of the system. The energy activities, where electricity is used as the main energy the energy consumption costssystem without minimizing the all the for energy consumption is associated with industrial amount of energy consumption of the system. The energy activities, where electricity used as as the the main energy consumption of a production is composed of the source manufacturing. Furthermore, electricity prices amount of energy consumption of the system. The energy activities, where electricity is used main energy of aa consumption production system is composed of the source for where manufacturing. Furthermore, electricity prices consumption amount of energy of the system. The energy activities, electricity is used as the main energy consumption of production system is composed of the source for manufacturing. Furthermore, electricity prices amount of energy consumed during non-processing states are rising rising continuously. That is why, why, improving improving theprices effi- amount consumption of a consumed productionduring systemnon-processing is composed of the source for continuously. manufacturing. Furthermore, electricity of energy states are That is the effiof a consumed production system is and composed of the source foracontinuously. manufacturing. Furthermore, electricity amountand of energy energy consumed during non-processing states are rising continuously. That is improving the effi(NPE) processing state (PE). Biel Glock (2016) ciency of production system system inwhy, electricity consumption amount of during non-processing states are rising That isin why, improving theprices effi- consumption (NPE) and processing state (PE). Biel and Glock (2016) ciency of a production electricity consumption amount of energy consumed during non-processing states are rising continuously. That is why, improving the effi(NPE) and anda processing processing state (PE). (PE). Biel and and Glock models (2016) ciency of aaproduction productioncosts system in environmental electricity consumption consumption presented literature review of decision support to reduce reduce andin pollution (NPE) state Biel Glock (2016) ciency of production system electricity presented literature review of decision support models to production costs and environmental pollution (NPE) andaaaefficient processing state (PE). Biel and Glockthe (2016) ciency of aproduction production system electricitytoconsumption presented literature review of decision support models to reduce production costs andin environmental pollution for energy production planning. Within pasimultaneously, has encouraged researchers study these presented literature review of decision support models to reduce costs and environmental pollution energy efficient production planning. Within the pasimultaneously, has encouraged researchers to study these for presented a literature review of decision support models to reduce production costs and environmental pollution for energy efficient production planning. Within the pasimultaneously, has encouraged researchers to study these per which consider energy consumption minimization, Liu issues. for energy efficient production planning. Within the pasimultaneously, has encouraged researchers to study these per which consider energy consumption minimization, Liu issues. for energy efficient production planning. Within the pasimultaneously, has encouraged researchers to study these per which consider energy consumption minimization, Liu issues. and Huang (2014) examined a multi-objective optimizaA comprehensive review of previous works shows that per which consider energy consumption minimization, Liu issues. and Huang (2014) examined a multi-objective optimizaA comprehensive review of previous works shows that per which consider energy consumption minimization, Liu issues. and Huang Huang (2014) examined examinedmachine multi-objective optimizaA comprehensive consumption review of of previous previous works shows shows that and tion on aa batch-processing scheduling problem energy/electricity minimization, in aa manumanu(2014) aa multi-objective optimizaA comprehensive review works that tion on batch-processing scheduling problem energy/electricity consumption minimization, in and Huang (2014) examinedmachine a multi-objective optimizaA comprehensive review of previous works shows that tion on aapower batch-processing machine scheduling problem energy/electricity consumption minimization, in a manumanuto reduce consumption (carbon footprint) and total facturing system, can be applied at different levels: mation on batch-processing machine scheduling problem energy/electricity consumption minimization, in a to reduce power consumption (carbon footprint) and total facturing system, can be applied at different levels: mation on a batch-processing machine scheduling problem energy/electricity consumption minimization, in a manuto reduce power consumption (carbon footprint) and total facturing system, can be applied at different levels: maweighted tardiness. Mouzon and Yildirim (2008) presented chine; product and system. Among these perspectives, to reduce power consumption (carbon footprint) and total facturing system, can be applied at different levels: maweighted tardiness. Mouzon and Yildirim (2008) presented chine; product and system. Among these perspectives, to reduce power consumption (carbon footprint) and total facturing system, can be applied at different levels: maweighted tardiness. Mouzon and Yildirim (2008) presented chine; product and system. Among these perspectives, a framework to minimize the total energy consumption machine-level and product-level need enormous financial weighted tardiness. Mouzon and Yildirim (2008) presented chine; productand andproduct-level system. Among these perspectives, a framework to minimize the total energy consumption machine-level need enormous financial weighted tardiness. Mouzon and Yildirim (2008) presented chine; product and system. Among these perspectives, a framework framework to minimize the total energy energy consumption machine-level and product-level product-level need or enormous financial the total tardiness of aa system with idle and setup investment to to redesign redesign the machine(s) machine(s) product(s). But, aand to minimize the total consumption machine-level and need enormous financial and the total tardiness of with idle and setup investment the or product(s). But, aand framework to minimize the total energy consumption machine-level and product-level need enormous financial the total tardiness of aa system system with idle and setup investment to redesign the machine(s) or product(s). But, and energy. Yildirim and Mouzon (2012) proposed a matheat the system-level, manufacturers may reduce the energy the total tardiness of system with idle and setup investment to redesign the machine(s) or product(s). But, energy. Yildirim and Mouzon (2012) proposed a matheat the system-level, manufacturers may reduce the energy and the total tardiness of a system with idle and setup investment to redesign the machine(s) or product(s). But, energy. Yildirim and Mouzon (2012) proposed a matheat the system-level, manufacturers may reduce the energy matical model to minimize energy consumption and total consumption of their system by using existing decision energy. Yildirim and Mouzon (2012) proposed a matheat the system-level, manufacturers may reduce the energy matical model to minimize energy consumption and total consumption of their system by using existing decision energy. Yildirim and Mouzon (2012) proposed a matheat the system-level, manufacturers may reduce the energy matical model to minimize energy consumption and totala consumption of their system by using existing decision completion time of a single machine system by using models, production planning and scheduling optimization matical model to minimize energy consumption and total consumption of their systemand by scheduling using existing decision completion time of a single machine system by using a models, production planning optimization matical model to minimize energy consumption total consumption ofthis their system by using existing decision completion time of a single single machine system by byand using a models, production planning and scheduling optimization multi-objective genetic algorithm and dominance rules. A techniques. In paper, the system-level is studied to completion time of a machine system using a models, production planning and scheduling optimization multi-objective genetic algorithm and dominance rules. A techniques. In this paper, the system-level is studied to completion timegenetic of with a single machine system and by rules. using a models, production planning and scheduling optimization multi-objective algorithm and dominance A techniques. In this thisefficient paper, the the system-level is studied to scheduling problem continuous resource energy create an energy system for a single machine multi-objective genetic algorithm and dominance rules. A techniques. In paper, system-level is studied to scheduling problem with continuous resource and energy create an energy efficient system for a single machine multi-objective genetic algorithm and dominance rules. A techniques. In this paper, the system-level is studied to scheduling problem with continuous resource and energy create an energy efficient system for a single machine constraint for the energy consumption minimization is manufacturing. scheduling for problem with continuous resource and energy create an energy efficient system for a single machine constraint the consumption minimization is manufacturing. problem with continuous resource and energy create an energy we efficient system for a of single machine scheduling constraint by for the energy energy consumption minimization is manufacturing. addressed Artigues et al. (2013). In the the following, following, give some some examples the previous previous constraint for the energy consumption minimization is manufacturing. addressed by Artigues et al. (2013). In we give examples of the constraint forArtigues the energy consumption minimization is manufacturing. addressed by et al. (2013). In the following, following, we give givethese sometypes examples of the the previous The variation of electricity prices during the time period studies that investigated of problems. addressed by Artigues et al. (2013). In the we some examples of previous The variation of electricity prices during the time period studies that investigated these types of problems. addressed by Artigues et al. (2013). In the following, we give some examples of the previous The variation of electricity prices during the time period studies that investigated these types of problems. may impact the total energy cost of a system. In practice, The variation of electricity prices during the time period studies that investigated these types of problems. may impact the energy cost aa system. In practice, The electricity the time period studies that investigated these types of problems. may variation impact theoftotal total energyprices cost of ofduring system. In practice, electricity suppliers propose variable to balance the may impact the total energy cost of apricing system. In practice, electricity suppliers propose variable pricing to balance the Recently, various various approaches approaches have have been been proposed proposed for for enen- may impact the total energy costtoofimprove apricing system. In practice, electricity suppliers propose variable pricing to balance the supply and demand the reliability Recently, electricity suppliers propose variable to balance the electricity supply and demand to improve the reliability Recently, various approaches have been proposed for energy efficient efficient scheduling in manufacturing manufacturing systems, such such as electricity Recently, various approaches have been proposed for ensuppliers propose variable pricing to balance the supply and demand to improve the reliability and efficiency of electrical power grids, like time-of-use ergy scheduling in systems, as electricity supply and demand to improve the reliability Recently, various approaches have been proposed forcost enefficiency of electrical power grids, like time-of-use ergy efficient scheduling in manufacturing manufacturing systems, such as and energy consumption minimization and total total energy ergy efficient scheduling in systems, such as electricity supply and demand to improve the reliability and efficiency of electrical power grids, like time-of-use pricing, real-time pricing, and critical peak pricing. For energy consumption minimization and energy cost efficiency of pricing, electrical grids, time-of-use ergy efficient scheduling invarying manufacturing systems, such as and pricing, real-time andpower critical peaklike pricing. For exexenergy consumption minimization and total total energy cost minimization in aa time time electricity price system. energy consumption minimization and energy cost and efficiency of pricing, electrical grids, like time-of-use pricing, real-time pricing, andpower critical peak pricing. For example, with time-of-use (TOU) tariffs, retail energy prices minimization in varying electricity price system. pricing, real-time and critical peak pricing. For exenergy consumption minimization and total energy cost ample, with time-of-use (TOU) tariffs, retail energy prices minimization in aa time timethat, varying electricity price system. It must must be be mentioned mentioned for the the problemprice withsystem. energy pricing, minimization in varying electricity real-time pricing, and critical peak pricing. For example, with time-of-use (TOU) tariffs, retail energy prices to customers vary hourly to reflect changes in wholesale It that, for problem with energy ample, with time-of-use (TOU) tariffs, retail energy prices minimization in a minimization, timethat, varying electricity price system. customers vary hourly to reflect changes in wholesale It must be be mentioned mentioned that, for the the problem with energy to consumption costs it is possible to minimize It must for problem with energy ample, with time-of-use (TOU) tariffs, retail energy prices to customers vary hourly to reflect changes in wholesale energy prices, which are typically announced aa day ahead consumption costs minimization, it isproblem possible with to minimize customers vary hourly to reflect changes in wholesale It must be mentioned that, for the energy to energy prices, which are typically announced ahead consumption costs minimization, minimization, it is is possible possible to to minimize minimize consumption costs it to customers vary hourly to(2014)). reflect changes in wholesale  energy prices, which are typically announced aa day day ahead or an hour ahead (Fang et al. One of the most popresearchcosts is supported by the Champagne-Ardenne region in energy prices, which are typically announced day ahead  This consumption minimization, it is possible to minimize or an hour ahead (Fang et al. (2014)). One of the most popThis research is supported by the Champagne-Ardenne region in  energy prices, which are typically announced a day ahead  or an hour ahead (Fang et al. (2014)). One of the most popFrance and FEDER (Fonds europen de dveloppement conomique et This research is supported by the Champagne-Ardenne region in ular solutions to minimize the total energy consumption or an hour ahead (Fang et al. (2014)). One of the most popThis and research is supported by the de Champagne-Ardenne region in France FEDER (Fonds europen dveloppement conomique et ular solutions to minimize the total energy consumption  This and or ansolutions hour ahead (Fang et al.the (2014)). One of the most popresearch is supported by the de Champagne-Ardenne region in rgional). France FEDER (Fonds europen dveloppement conomique et ular solutions to minimize the total energy consumption France and FEDER (Fonds europen de dveloppement conomique et ular to minimize total energy consumption rgional). France ular solutions to minimize the total energy consumption rgional).and FEDER (Fonds europen de dveloppement conomique et rgional).

rgional). 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 1072Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2018 IFAC 1072Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 1072 Copyright © 2018 IFAC 1072 10.1016/j.ifacol.2018.08.468 Copyright © 2018 IFAC 1072

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cost of any system, is to investigate the NPE consumption and use a scheduling method to shift electricity use from peak periods to off-peak periods. Within the paper which deals with energy consumption costs minimization, Masmoudi et al. (2017a) and Masmoudi et al. (2017b) considered energy constraints and different energy costs during the planning horizon of a flowshop system in a lot-sizing problem. Gong et al. (2016a) proposed a generic mixed-integer programming model for a single machine scheduling problem to minimize total energy costs at volatile energy prices without exceeding the due date. The kind of machine or job, as well as the state of the machine and its speed during each state may modify the energy consumption of a machine. Among the papers that consider different energy consumptions for the machine, Albers and Fujiwara (2007) investigated online and offline energy-efficient algorithms for flow time and total cost minimization when the machine has variable speed and energy consumption. Antoniadis et al. (2015) analyzed the complexity of a decision problem that consists of minimizing the total electricity cost of processing jobs preemptively and non-preemptively, on a classical deadlines scheduling problem, when the processor has a variablespeed. Bampis et al. (2015) designed approximation algorithms for single processor and parallel processors speedscaling systems based on the idea of transforming preemptive schedules into non-preemptive schedules. Fang et al. (2014) considered the scheduling problem of processing jobs with arbitrary power demands that must be processed at a single uniform speed or speed-scalable machine to minimize total electricity cost under a time of use electricity tariffs. They analyzed the complexity of these two problems in preemptive and non-preemptive cases. Mikhaylidi et al. (2015) investigated a preemptive scheduling problem with an energy constraint in each period and different energy consumption for each job. They assumed the electricity time-varying prices to minimize the total electricity consumption and operations’ postponement penalties costs. Che et al. (2016) assumed an energyconscious single machine scheduling problem, when each processing job has its power consumption and electricity prices may vary from hour to hour throughout a day. Moreover, within some papers, authors consider several states for the machine. Each state consumes specific amount of energy. Shrouf et al. (2014) proposed a mathematical model to minimize total energy consumption costs of the processing jobs, considering variable energy prices during one day and different possible states for the machine with different energy consumption. Aghelinejad et al. (2016) and Aghelinejad et al. (2017b) studied the same problem as Shrouf et al. (2014), proposing two mathematical models. The first one considers a predetermined fixed order for the processing jobs, and the second one finds the optimal schedule for the machine state and job’s sequence simultaneously. They also presented a heuristic algorithm and a genetic algorithm to solve the second problem. Moreover, Aghelinejad et al. (2017a) analyzed the complexity of the preemption case of the problem with jobs and machine scheduling. Gong et al. (2016b)

Fig. 1. The considered states and transitions for the machine (Aghelinejad et al. (2017a)). addressed a novel production scheduling method to minimize the energy cost when finite state machine, multiple processes, idle modes and time varied electricity price are considered. The remainder of the paper is organized as follows. Section 2 describes the problem with uniform speed and presents a new modelling approach. Section 3 considers the speed scalable case, and proposes a modeling approach by developing previous one. Finally, Section 4 concludes the paper and gives out some future directions. 2. UNIFORM SPEED PROBLEM 2.1 problem statement This section deals with a non-preemption scheduling problem of n jobs on a single machine which may switch between three different states, namely ON (processing), OFF or Idle. Moreover, a fixed number of periods must elapse between states OFF and ON (i.e. Turn on and Turn off). As shown in Fig. 1, we assume that: • A production shift consists of T periods. • Each period t (0 ≤ t ≤ T ) is characterized by an electricity price (ct ) (T OU ). • n jobs (J1 , · · · , Jn ) must be processed by the machine in a predefined order (sequence = J1 − J2 − · · · − Jn−1 − Jn ), with their related processing time (pj ) and energy consumption (qj )(j = 1, · · · , n). • Each state of the machine is characterized by an energy consumption (eON = qj , eOF F , eIdle , eT on , eT of f ), where their values are the problem’s input (states). • When the machine goes to state i, it must stay in this state during a fixed number of periods (di )(Fig. 1). • The machine must be in OFF state during the initial (t = 0) and final periods (t = T ). This problem can be denoted by: 1, T OU |states, sequence, qj |T EC, where ‘T OU ’ represents the time of use policy of energy price, ‘states’ represents different states for the machine, ‘sequence’ represents predefined sequence of the jobs, ‘qj ’ represents jobs’ power demand, and ‘T EC’ represents the total energy cost. In the next section, a finite graph based on a dynamic programming approach is proposed to model this problem. It must be mentioned that, the complexity of the preemption version of this problem, when the jobs consume the same amount of energy (1, T OU |states, pmtn|T EC), is before analysed in Aghelinejad et al. (2017a).

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2.2 Dynamic programming modelling approach In this approach, the problem with T periods and n jobs is modelled by a graph with V nodes and E edges, which is composed of T +1 decision levels. Each level corresponds to a period of the time horizon (from 0 to T ). A set of nodes (Nt ) is allocated to each decision level (t) representing the possible last jobs in the given sequence (Jj ), which has been processed until period t. The initial-OFF and finalOFF states are respectively specified by the nodes I and F . Each node is identified by a (m, t) notation, where, m ∈ {I, J1 , · · · , Jn , F } and t ∈ {0, · · · , T }. Therefore, the graph is composed of n + 2 different kind of nodes. For example, in Fig. 2, N4 = {I, J1 }; N10 = {J1 , J2 , J3 , F }. In this problem, the number of periods is fixed, and the required time for processing all the jobs, transitions and setups are given. So, the decision makers are facing x extra periods (x = T − (P + β1 + β2 + 1), where P =  n p ), corresponding to x non-processing periods. That j=1 j is why each node of the graph can appear in x + 1 consecutive periods. Thus, sets τm are defined for any m ∈ {I, J1 , · · · , Jn , F }, which contains the possible periods (decision levels) that machine can be in state m during them. For example, τI = {0, · · · , x}; τJ1 = {β1 + p1 , · · · , β1 + p1 + x}; τF = {T − x, · · · , T }. So, the total number of nodes for a problem with T periods and n jobs is: |V | = (n + 2) ∗ (x + 1) ∼ = nT

(1)

The edges of the graph can be divided into three sets. In this approach, the value of each edge (V(m,t)−(m ,t ) ), represents the total energy consumption cost (positive value) of related transition between two connecting nodes ((m, t) and (m , t )). The first set of edges (E1 ) connects two nodes with the same m value in two consecutive decision levels (t and t + 1). These edges can indicate the initial and final OFF state with the edge value of 0 (if eOF F = 0) or the idle state with the edge value of: V(m,t)−(m,t+1) = ct+1 ∗ eIdle

; ∀m ∈ {J1 , · · · , Jn }; ∀t, t + 1 ∈ τm (2)

where, ct is the energy unit price in period t, and eIdle is the machine’s energy consumption in idle state. The cardinal of E1 is |E1 | = (n + 1) × x. The second set of edges (E2 ), connects a node m at period t with a node m + 1 at period t (t > t), which illustrates three transitions cases: • initial turning on and processing the first job (J1 ) with the edge value of: t+β1 t+β1 +p1   V(I,t)−(J1 ,t ) =

(ci ∗ eT on ) +

i=t+1

i=t+β1 +1

ci ∗ q 1

(3)

; ∀t ∈ τI

The last set of the edges (E3 ) corresponds to middle scheduling shutdowns between two processing jobs. Each middle shutdown consists of Toff, OFF, and Ton states. These edges connect nodes m in level t, and node m + 1 in level t , where, t ∈ {t+β1 +β2 +1+pm+1 , · · · , t+x+1}|t ∈ τm+1 with the edge value of: t+β2

V(m,t)−(m+1,t ) = t −pm+1



i=t −pm+1 −β1 −1

i=t+1

ci ∗ qm+1

• final turning off:

V(Jn ,t)−(F,t ) =

t −1



(ci ∗ eT of f ) + ct ∗ eOF F

(5)

i=t+1

; ∀t ∈ τJn ; t = t + β2 + 1

The cardinal of this set of edges is equal to |E2 | = (n + 1) ∗

(x + 1).

t 

i=t −pm+1 +1

(6) (ci ∗ qm+1 )

x−(β1 +β2 )

|E3 | = =

(x −



i=1 (β1 +

i ∗ (n − 1)

(7)

β2 )) ∗ (x − (β1 + β2 ) + 1) ∗ (n − 1) 2

Therefore, the total number of edges for a problem with T periods and n jobs is: |E| = |E1 | + |E2 | + |E3 | ∼ = T 2n

(8)

To illustrate the construction of this modelling approach, we consider an example with 3 jobs and 15 periods. The related graph for a problem with n = 3, pj = {2, 1, 2}, qj = {3, 5, 7}, T = 15, β1 = 2, β2 = 1, x = 6, and ct ∈ [0, 10] is presented in Fig. 2, which consists of 35 vertices and 64 edges. 2.3 Problem complexity analysis Based on the presented modelling approach, each path of the graph which starts from node I in level 0 and ends at node F in level T , represents a feasible solution of the problem. Since the objective is to find the minimum total energy consumption costs, the shortest path that starts at node (I, 0) and ends in node (F, T ) represents the optimal solution of the problem. For this purpose, Dijkstra’s algorithm, which is one of the most efficient algorithms to find the shortest path of a graph between the source node and every other node (when all the graph’s edge values are positive), is used to find the shortest path of the graph as the optimal solution. Let us consider that the cost C(m,t) associated to node (m, t), indicates the minimum cost to obtain production level m at period t. The recurrence relationship used to evaluate it for each node, is as follows: min

(m ,t )∈Am,t

; ∀m ∈ {J1 , J2 , · · · , Jn }; ∀t ∈ τm



(ci ∗ eT on ) +

The total number of the third set of edges is equal to:

C(I,0) = 0 C(m,t) = (4)

(ci ∗ eT of f )+

; ∀m ∈ {J1 , · · · , Jn−1 }; ∀t ∈ τm

t+pm+1

V(m,t)−(m+1,t+pm+1 ) =



i=t+1

• processing the job m + 1 (m = J1 , · · · , Jn−1 ): 

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{C(m ,t ) + V(m ,t )−(m,t) }

(9)

where Am,t is set of the precedent nodes connected to node (m, t) directly. For example, in Fig. 2 AJ2 ,9 = {(J1 , 4), (J1 , 8), (J2 , 8)}. Finally, C(F,T ) represents the value of the optimal solution for the considered problem. The application of Dijkstra’s algorithm for the considered example, gives the optimal solution with turning on during period 1 and 2, then processing all the jobs based on their order, within period 3 to 7, and finally turning off at period 9, with the total cost of 133 (Fig. 2). According to Fredman and Tarjan (1987), the worst case

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Fig. 2. Graph representation for an instance of uniform-speed problem 3. SPEED SCALABLE PROBLEM

Table 1. The obtained results for CPLEX and Dynamic programming approach (DP) (n,T) (3,15) (5,25) (7,35) (10,48) (13,52) (15,59) (17,65) (20,72) (25,92) (30,107)

ObjCP LEX 133 199 502 784 984 989 1114 1024 1429 1559

CP UCP LEX 0.78 1.63 3.56 10.24 27.14 36.78 37.92 133.64 442.94 858.60

ObjDP 133 199 502 784 984 989 1114 1024 1429 1559

3.1 problem statement

CP UDP 0.03 0.03 0.12 0.12 0.11 0.12 0.32 0.28 0.35 0.59

implementation of Dijkstra’s algorithm is based on a minpriority queue, that runs in O(|E|+|V | log |V |). Consequently, the complexity of the developed approach is equal to: O(T 2 n + T n log T n) = O(T 2 n + T n log T + T n log n) ∼ = O(T 2 n)

(10)

Since n < T (worst case for any feasible problem), we have log n < log T < T . So, we can conclude that the problem is polynomial of degree 3 or a cubic polynomial problem (O(T 3 )). 2.4 Numerical experiments Some numerical experiments are presented to show the effectiveness of the proposed approach. Also, the results of this approach are compared with the linear programming method which is implemented on ILOG CPLEX Software. In table 1, a part of the obtained results for the problem with n jobs and T periods is given. The results show that the proposed dynamic programming approach finds the same solution as CPLEX in all the cases with less computation time.

As the second contribution of this paper, a speed scalable version of the problem is studied. Using the three-field notation, this problem can be denoted by: 1, T OU |states, speed, sequence, qj |T EC, where ‘speed’ represents the speed scalability of the machine. In this case, for each job, we have to choose its execution speed among a given set values. Each speed corresponds to a given energy consumption and a processing time. Consequently, it could be interesting to process the jobs faster when the energy cost is low and process them slower when the energy cost is high. Thus, assuming that the machine has s different processing speeds, each job j has s different possibilities for its processing time and power consumption. Without loss of generality, we assume that each job has a set of processing time and power consumption (Qj = {(p1j , qj1 ), (p2j , qj2 ), · · · , (psj , qjs )}), with the following relations: p1j > p2j > · · · > psj qj1 < qj2 < · · · < qjs

; ∀j ∈ {1, · · · , n}

(11)

; ∀j ∈ {1, · · · , n}

(12)

Note that to satisfy the non-preemption in this case, the solution must be composed of an unique speed i for each job j to process it with the related process time pij and power consumption qji non-preemptively. In the following, an adaptation of the proposed dynamic programming approach for uniform speed problem is used to model the speed scalable case. 3.2 Dynamic programming modelling approach As the uniform-speed case of this problem, a graph with T +1 decision levels is depicted. Each decision level (t) has a set of nodes (Nt ) which represents the possible last jobs in the given sequence Jj , that has been processed with

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speed i until moment t (Jji ; ∀j ∈ {1, · · · , n}, i ∈ {1, · · · , s}) (Fig. 3). Therefore, by considering initial-OFF and finalOFF states, the graph is composed of (n ∗ s) + 2 different kind of nodes, where n represents number of jobs, and s represents the number of speeds. The number of non-processing periods for this case is obtained by the following formulation: n 

The total number of these edges is equal to:

In the whole graph, there are at most (x + 1) nodes with the same node’s number (m ∈ {I, J1i , J2i , · · · , Jni , F }). So, the total number of nodes for a problem with T periods, n jobs, and s speeds is:

A part of this graph for a problem with T periods, n jobs, and s possible processing speeds for each job is illustrated in Fig. 3.

x = T −

j=1

psj − (β1 + β2 + 1)

(13)

|V | ≤ ((n ∗ s) + 2) ∗ (x + 1) ∼ = nsT

(14)

The first set of edges (E1 ), indicates the initial and final OFF state with the edge value of 0 (if eOF F = 0) or the idle state with the edge value of: V(m,t)−(m,t+1) = ct+1 ∗ eIdle ; ∀m ∈ Jji ; ∀t, t + 1 ∈ τm ; ∀j ∈ {1, · · · , n}, i ∈ {1, · · · , s}

(15)

The cardinal of E1 is: |E1 | ≤ [((n − 1) ∗ s) + 2] ∗ x

(16)

The second set of edges (E2 ) illustrates three transitions cases: • initial turning on and processing the first job with speed i (J1i ) with the edge value of: V(I,t)−(J i ,t ) = 1

(ck ∗ eT on ) +

k=t+1



k=t+β1 +1

; ∀t ∈ τI ; ∀i ∈ {1, · · · , s}

ck ∗ q1i

t+pm+1



V(m,t)−(m+1,t+pm+1 ) =

ck ∗ qm+1

k=t+1

i {J1i , J2i , · · · , Jn−1 }; ∀t

• final turning off:

(18)

∈ τm ; ∀i ∈ {1, · · · , s}

t −1

V(J i ,t)−(F,t ) = n



(ck ∗ eT of f ) + ct ∗ eOF F

k=t+1

(19)



; ∀t ∈ τJ i ; ∀i ∈ {1, · · · , s}; t = t + β2 + 1 n

The cardinal of this set of edges is equal to: |E2 | ≤ [((n − 1) ∗ s) + 2] ∗ x

(20)

The last set of the edges (E3 ) corresponds to middle scheduling shutdowns between two processing jobs, and processing the second ones. These edges connect nodes m in level t, and node m + 1 in level t , where, t ∈ {t + β1 + β2 + 1 + pm+1 , · · · , t + x + 1}|t ∈ τm+1 with the edge value of: t+β2

V(m,t)−(m+1,t ) =



=



k=1 (β1 +

k ∗ (n − 1) ∗ s

(22)

β2 )) ∗ (x − (β1 + β2 ) + 1) ∗ (n − 1) ∗ s 2

|E| = |E1 | + |E2 | + |E3 | ∼ = T 2 ns

(23)

3.3 Problem complexity analysis As it is mentioned in previous section, the shortest path that starts at node (I, 0) and ends in node (F, T ) represents the optimal solution for the speed scalable problem too. In this case, the recurrence relationship for evaluating the cost of each node is the same as uniform-speed case (equation 9). According to Fredman and Tarjan (1987), the complexity of Dijkstra’s algorithm for the speed-scalable problem can be calculate by the following formulation: O(T 2 ns + T ns log T ns) ∼ = O(T 2 s(T + s))

(24)

Since n < T , we have log n < log T < T , also log s < s, and there is not any limitation for s. So, we can conclude that the speed-scalable case of this problem is pseudo polynomial (O(T 3 s + T 2 s2 )). 4. CONCLUSION

(ck ∗ eT of f )+

This paper introduced a complexity analysis of single machine scheduling problems with several states and time varying electricity price. In the considered system, the machine has three main states and two transition states. The objective function consists of minimizing the total energy consumption costs, for processing several jobs in a pre-defined order with different processing times and energy demands. For this purpose, we proposed new dynamic programming approaches to model the problems by using a finite graph. To analyze the complexity of these two problems (1, T OU |states, sequence, qj |T EC and 1, T OU |states, speed, sequence, qj |T EC), Dijkstra’s algorithm is used to obtain the shortest path between the first node and last node of each graph which represents the optimal sequence of the machine states during the horizon time for each problem. Computing the complexity of Dijkstra’s algorithm for these problems, proved that the uniform speed case of this problem is polynomial, and the speed scalable ones is pseudo polynomial. For future works, the proposed approaches have some possible extensions for the more complex problems. Also, the general version of the speed scalable problem (without fixed-sequence of the jobs, and when the number of possible speeds for each job depend on the job) can be considered as our next studies.

k=t+1 t −pm+1

(x −



Therefore, the total number of edges for a speed scalable problem with T periods, n jobs, and s speeds is:

(17)

• processing the jobs (except J1 ):

; ∀m ∈

|E3 | =

t+β1 +pi1

t+β1



x−(β1 +β2 )



(ck ∗ eT on ) +

k=t −pm+1 −β1 −1 i }; ∀t ; ∀m ∈ {J1i , · · · , Jn−1

t 

k=t −pm+1 +1

REFERENCES

(21) (ck ∗ qm+1 )

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