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A new approach to scheduling in manufacturing for power consumption and carbon footprint reduction
Journal of Manufacturing Systems 30 (2011) 234–240
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Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys
Technical paper
A new approach to scheduling in manufacturing for power consumption and carbon footprint reduction Kan Fang a , Nelson Uhan a , Fu Zhao b,∗ , John W. Sutherland b a b
School of Industrial Engineering, Purdue University, West Lafayette, IN, USA School of Mechanical Engineering and Division of Environmental and Ecological Engineering, Purdue University, West Lafayette, IN, USA
a r t i c l e
i n f o
Article history: Received 12 July 2011 Received in revised form 3 August 2011 Accepted 5 August 2011 Available online 1 September 2011 Keywords: Scheduling Peak load Carbon footprint Makespan
a b s t r a c t Manufacturing scheduling strategies have historically emphasized cycle time; in almost all cases, energy and environmental factors have not been considered in scheduling. This paper presents a new mathematical programming model of the flow shop scheduling problem that considers peak power load, energy consumption, and associated carbon footprint in addition to cycle time. The new model is demonstrated using a simple case study: a flow shop where two machines are employed to produce a variety of parts. In addition to the processing order of the jobs, the proposed scheduling problem considers the operation speed as an independent variable, which can be changed to affect the peak load and energy consumption. Even with a single objective, finding an optimal schedule is notoriously difficult, so directly applying commercial software to this multi-objective scheduling problem requires significant computation time. This paper calls for the development of more specialized algorithms for this new scheduling problem and examines computationally tractable approaches for finding near-optimal schedules. © 2011 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction The industrial sector currently accounts for about one-half of the world’s total energy consumption [1], and the consumption of energy by the sector has almost doubled over the last 60 years [2]. Furthermore, industrial energy consumption is expected to increase 40% from 175 quadrillion Btu in 2006 to 246 quadrillion Btu in 2030 [3]. In the U.S., the industrial sector was responsible for 34% of energy consumption in 2006 [4], and the associated energy costs were about $100 billion [5]. According to the Energy Information Administration [4] fossil fuels dominate the U.S. primary energy supply (with more than 85% of energy coming from such sources as coal, petroleum, and natural gas); thus, the industrial sector contributes 27% of the U.S. greenhouse gas emissions, which is second only to transportation among all end-use sectors [6]. Given mounting concerns related to climate change, manufacturing enterprises are facing growing pressure to reduce their carbon footprint. This pressure will become even more significant in the future due to the increasing cost of energy, resulting from both likely taxes and regulations related to carbon emissions as well as increasing energy demands from developing countries. These environmental and economic factors provide motivation for substantial initiatives directed at minimizing energy consumption and GHG emissions from manufacturing enterprises [7–9].
∗ Corresponding author. Tel.: +1 765 494 6637; fax: +1 765 494 0539. E-mail address: [email protected] (F. Zhao).
Previous research related to reducing manufacturing energy consumption has largely centered on developing more energy efficient machines and processes [10–14]. However, for metal working operations, the amount of energy needed for the active deformation and removal of material is often very small relative to the energy required for manufacturing equipment support functions [15]. This observation is supported by Drake et al. [16] who reported significant energy consumption during machine idling. As illustrated in Fig. 1, in a mass production environment, more than 85% of the energy is used for functions that are not directly related to the actual production of parts [17]. This implies that rather than focusing on updating individual machines or processes to be more energy efficient, attention should be directed at system-level changes that could realize significant energy benefits. Such efforts have not yet received adequate consideration. For example, current job shop scheduling approaches focus mainly on productivity and total cycle time, and nearly always neglect energy and environment considerations. Although criteria related to energy and the environment have rarely been major factors in manufacturing system decision making, this has not been the case in other industries. For example, system-oriented approaches have been successful in minimizing energy consumption in computer and embedded electronics systems [e.g. 18, 19]. It seems clear that comparable system-focused initiatives are needed for manufacturing enterprises. Such efforts are likely to achieve even greater reductions in energy consumption and associated greenhouse gas emissions relative to computer systems, given the scale of the related industrial activities. In addition,
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K. Fang et al. / Journal of Manufacturing Systems 30 (2011) 234–240
Fig. 1. Energy use breakdown for machining (Gutowski et al. [17]).
since electricity costs for production facilities are often partially based on the maximum power needed (referred to as “power consumption” in this paper), manufacturing schedules that reduce peak power will further reduce energy costs. Moreover, optimized job shop schedules should require fairly small capital investments relative to hardware changes in process/system equipment. With the forgoing discussion in mind, a mixed integer programming formulation of a flow shop scheduling problem is presented. This formulation simultaneously addresses productivity, carbon footprint, and peak (power) load. Peak load is included here since industrial facilities are usually billed based on both actual electricity consumption and peak load, with the proportion between the two depending on rate structures. In order to demonstrate the potential trade-offs among these criteria, a simple instance of the shop scheduling problem is considered. Some of the inherent complexities of this new scheduling challenge are also examined.
2. Literature review Research on reducing environmental impacts through production operation and scheduling has been quite limited. For the operation of a single machine, only a handful of papers were found following a rigorous literature search. Among these, the work done by Mouzon et al. [20] seems to be the most relevant: they studied the scheduling of a CNC machine in a machine shop making small aircraft parts in order to minimize total energy consumption. They reported that up to 80% of the total energy consumed during idling, start up, and shut down could be saved if the machine was turned off until needed, instead of being left on all the time. The paper also pointed out that if the machine operator can predict the next arrival of a job (e.g., the inter-arrival times between jobs), more energy efficient dispatching rules (for example, batching vs. non-batching) can be adopted. In a follow-up work, Mouzon and Yildirim [21] proposed a metaheuristic framework to compute schedules that minimize the total energy consumption and the total tardiness on a single machine. In a manufacturing facility, due to its hierarchical structure, clearly the utilization of a machine is largely determined by the shop floor schedule. Thus it is safe to argue that shop floor schedules can greatly affect energy consumption as well as other environmental impacts of the entire facility. Unfortunately, until now, the performance of a shop floor schedule has almost always been measured in term of cost, cycle time, and operational feasibility. For example, minimizing the setup cost or the transition time between jobs is one common objective in such scheduling problems. For just-in-time manufacturing systems, shop floor schedules are often
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designed to keep the usage rate of all parts constant, or to distribute the overall load evenly to each workstation. Similar to the case of single machine operation, a rigorous search for literature in environmentally conscious shop floor scheduling revealed that limited research had been conducted in this area. The most relevant work found was work reported by Subai and Baptiste [22] where energy consumption and waste generation were considered when designing hoist schedules for several surface treatment processes in an electroplating line. It should be noted, however, that research in energy-aware scheduling is growing. For example, an optimal scheduling procedure for vehicle sequencing has been proposed by Wang et al. [23] to reduce energy consumption in an automotive paint shop. Along with energy consumption reduction, it was found that the paint quality can be improved and repaints can be reduced if appropriate batch and sequence rules are used. Our literature review also suggests that no research has considered environment or energy related objectives in the scheduling of a flexible manufacturing system (FMS). An FMS consists of machines or workstations that can perform different operations with the appropriate tooling setups when needed. FMSs are preferred by many large OEMs since an FMS can better meet customers demand for greater product variety than dedicated equipment such as transfer lines. One interesting fact about scheduling an FMS is that in an FMS an operation may be performed at any one of the machines. This makes the routing of a job flexible. However, this flexible routing is subject to the capacity limits of material handling systems and buffers, which further increases the complexity when determining a schedule. As a result, much of the research on FMS scheduling has focused on specific problems that arise in industry [e.g. 24–27]. Various optimization-based approaches have been considered, including agent-based approaches [e.g. 28], metaheuristics [e.g. 29, 30], constraint programming [e.g. 31], and mathematical programming [e.g. 32]. Although most of the research so far has not considered environment and energy related objectives, these efforts provide a starting point for research on environmentally conscious schedule optimization, particularly from a modeling and algorithm development perspective. In the next two sections, an optimization model with productivity and environmental objectives will be presented first, followed by a two-machine flow shop case study. 3. A multi-objective mixed integer programming formulation for flow shop scheduling The scheduling problem studied here can be described mathematically as follows. There is a set of machines M = {1, 2, . . . , m} in a flow shop environment: the machines are ordered so that a job cannot start on machine i until it is completed on machine i − 1, for i = 2, . . ., m. In addition, there is a set of jobs J = {1, 2, . . . , n}, and each job needs to be processed on each machine in M. There is unlimited intermediate storage between the machines: that is, machine i is available for processing immediately after completing a job, regardless of whether machine i + 1 is ready to process this job. This setup can be modified to reflect other types of process/job interdependency (note that a flow shop is a special case of a job shop). There is also a finite and discrete set of speeds S = {1, . . . , d} that affect the power consumption; every job runs at a particular speed on each machine. The processing time of job j ∈ J on machine i ∈ M at speed s ∈ S is pijs ≥ 0. In addition, at speed s ∈ S the power consumption by job j ∈ J on machine i ∈ M is qijs ≥ 0. Ten sets of decision variables are defined below: • Cmax is the makespan of the schedule, i.e. the completion time of the last job in the schedule; • Pmax is the peak total power consumption;
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• • • • •
•
•
•
K. Fang et al. / Journal of Manufacturing Systems 30 (2011) 234–240
Gmax is the carbon footprint of the schedule; Cij is the completion time of the jth job processed on machine i; Sij is the start time of the jth job processed on machine i; xijks is equal to 1 if job j is the kth job processed on machine i with speed s, and 0 otherwise; uhkij is equal to 1 if the start time of the kth job processed on machine h is before the start time of the jth job processed on machine i, and 0 otherwise; in other words, Shk ≤ Sij ; vhkij is equal to 1 if the completion time of the kth job processed on machine h is after the start time of the jth job processed on machine i, and 0 otherwise; in other words, Chk > Sij ; yhkij is equal to 1 if the start time of the jth job processed on machine i occurs during the processing of the kth job on machine h, and 0 otherwise; in other words, Shk ≤ Sij < Chk ; zhlksij is equal to 1 if job l is the kth job processed on machine h with speed s, and starts while the jth job is running on machine i; in other words, if xhlks = 1 and yhkij = 1, and 0 otherwise.
The following is a valid multi-objective mixed integer programming model that finds a schedule that minimizes (i) the makespan, (ii) the peak total power consumption, and (iii) the carbon footprint. Below, K represents a very large constant and is the carbon footprint per unit of electricity consumed (kg CO2 equivalent/kWh): Cmax Pmax Gmax
minimize minimize minimize subject to Cmax ≥ Cmn C11 ≥
j∈J s∈S
Cik ≥ Ci−1,k +
(1) p1js x1j1s
(2)
pijs xijks
i ∈ {2, . . . , m}; k ∈ {1, 2, . . . , n}, (3)
pijs xijks
i ∈ M; k ∈ {2, . . . , n},
n
qijs pijs xijks ≤ Gmax
(14)
i∈M j∈J k=1 s∈S
xijks , uhkij , vhkij , yhkij , zhlksij ∈ {0, 1} i, h ∈ M; j, l, k ∈ {1, 2, . . . , n}; s ∈ S,
(15)
Constraint (1) ensures that the makespan of the schedule is greater than the completion time of the last job on the last machine. Constraints (2)–(4) ensure that a job cannot start on machine i until it is completed on machine i − 1. Constraints (5) ensure that a job cannot be processed preemptively. Constraints (6)–(9) ensure that the binary decision variables u, v, y and z take the intended values. Constraints (10)–(12) ensure that each job is processed with exactly one speed on each machine, and that the jobs are processed in the same order on each machine. Constraint (13) ensures that at any time, the total power consumption across machines is at most Pmax . Constraint (14) ensures that the carbon footprint of the schedule is at most Gmax . Finally, constraint (15) specifies that decision variables x, u, v, y and z are binary. Pareto efficient schedules, that is, feasible schedules not dominated by any single objective or performance measure, can be identified for the multi-objective mixed integer program described above. It should be noted that the associated traditional scheduling problem, i.e. finding a schedule with minimum makespan in a flow shop, is already computationally difficult: when m = 3, the problem is NP-hard [33]. Therefore, it is expected that the formulation introduced above represents an even more computationally challenging scheduling problem, with additional objectives, and constraints related to power/energy consumption and carbon footprint. To gain a better understanding of the problem, a preliminary case study on a hypothetical flow shop is given below.
j∈J s∈S
Cik ≥ Ci,k−1 +
(4)
j∈J s∈S
Cij = Sij +
i ∈ M; j ∈ {1, 2, . . . , n},
(5)
Sij − Shk ≤ Kuhkij − 1 i, h ∈ M; j, k ∈ {1, 2, . . . , n},
(6)
pirs xirjs
r∈J s∈S
Shk − Sij +
phls xhlks ≤ Kvhkij
i, h ∈ M; j, k ∈ {1, 2, . . . , n},
l∈J s∈S
(7)
uhkij + vhkij = 1 + yhkij xhlks + yhkij ≤ 1 + zhlksij
i, h ∈ M; j, k ∈ {1, 2, . . . , n},
4. Case study: a two machine flow shop Consider the following simple two-machine flow shop that makes cast iron plates with slots. Cast iron plates that come into the shop undergo two machining processes, with each process performed on a separate machine: face milling to prepare the surface, followed by profile milling to cut the slots. As shown in Fig. 2, the plates can have different lengths, different total depths of milling on the surface, and different numbers of slots and slot depth. Fig. 3 shows the Gantt chart of a simple case with only 3 different parts (jobs) that can be machined at 5 possible cutting speeds. It is clear that there is a significant trade-off between makespan and peak (power) load. To achieve the shortest possible makespan, jobs
(8)
i, h ∈ M; j, k, l ∈ {1, 2, . . . , n}; s ∈ S, (9)
n
xijks = 1 i ∈ M; j ∈ J,
(10)
xijks = 1 i ∈ M; k ∈ {1, 2, . . . , n},
(11)
k=1 s∈S
j∈J s∈S
s∈S
xijks =
xhjks
i, h ∈ M; j, k ∈ {1, 2, . . . , n},
(12)
s∈S n
h∈M,h = / i l∈J k=1 s∈S
j ∈ {1, 2, . . . , n},
qhls zhlksij +
qirs xirjs ≤ Pmax
i ∈ M;
r∈J s∈S
(13)
Fig. 2. A cast iron plate with slots.
K. Fang et al. / Journal of Manufacturing Systems 30 (2011) 234–240
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Fig. 3. Power profile of 2-machine, 3-job case.
should be processed at the highest possible cutting speed and the two machines should be run simultaneously (Fig. 3a). This leads to a high peak load (16.8 kW). On the other hand, if the objective is to minimize peak load, jobs should be processed using the lowest cutting speed and the two machines should operate without overlap (Fig. 3b). Even for this simple instance, identifying the Pareto efficient frontier is not trivial: there are a total of 93,750 combinations of job permutations and cutting speeds (as compared to only 6 combinations if the cutting speed is fixed), along with an infinite number of job start times to be considered. Apparently, even for a two-machine flow shop, a principled mathematical optimization-based approach has to be adopted when scheduling realistic numbers of jobs. Below we will explore a case of 36 jobs in this two machine flow shop. The 36 different parts correspond to the combinations of three different plate lengths, two different depths of milling on surface, three different numbers of slots, and two different depths of slots. Cutting speeds for both face milling and profile milling are chosen to be within the range recommended by the Machinery’s Handbook [34]. Tables 1 and 2 show the geometry of the parts and the cutting conditions respectively.
Fig. 4. Power profile for a representative machining operation (e.g. face milling).
For a job processed on one of the machines, the processing time, power consumption, and energy consumption can be calculated based on following equations and Fig. 4: Pcutting = w · d · V · Up , tcutting =
(16)
L + Dcutter L + Dcutter , = nt · d · ns nt · d · (V/( · Dcutter ))
(for face milling) (17)
Table 1 Geometry of parts. Width (cm) Length (cm) Thickness (cm) Depth of milling on surface (mm) of slots Slot depth (cm) Slot width (cm)
20 35 or 45 or 55 10 2 or 4 4 or 6 or 8 0.5 or 1.0 3.0
Table 2 Cutting conditions. Face milling Cutter diameter (cm) of teeth Feed per tooth (mm) Cutting speed (m/min) Profile milling of flutes Feed per flute (mm) Cutting speed (m/min)
30 15 0.2 18/24/30/36/45 4 0.1 18/24/30/36/45
tcutting = nslot ·
w + Dcutter , nf · d · (V/( · Dcutter ))
(for profile milling)
(18)
Ppeak = Pbasic + Pidle + Pcutting ,
(19)
Etotal = Pbasic · tbasic + Pidle · tidle + Pcutting · tcutting ,
(20)
where • • • • • • • • • • •
E is the electricity consumption (in MJ or kWh), P is the power consumption (in kW), w is the part width or slot width, d is the feed per tooth (or flute), V is the cutting speed, Up is the specific cutting energy, L is the length of the workpiece, Dcutter is the cutter diameter, nt (nf ) is the number of teeth (flutes) per cutter, ns is the spindle speed, nslot is the number of slots to be cut.
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K. Fang et al. / Journal of Manufacturing Systems 30 (2011) 234–240
Fig. 5. Pareto frontier between peak load and makespan with carbon footprint (6 job case).
Here the subscript “basic” corresponds to activities such as workpiece loading/unloading, positioning, and fixturing. At this stage, the lighting, the NC controller, the chiller system, the oil pump, and the way lube system are all turned on, and thus their power consumption is also included in basic load. The subscript “idle” corresponds to the processes of tool approaching and retracting from the workpiece, tool movements between features, adjusting machine settings, and changing the tools. At this stage the main spindle is turned on along with the tool changer and cutting fluid pump. Finally the subscript “cutting” corresponds to the actual material removal process. It should be noted that the energy consumption corresponding to actual material removal is a constant for different cutting speeds if it is assumed the specific cutting energy is independent of feed rate and cutting speed. Attempts were first made to use the commercial optimization software package Gurobi Optimizer 4.0, in conjunction with goal programming techniques to identify points on the Pareto efficient frontier for makespan (Cmax ) and peak load (Pmax ) for the multiobjective mixed integer program above. Unfortunately, for this two-machine 36-job problem, a provably optimal schedule for just one point on the Pareto efficient frontier could not be found within 24 h on a computer with two 2.5 GHz Quad-Core AMD 2380 processors and 32 GB of RAM. Also, reducing the available speeds on each machine to two (i.e. highest and lowest) from five did not help. Since it is known that job scheduling with only a single objective for a flow shop tends to be difficult to solve, this is somewhat expected. However, for a smaller, similar instance in the two-machine flow shop described above with 6 jobs and 5 speeds, the Pareto frontier was successfully computed. The Pareto frontier was computed by specifying 21 different upper bounds on the peak load and then searching for the minimum makespan. Each schedule on the Pareto efficient frontier was computed in anywhere between 3 and 958 s. Fig. 5 shows the Pareto efficient frontier for this case. It can be seen that there exists a significant trade-off between makespan and peak load. If makespan is the dominating consideration, a completion time as short as approximately 34 min can be achieved at the cost of high peak load, i.e., 16 kW. On the other hand, if a longer completion time can be tolerated, the peak load can be reduced to 5 kW with a makespan approximately four times larger. Fig. 5 also shows the carbon footprints corresponding to the 21 optimal schedules identified on the Pareto efficient frontier. The carbon footprint decreases as the upper bound of peak load increases initially, just as the makespan does, but levels off when the upper bound on peak load reaches a certain level. It turns out that various special cases and variants of the flow shop problem described above are easier to solve. In certain applications, it may be appropriate to assume that there is no intermediate
Fig. 6. Pareto frontier between peak load and makespan with carbon footprint (36 job, blocking case).
storage between machines (e.g. some assembly lines). In the context of this case study, this means that Machine 1 (face milling in this case) cannot start processing the next part if the part it has just completed cannot start on Machine 2 (profile milling). This is sometimes known as “blocking” in the scheduling literature. If there is no intermediate storage between machines, then the flow shop scheduling problem becomes easier, and a Pareto efficient frontier can be computed for the 36 job instance described above. In particular, it can be shown that when there is no intermediate storage between machines, the two-machine flow shop scheduling problem is equivalent to an asymmetric traveling salesman problem (TSP). Using this transformation in conjunction with the Concorde TSP solver, the Pareto efficient frontier for makespan and peak load was computed for the 36 job instance. Similar to the 6 job instance, the Pareto efficient frontier was computed by searching for the minimum makespan, given 21 different specified upper bounds on the peak load. Each schedule corresponding to the minimum makespan was computed in 4–5 s. Fig. 6 shows the Pareto frontier for this case, along with the associated carbon footprints of the schedules on the Pareto frontier. Again, like the 6 job instance, there is a significant trade-off between makespan and peak load, and both carbon footprint and makespan decrease as the upper bound of power consumption increases. However, carbon footprint levels off when the upper bound on peak load reaches a certain level. In a companion paper [35], we show that if the cutting speed is assumed to be continuous and can take any value within a recommended range, then we can give efficient polynomial-time algorithms and theoretical results on the structure of optimal solutions for different variants of this flow shop scheduling problem. These results are as follows: Case 1. No intermediate storage between the two machines: • Suppose the amount of work is consistent across machines: that is, for any two jobs j, k ∈ J, p1j ≤ p1k implies p2j ≤ p2k . Then an optimal schedule can be found in polynomial time by transforming the scheduling problem into an equivalent asymmetric traveling salesman problem that satisfies the so-called Demidenko conditions [36]. • Furthermore, if the work for each job is the same on each machine – that is, for any job j ∈ J, p1j = p2j – then, an optimal schedule can be computed by a simple priority-list algorithm. In particular, if p11 ≤ p12 ≤ · · · ≤ p1n then scheduling the jobs according to the permutation (1, 3, 5, n, 6, 4, 2) yields an optimal schedule.
K. Fang et al. / Journal of Manufacturing Systems 30 (2011) 234–240
Case 2. Unlimited machines:
intermediate
storage
between
the
two
• If the total power consumption is Pmax at any time in the schedule, an optimal schedule (i.e., job completion times and job/machine speeds) can be found in polynomial time given a fixed permutation of the jobs. In addition, the sequence of jobs can be decomposed into sub-sequences of jobs that are processed at the same speed. This kind of result on the mathematical structure of the scheduling problem is useful in the design of neighborhood search algorithms to find near-optimal solutions. 5. Summary, conclusions, and future work This paper presents a general multi-objective mixed integer linear programming formulation for optimizing the operating schedule of a flow shop that considers both productivity (i.e., makespan) and energy (i.e., peak load and carbon footprint) related criteria. In this new formulation, operation speed is allowed to vary in order to affect the peak load and energy consumption. This is different from traditional schedule optimization approaches where highest possible operation speed is preferable in order to maximize productivity. Although the formulation given is for a flow shop, it is possible to construct formulations suitable for shop floors with other types of process/job interdependency (e.g., a job shop) by modifying or relaxing its constraints. Scheduling cast iron plates with slots in a two-machine flow shop is used as a simple case study to demonstrate the complexity of the problem. For this simple case, even when only a moderate number of jobs are considered (36 parts) it is almost time prohibitive to compute even one point on the Pareto frontier using commercial optimization software tools. Without any further intervention, it is only feasible to compute the Pareto frontiers for a small size problem (6 jobs). It should be pointed out that although interesting structural and algorithmic properties can be derived for the two-machine flow shop problem, they will not likely hold for a shop of industrial scale where perhaps hundreds of machines are involved. Moreover, for such industrial sized problems the computation time will be prohibitive. Therefore, for industrial scale shop floors, in order to find practical schedules that balance economic and environmental performance, the following approaches are suggested: (1) construct a simplified model for the shop floor while maintaining fidelity with regard to scheduling; and (2) design algorithms specialized for this new model, either through mathematical programming, or alternate approaches such as neighborhood search. With these approaches acceptable schedules may be identified with reasonable computational effort. It should also be noted that it is possible to expand the formulation to consider other environmental and occupational health impacts, such as water consumption, air emissions, and noise, although only energy related criteria (i.e., peak load, energy consumption, and carbon footprints) are introduced beyond productivity as new objectives for the shop floor schedule optimization problem in this paper. Doing so will likely make the problem even more challenging, especially if nonlinear correlations among decision variables and constraints are introduced. Numerous opportunities exist in this new area, which are expected to have broad impacts on U.S. manufacturing enterprises and even the entire industrial sector, given the fact that environmentally conscious shop floor scheduling has the potential to significantly reduce the energy and environmental footprints of manufacturing facilities without incurring large capital investment. This paper represents one of the first ever attempts on environmentally conscious shop scheduling and serves as a starting point to develop a research roadmap and agenda for this emerging area.
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Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys
Technical paper
A new approach to scheduling in manufacturing for power consumption and carbon footprint reduction Kan Fang a , Nelson Uhan a , Fu Zhao b,∗ , John W. Sutherland b a b
School of Industrial Engineering, Purdue University, West Lafayette, IN, USA School of Mechanical Engineering and Division of Environmental and Ecological Engineering, Purdue University, West Lafayette, IN, USA
a r t i c l e
i n f o
Article history: Received 12 July 2011 Received in revised form 3 August 2011 Accepted 5 August 2011 Available online 1 September 2011 Keywords: Scheduling Peak load Carbon footprint Makespan
a b s t r a c t Manufacturing scheduling strategies have historically emphasized cycle time; in almost all cases, energy and environmental factors have not been considered in scheduling. This paper presents a new mathematical programming model of the flow shop scheduling problem that considers peak power load, energy consumption, and associated carbon footprint in addition to cycle time. The new model is demonstrated using a simple case study: a flow shop where two machines are employed to produce a variety of parts. In addition to the processing order of the jobs, the proposed scheduling problem considers the operation speed as an independent variable, which can be changed to affect the peak load and energy consumption. Even with a single objective, finding an optimal schedule is notoriously difficult, so directly applying commercial software to this multi-objective scheduling problem requires significant computation time. This paper calls for the development of more specialized algorithms for this new scheduling problem and examines computationally tractable approaches for finding near-optimal schedules. © 2011 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction The industrial sector currently accounts for about one-half of the world’s total energy consumption [1], and the consumption of energy by the sector has almost doubled over the last 60 years [2]. Furthermore, industrial energy consumption is expected to increase 40% from 175 quadrillion Btu in 2006 to 246 quadrillion Btu in 2030 [3]. In the U.S., the industrial sector was responsible for 34% of energy consumption in 2006 [4], and the associated energy costs were about $100 billion [5]. According to the Energy Information Administration [4] fossil fuels dominate the U.S. primary energy supply (with more than 85% of energy coming from such sources as coal, petroleum, and natural gas); thus, the industrial sector contributes 27% of the U.S. greenhouse gas emissions, which is second only to transportation among all end-use sectors [6]. Given mounting concerns related to climate change, manufacturing enterprises are facing growing pressure to reduce their carbon footprint. This pressure will become even more significant in the future due to the increasing cost of energy, resulting from both likely taxes and regulations related to carbon emissions as well as increasing energy demands from developing countries. These environmental and economic factors provide motivation for substantial initiatives directed at minimizing energy consumption and GHG emissions from manufacturing enterprises [7–9].
∗ Corresponding author. Tel.: +1 765 494 6637; fax: +1 765 494 0539. E-mail address: [email protected] (F. Zhao).
Previous research related to reducing manufacturing energy consumption has largely centered on developing more energy efficient machines and processes [10–14]. However, for metal working operations, the amount of energy needed for the active deformation and removal of material is often very small relative to the energy required for manufacturing equipment support functions [15]. This observation is supported by Drake et al. [16] who reported significant energy consumption during machine idling. As illustrated in Fig. 1, in a mass production environment, more than 85% of the energy is used for functions that are not directly related to the actual production of parts [17]. This implies that rather than focusing on updating individual machines or processes to be more energy efficient, attention should be directed at system-level changes that could realize significant energy benefits. Such efforts have not yet received adequate consideration. For example, current job shop scheduling approaches focus mainly on productivity and total cycle time, and nearly always neglect energy and environment considerations. Although criteria related to energy and the environment have rarely been major factors in manufacturing system decision making, this has not been the case in other industries. For example, system-oriented approaches have been successful in minimizing energy consumption in computer and embedded electronics systems [e.g. 18, 19]. It seems clear that comparable system-focused initiatives are needed for manufacturing enterprises. Such efforts are likely to achieve even greater reductions in energy consumption and associated greenhouse gas emissions relative to computer systems, given the scale of the related industrial activities. In addition,
0278-6125/$ – see front matter © 2011 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jmsy.2011.08.004
K. Fang et al. / Journal of Manufacturing Systems 30 (2011) 234–240
Fig. 1. Energy use breakdown for machining (Gutowski et al. [17]).
since electricity costs for production facilities are often partially based on the maximum power needed (referred to as “power consumption” in this paper), manufacturing schedules that reduce peak power will further reduce energy costs. Moreover, optimized job shop schedules should require fairly small capital investments relative to hardware changes in process/system equipment. With the forgoing discussion in mind, a mixed integer programming formulation of a flow shop scheduling problem is presented. This formulation simultaneously addresses productivity, carbon footprint, and peak (power) load. Peak load is included here since industrial facilities are usually billed based on both actual electricity consumption and peak load, with the proportion between the two depending on rate structures. In order to demonstrate the potential trade-offs among these criteria, a simple instance of the shop scheduling problem is considered. Some of the inherent complexities of this new scheduling challenge are also examined.
2. Literature review Research on reducing environmental impacts through production operation and scheduling has been quite limited. For the operation of a single machine, only a handful of papers were found following a rigorous literature search. Among these, the work done by Mouzon et al. [20] seems to be the most relevant: they studied the scheduling of a CNC machine in a machine shop making small aircraft parts in order to minimize total energy consumption. They reported that up to 80% of the total energy consumed during idling, start up, and shut down could be saved if the machine was turned off until needed, instead of being left on all the time. The paper also pointed out that if the machine operator can predict the next arrival of a job (e.g., the inter-arrival times between jobs), more energy efficient dispatching rules (for example, batching vs. non-batching) can be adopted. In a follow-up work, Mouzon and Yildirim [21] proposed a metaheuristic framework to compute schedules that minimize the total energy consumption and the total tardiness on a single machine. In a manufacturing facility, due to its hierarchical structure, clearly the utilization of a machine is largely determined by the shop floor schedule. Thus it is safe to argue that shop floor schedules can greatly affect energy consumption as well as other environmental impacts of the entire facility. Unfortunately, until now, the performance of a shop floor schedule has almost always been measured in term of cost, cycle time, and operational feasibility. For example, minimizing the setup cost or the transition time between jobs is one common objective in such scheduling problems. For just-in-time manufacturing systems, shop floor schedules are often
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designed to keep the usage rate of all parts constant, or to distribute the overall load evenly to each workstation. Similar to the case of single machine operation, a rigorous search for literature in environmentally conscious shop floor scheduling revealed that limited research had been conducted in this area. The most relevant work found was work reported by Subai and Baptiste [22] where energy consumption and waste generation were considered when designing hoist schedules for several surface treatment processes in an electroplating line. It should be noted, however, that research in energy-aware scheduling is growing. For example, an optimal scheduling procedure for vehicle sequencing has been proposed by Wang et al. [23] to reduce energy consumption in an automotive paint shop. Along with energy consumption reduction, it was found that the paint quality can be improved and repaints can be reduced if appropriate batch and sequence rules are used. Our literature review also suggests that no research has considered environment or energy related objectives in the scheduling of a flexible manufacturing system (FMS). An FMS consists of machines or workstations that can perform different operations with the appropriate tooling setups when needed. FMSs are preferred by many large OEMs since an FMS can better meet customers demand for greater product variety than dedicated equipment such as transfer lines. One interesting fact about scheduling an FMS is that in an FMS an operation may be performed at any one of the machines. This makes the routing of a job flexible. However, this flexible routing is subject to the capacity limits of material handling systems and buffers, which further increases the complexity when determining a schedule. As a result, much of the research on FMS scheduling has focused on specific problems that arise in industry [e.g. 24–27]. Various optimization-based approaches have been considered, including agent-based approaches [e.g. 28], metaheuristics [e.g. 29, 30], constraint programming [e.g. 31], and mathematical programming [e.g. 32]. Although most of the research so far has not considered environment and energy related objectives, these efforts provide a starting point for research on environmentally conscious schedule optimization, particularly from a modeling and algorithm development perspective. In the next two sections, an optimization model with productivity and environmental objectives will be presented first, followed by a two-machine flow shop case study. 3. A multi-objective mixed integer programming formulation for flow shop scheduling The scheduling problem studied here can be described mathematically as follows. There is a set of machines M = {1, 2, . . . , m} in a flow shop environment: the machines are ordered so that a job cannot start on machine i until it is completed on machine i − 1, for i = 2, . . ., m. In addition, there is a set of jobs J = {1, 2, . . . , n}, and each job needs to be processed on each machine in M. There is unlimited intermediate storage between the machines: that is, machine i is available for processing immediately after completing a job, regardless of whether machine i + 1 is ready to process this job. This setup can be modified to reflect other types of process/job interdependency (note that a flow shop is a special case of a job shop). There is also a finite and discrete set of speeds S = {1, . . . , d} that affect the power consumption; every job runs at a particular speed on each machine. The processing time of job j ∈ J on machine i ∈ M at speed s ∈ S is pijs ≥ 0. In addition, at speed s ∈ S the power consumption by job j ∈ J on machine i ∈ M is qijs ≥ 0. Ten sets of decision variables are defined below: • Cmax is the makespan of the schedule, i.e. the completion time of the last job in the schedule; • Pmax is the peak total power consumption;
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• • • • •
•
•
•
K. Fang et al. / Journal of Manufacturing Systems 30 (2011) 234–240
Gmax is the carbon footprint of the schedule; Cij is the completion time of the jth job processed on machine i; Sij is the start time of the jth job processed on machine i; xijks is equal to 1 if job j is the kth job processed on machine i with speed s, and 0 otherwise; uhkij is equal to 1 if the start time of the kth job processed on machine h is before the start time of the jth job processed on machine i, and 0 otherwise; in other words, Shk ≤ Sij ; vhkij is equal to 1 if the completion time of the kth job processed on machine h is after the start time of the jth job processed on machine i, and 0 otherwise; in other words, Chk > Sij ; yhkij is equal to 1 if the start time of the jth job processed on machine i occurs during the processing of the kth job on machine h, and 0 otherwise; in other words, Shk ≤ Sij < Chk ; zhlksij is equal to 1 if job l is the kth job processed on machine h with speed s, and starts while the jth job is running on machine i; in other words, if xhlks = 1 and yhkij = 1, and 0 otherwise.
The following is a valid multi-objective mixed integer programming model that finds a schedule that minimizes (i) the makespan, (ii) the peak total power consumption, and (iii) the carbon footprint. Below, K represents a very large constant and is the carbon footprint per unit of electricity consumed (kg CO2 equivalent/kWh): Cmax Pmax Gmax
minimize minimize minimize subject to Cmax ≥ Cmn C11 ≥
j∈J s∈S
Cik ≥ Ci−1,k +
(1) p1js x1j1s
(2)
pijs xijks
i ∈ {2, . . . , m}; k ∈ {1, 2, . . . , n}, (3)
pijs xijks
i ∈ M; k ∈ {2, . . . , n},
n
qijs pijs xijks ≤ Gmax
(14)
i∈M j∈J k=1 s∈S
xijks , uhkij , vhkij , yhkij , zhlksij ∈ {0, 1} i, h ∈ M; j, l, k ∈ {1, 2, . . . , n}; s ∈ S,
(15)
Constraint (1) ensures that the makespan of the schedule is greater than the completion time of the last job on the last machine. Constraints (2)–(4) ensure that a job cannot start on machine i until it is completed on machine i − 1. Constraints (5) ensure that a job cannot be processed preemptively. Constraints (6)–(9) ensure that the binary decision variables u, v, y and z take the intended values. Constraints (10)–(12) ensure that each job is processed with exactly one speed on each machine, and that the jobs are processed in the same order on each machine. Constraint (13) ensures that at any time, the total power consumption across machines is at most Pmax . Constraint (14) ensures that the carbon footprint of the schedule is at most Gmax . Finally, constraint (15) specifies that decision variables x, u, v, y and z are binary. Pareto efficient schedules, that is, feasible schedules not dominated by any single objective or performance measure, can be identified for the multi-objective mixed integer program described above. It should be noted that the associated traditional scheduling problem, i.e. finding a schedule with minimum makespan in a flow shop, is already computationally difficult: when m = 3, the problem is NP-hard [33]. Therefore, it is expected that the formulation introduced above represents an even more computationally challenging scheduling problem, with additional objectives, and constraints related to power/energy consumption and carbon footprint. To gain a better understanding of the problem, a preliminary case study on a hypothetical flow shop is given below.
j∈J s∈S
Cik ≥ Ci,k−1 +
(4)
j∈J s∈S
Cij = Sij +
i ∈ M; j ∈ {1, 2, . . . , n},
(5)
Sij − Shk ≤ Kuhkij − 1 i, h ∈ M; j, k ∈ {1, 2, . . . , n},
(6)
pirs xirjs
r∈J s∈S
Shk − Sij +
phls xhlks ≤ Kvhkij
i, h ∈ M; j, k ∈ {1, 2, . . . , n},
l∈J s∈S
(7)
uhkij + vhkij = 1 + yhkij xhlks + yhkij ≤ 1 + zhlksij
i, h ∈ M; j, k ∈ {1, 2, . . . , n},
4. Case study: a two machine flow shop Consider the following simple two-machine flow shop that makes cast iron plates with slots. Cast iron plates that come into the shop undergo two machining processes, with each process performed on a separate machine: face milling to prepare the surface, followed by profile milling to cut the slots. As shown in Fig. 2, the plates can have different lengths, different total depths of milling on the surface, and different numbers of slots and slot depth. Fig. 3 shows the Gantt chart of a simple case with only 3 different parts (jobs) that can be machined at 5 possible cutting speeds. It is clear that there is a significant trade-off between makespan and peak (power) load. To achieve the shortest possible makespan, jobs
(8)
i, h ∈ M; j, k, l ∈ {1, 2, . . . , n}; s ∈ S, (9)
n
xijks = 1 i ∈ M; j ∈ J,
(10)
xijks = 1 i ∈ M; k ∈ {1, 2, . . . , n},
(11)
k=1 s∈S
j∈J s∈S
s∈S
xijks =
xhjks
i, h ∈ M; j, k ∈ {1, 2, . . . , n},
(12)
s∈S n
h∈M,h = / i l∈J k=1 s∈S
j ∈ {1, 2, . . . , n},
qhls zhlksij +
qirs xirjs ≤ Pmax
i ∈ M;
r∈J s∈S
(13)
Fig. 2. A cast iron plate with slots.
K. Fang et al. / Journal of Manufacturing Systems 30 (2011) 234–240
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Fig. 3. Power profile of 2-machine, 3-job case.
should be processed at the highest possible cutting speed and the two machines should be run simultaneously (Fig. 3a). This leads to a high peak load (16.8 kW). On the other hand, if the objective is to minimize peak load, jobs should be processed using the lowest cutting speed and the two machines should operate without overlap (Fig. 3b). Even for this simple instance, identifying the Pareto efficient frontier is not trivial: there are a total of 93,750 combinations of job permutations and cutting speeds (as compared to only 6 combinations if the cutting speed is fixed), along with an infinite number of job start times to be considered. Apparently, even for a two-machine flow shop, a principled mathematical optimization-based approach has to be adopted when scheduling realistic numbers of jobs. Below we will explore a case of 36 jobs in this two machine flow shop. The 36 different parts correspond to the combinations of three different plate lengths, two different depths of milling on surface, three different numbers of slots, and two different depths of slots. Cutting speeds for both face milling and profile milling are chosen to be within the range recommended by the Machinery’s Handbook [34]. Tables 1 and 2 show the geometry of the parts and the cutting conditions respectively.
Fig. 4. Power profile for a representative machining operation (e.g. face milling).
For a job processed on one of the machines, the processing time, power consumption, and energy consumption can be calculated based on following equations and Fig. 4: Pcutting = w · d · V · Up , tcutting =
(16)
L + Dcutter L + Dcutter , = nt · d · ns nt · d · (V/( · Dcutter ))
(for face milling) (17)
Table 1 Geometry of parts. Width (cm) Length (cm) Thickness (cm) Depth of milling on surface (mm) of slots Slot depth (cm) Slot width (cm)
20 35 or 45 or 55 10 2 or 4 4 or 6 or 8 0.5 or 1.0 3.0
Table 2 Cutting conditions. Face milling Cutter diameter (cm) of teeth Feed per tooth (mm) Cutting speed (m/min) Profile milling of flutes Feed per flute (mm) Cutting speed (m/min)
30 15 0.2 18/24/30/36/45 4 0.1 18/24/30/36/45
tcutting = nslot ·
w + Dcutter , nf · d · (V/( · Dcutter ))
(for profile milling)
(18)
Ppeak = Pbasic + Pidle + Pcutting ,
(19)
Etotal = Pbasic · tbasic + Pidle · tidle + Pcutting · tcutting ,
(20)
where • • • • • • • • • • •
E is the electricity consumption (in MJ or kWh), P is the power consumption (in kW), w is the part width or slot width, d is the feed per tooth (or flute), V is the cutting speed, Up is the specific cutting energy, L is the length of the workpiece, Dcutter is the cutter diameter, nt (nf ) is the number of teeth (flutes) per cutter, ns is the spindle speed, nslot is the number of slots to be cut.
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Fig. 5. Pareto frontier between peak load and makespan with carbon footprint (6 job case).
Here the subscript “basic” corresponds to activities such as workpiece loading/unloading, positioning, and fixturing. At this stage, the lighting, the NC controller, the chiller system, the oil pump, and the way lube system are all turned on, and thus their power consumption is also included in basic load. The subscript “idle” corresponds to the processes of tool approaching and retracting from the workpiece, tool movements between features, adjusting machine settings, and changing the tools. At this stage the main spindle is turned on along with the tool changer and cutting fluid pump. Finally the subscript “cutting” corresponds to the actual material removal process. It should be noted that the energy consumption corresponding to actual material removal is a constant for different cutting speeds if it is assumed the specific cutting energy is independent of feed rate and cutting speed. Attempts were first made to use the commercial optimization software package Gurobi Optimizer 4.0, in conjunction with goal programming techniques to identify points on the Pareto efficient frontier for makespan (Cmax ) and peak load (Pmax ) for the multiobjective mixed integer program above. Unfortunately, for this two-machine 36-job problem, a provably optimal schedule for just one point on the Pareto efficient frontier could not be found within 24 h on a computer with two 2.5 GHz Quad-Core AMD 2380 processors and 32 GB of RAM. Also, reducing the available speeds on each machine to two (i.e. highest and lowest) from five did not help. Since it is known that job scheduling with only a single objective for a flow shop tends to be difficult to solve, this is somewhat expected. However, for a smaller, similar instance in the two-machine flow shop described above with 6 jobs and 5 speeds, the Pareto frontier was successfully computed. The Pareto frontier was computed by specifying 21 different upper bounds on the peak load and then searching for the minimum makespan. Each schedule on the Pareto efficient frontier was computed in anywhere between 3 and 958 s. Fig. 5 shows the Pareto efficient frontier for this case. It can be seen that there exists a significant trade-off between makespan and peak load. If makespan is the dominating consideration, a completion time as short as approximately 34 min can be achieved at the cost of high peak load, i.e., 16 kW. On the other hand, if a longer completion time can be tolerated, the peak load can be reduced to 5 kW with a makespan approximately four times larger. Fig. 5 also shows the carbon footprints corresponding to the 21 optimal schedules identified on the Pareto efficient frontier. The carbon footprint decreases as the upper bound of peak load increases initially, just as the makespan does, but levels off when the upper bound on peak load reaches a certain level. It turns out that various special cases and variants of the flow shop problem described above are easier to solve. In certain applications, it may be appropriate to assume that there is no intermediate
Fig. 6. Pareto frontier between peak load and makespan with carbon footprint (36 job, blocking case).
storage between machines (e.g. some assembly lines). In the context of this case study, this means that Machine 1 (face milling in this case) cannot start processing the next part if the part it has just completed cannot start on Machine 2 (profile milling). This is sometimes known as “blocking” in the scheduling literature. If there is no intermediate storage between machines, then the flow shop scheduling problem becomes easier, and a Pareto efficient frontier can be computed for the 36 job instance described above. In particular, it can be shown that when there is no intermediate storage between machines, the two-machine flow shop scheduling problem is equivalent to an asymmetric traveling salesman problem (TSP). Using this transformation in conjunction with the Concorde TSP solver, the Pareto efficient frontier for makespan and peak load was computed for the 36 job instance. Similar to the 6 job instance, the Pareto efficient frontier was computed by searching for the minimum makespan, given 21 different specified upper bounds on the peak load. Each schedule corresponding to the minimum makespan was computed in 4–5 s. Fig. 6 shows the Pareto frontier for this case, along with the associated carbon footprints of the schedules on the Pareto frontier. Again, like the 6 job instance, there is a significant trade-off between makespan and peak load, and both carbon footprint and makespan decrease as the upper bound of power consumption increases. However, carbon footprint levels off when the upper bound on peak load reaches a certain level. In a companion paper [35], we show that if the cutting speed is assumed to be continuous and can take any value within a recommended range, then we can give efficient polynomial-time algorithms and theoretical results on the structure of optimal solutions for different variants of this flow shop scheduling problem. These results are as follows: Case 1. No intermediate storage between the two machines: • Suppose the amount of work is consistent across machines: that is, for any two jobs j, k ∈ J, p1j ≤ p1k implies p2j ≤ p2k . Then an optimal schedule can be found in polynomial time by transforming the scheduling problem into an equivalent asymmetric traveling salesman problem that satisfies the so-called Demidenko conditions [36]. • Furthermore, if the work for each job is the same on each machine – that is, for any job j ∈ J, p1j = p2j – then, an optimal schedule can be computed by a simple priority-list algorithm. In particular, if p11 ≤ p12 ≤ · · · ≤ p1n then scheduling the jobs according to the permutation (1, 3, 5, n, 6, 4, 2) yields an optimal schedule.
K. Fang et al. / Journal of Manufacturing Systems 30 (2011) 234–240
Case 2. Unlimited machines:
intermediate
storage
between
the
two
• If the total power consumption is Pmax at any time in the schedule, an optimal schedule (i.e., job completion times and job/machine speeds) can be found in polynomial time given a fixed permutation of the jobs. In addition, the sequence of jobs can be decomposed into sub-sequences of jobs that are processed at the same speed. This kind of result on the mathematical structure of the scheduling problem is useful in the design of neighborhood search algorithms to find near-optimal solutions. 5. Summary, conclusions, and future work This paper presents a general multi-objective mixed integer linear programming formulation for optimizing the operating schedule of a flow shop that considers both productivity (i.e., makespan) and energy (i.e., peak load and carbon footprint) related criteria. In this new formulation, operation speed is allowed to vary in order to affect the peak load and energy consumption. This is different from traditional schedule optimization approaches where highest possible operation speed is preferable in order to maximize productivity. Although the formulation given is for a flow shop, it is possible to construct formulations suitable for shop floors with other types of process/job interdependency (e.g., a job shop) by modifying or relaxing its constraints. Scheduling cast iron plates with slots in a two-machine flow shop is used as a simple case study to demonstrate the complexity of the problem. For this simple case, even when only a moderate number of jobs are considered (36 parts) it is almost time prohibitive to compute even one point on the Pareto frontier using commercial optimization software tools. Without any further intervention, it is only feasible to compute the Pareto frontiers for a small size problem (6 jobs). It should be pointed out that although interesting structural and algorithmic properties can be derived for the two-machine flow shop problem, they will not likely hold for a shop of industrial scale where perhaps hundreds of machines are involved. Moreover, for such industrial sized problems the computation time will be prohibitive. Therefore, for industrial scale shop floors, in order to find practical schedules that balance economic and environmental performance, the following approaches are suggested: (1) construct a simplified model for the shop floor while maintaining fidelity with regard to scheduling; and (2) design algorithms specialized for this new model, either through mathematical programming, or alternate approaches such as neighborhood search. With these approaches acceptable schedules may be identified with reasonable computational effort. It should also be noted that it is possible to expand the formulation to consider other environmental and occupational health impacts, such as water consumption, air emissions, and noise, although only energy related criteria (i.e., peak load, energy consumption, and carbon footprints) are introduced beyond productivity as new objectives for the shop floor schedule optimization problem in this paper. Doing so will likely make the problem even more challenging, especially if nonlinear correlations among decision variables and constraints are introduced. Numerous opportunities exist in this new area, which are expected to have broad impacts on U.S. manufacturing enterprises and even the entire industrial sector, given the fact that environmentally conscious shop floor scheduling has the potential to significantly reduce the energy and environmental footprints of manufacturing facilities without incurring large capital investment. This paper represents one of the first ever attempts on environmentally conscious shop scheduling and serves as a starting point to develop a research roadmap and agenda for this emerging area.
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