Energy-optimal programming and scheduling of the manufacturing operations

The shop floor energy system covers the energy consumed for both the air conditioning and manufacturing processes. At the same time, most of energy consumed in manufacturing processes is converted in heat released in the shop floor interior and has a significant influence on the microclimate. Both these components of the energy consumption have a time variation that can be realistic assessed. Moreover, the consumed energy decisively determines the environmental sustainability of the manufacturing operation, while the expenditure for running the shop floor energy system is a significant component of the manufacturing operations cost. Finally yet importantly, the energy consumption can be fundamentally influenced by properly programming and scheduling of the manufacturing operations. In this paper, we present a method for modeling and energy-optimal programming & scheduling the manufacturing operations. In this purpose, we have firstly identified two optimization targets, namely the environmental sustainability and the economic efficiency. Then, we have defined three optimization criteria, which can assess the degree of achieving these targets. Finally, we have modeled the relationship between the optimization criteria and the parameters of programming and scheduling. In this way, it has been revealed that by adjusting these parameters one can significantly improve the sustainability and efficiency of manufacturing operations. A numerical simulation has proved the feasibility and the efficiency of the proposed method.


Introduction
On one hand, in today's manufacturing environment, all participants have to fight in order to meet the ever-changing competitive market requirements. To face this challenge, it is highly important to work in optimal conditions, and this is the reason why a very large number of researches has already been dedicated to formulate and to solve the problem of optimizing different types of manufacturing processes. As optimization criteria, the most considered are the manufacturing cost, e.g. [1,2] and the metal removing rate (MRR), e.g. [3,4].
On the other hand, the restrictions induced by the sustainable development concept become more and more seriously considered when addressing the manufacturing activities [5]. In connection to this, energy efficiency became an important optimization criterion [6,7].
The expenditures for running the shop floor energy system are a significant component of the overhead expenses, hence of the manufacturing operations cost also. The natural energy transfer of the Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 shop floor building has an hourly variation. A realistic assessment of the natural energy transfer from the shop floor building can be realized. Once programming a manufacturing operation, all data regarding both its duration and energy consumption are known. Most of this energy is converted in heat released in the shop floor interior and has a significant influence on the microclimate [8]. After all the current operations are programmed, they are also scheduled. If the energetic income and the operation schedule are determined, then the energy flow can be found for each manufacturing system. Various models for energy-efficient process modeling, planning and scheduling are presented in literature [9,10]. Their main purpose is to improve the energy efficiency by realizing uniform energy consumption through a better distribution of production tasks between the existing capacities, sometimes located in different interconnected plants, but without effectively making an optimization.
Here we present a method for modeling and energy-optimal programming & scheduling of the manufacturing operations, based on a holistic approach of the entire energy consumption. It means, for that, besides the energy directly consumed in the manufacturing process, the energy embedded in the assets used in the manufacturing operation and the energy required for running the shop floor climatization system are also taken into account. The paper is structured on five sections. The next one defines the concept of energy-based optimization. The third section is dedicated to modeling of the optimization problem by defining three criteria and expressing the corresponding objective functions. The fourth section presents a numerical simulation in the case of a turning process, while the last one is for conclusions & perspectives.

Energy-based optimization
Let us consider a generic optimization action. It supposes four successive stages. The first one is the motivation stage, i.e. the implementation of a politics. Here the targets are defined and the actions needed to reach them are identified. The second stage is dedicated to optimization problem formulation. The decision-maker establishes, through directives, which are the optimization criteria enabling to assess the target reaching moment and, starting from here, the appropriate features that will be adjusted are chosen. In the third stage, the optimization problem is modeled. Causal relations between criteria and selected features are determined. Hereby, the criteria are used to build objective functions, having as variables the features values. Finally, the last stage means the optimization problem solving, by extremizing the objective function. The solution gives the optimal values for being assigned to the adjustable features.
We particularize below the considerations from above in the case of a generic manufacturing operation, submitted to an energy-based optimization.
Here the targets are the environmental sustainability and the economic efficiency, while the actions for reaching them are energy-optimal programming and scheduling. Three optimization criteria may be chosen as the most relevant: the energy consumption (in connection to the first target), the profit per operation (related to the second target) and the specific profit (meaning the profit obtained for each Kwh of consumed energy), as synthetic criterion, making a trade off between the two targets. In the addressed problem, the programming action refers to processes intensity and to the choice of the assets used to fulfill the operation task. The scheduling action has in view the processes timing (assessed through the calendar time fraction used for the processes) and location (the energy efficiency of insulating the shop floor from the external environment).

Modeling of the optimization problem
Three criteria are considered in order to define the approached optimization problem: the energy consumption (EC), the profit per operation (PO) and the specific profit (SP). The optimization variables are the working process duration (T w ) and the scheduling effectiveness coefficient (β). The relation between criteria and variables are deducted below.

The energy consumption
The energy consumption criterion, issued from environmental sustainability policy, can be defined, at the most general level, as equation (1): Note: Here, by operation we mean the action of manufacturing a batch of parts from same product order, consisting in a unique process performed on a given workstation. For sample, the teeth machining on a given machine tool of the 20 gears needed for a 10 gearboxes order, each gearbox including two identical gears, means an operation.
In the case of a manufacturing operation, EC has three main components, equation (2): where EC w , EC h/c and EC e mean the energy consumptions supposed, respectively, by the working process, the shop floor heating & cooling system, and the embedded energy into involved assets (e.g. equipments, tools, devices).  The schedules of workstations from a shop floor, drawn for a T schedule time interval, are presented in figure 1. Each row corresponds to a workstation. The blank cells mean performed operations of T op duration, while the hatched onesinactivity periods. The coefficient β (i) reflecting the effectiveness of using the workstation "i" along a T schedule time interval is defined as equation (3): The energy natural transfer of the shop floor building is measured through the corresponding transfer power ( total transfer P ), time varying. The nominal power of workstation "i" driving system is   i P 0 .
Note: The below presented relations will refer to a generic "j" operation, performed on the "i" workstation. From simplification reasons, we gave up to attach these indexes where needed (e.g. we have written only T op ). If accepting an inverse proportional dependence between the working power P w and the working duration T w of an operation, the proportionality constant being K w , and a quadratic dependence between the energy efficiency ratio η and P w (see figure 2), then EC w can be expressed as equation (4): with a and b specific constant to be identified.
The equivalent time per operation TO can be calculated with equation (5): where T a means auxiliary time, consumed on handlings and other preparations required for effectively running the operation, τ rftime consumed for refreshing a resource needed in the process (e.g. for replacing a cutting tool), and T rf -time interval between two successive refreshes (e.g. tool durability). Between T rf and T w we suppose the following dependence relation, equation (6): Regarding EC h/c , it has different expressions for heating and for cooling. When heating is needed, the heat released by the functioning workstations diminishes the heat amount to be furnished by the climatization system EC h , therefore we have equation (7): with P transfer meaning the share γ (i) of total transfer P assigned to the workstation "i", calculated as equation (8): By making the required replacements, EC h expression becomes (equation (9) When cooling is needed, the heat released by the functioning workstations increases the heat amount to be eliminated by the climatization system EC c , similarly calculated (equation (10) In relation (10), ε means the frigorific efficiency of the cooling system.
The consumption of embedded energy corresponding to an operation can be determined with equation (11): where EC e/hour is the "cost" from embedded energy point of view for an hour of workstation use. If a generic asset "k" from the set of assets required for performing the considered operation is characterized by the EC e total (k) amount of embedded energy and the LC (k) lifecycle length, then (equation (12)

The profit per operation
This criterion reflects the economical efficiency of a manufacturing operation. Its basic form is (equation (13)): The price P of an operation can be conventionally established as a share of the product price, which means that it is a commercial choice of the producer. The cost C of an operation is calculated as: In relation (14), C non-op means the non-operational expenditures, namely those needed by the manufacturing process but not affected by operation programming & scheduling (e.g. the worked material cost). The other three components are the costs of wage, consumables refresh and energy, respectively and depend on operation programming & scheduling. A more detailed form of (14) is equation (15): Here, besides the already introduced notations, c τ is the unitary wage cost [Euro/hour], c kthe refill kit cost [Euro], and c ethe energy price [Euro/Kwh].

The specific profit
This is a synthetic criterion, making a tradeoff between the profit per operation and the afferent consumption of energy, equation (16):

Profile of the dependence between optimization criteria and optimization variables
We further discuss, at conceptual level, the aspect of the optimization problem, in the case of the above-mentioned criteria. If we define the process intensity q as, equation (17): then the three objective functions (EC, PO, SP) can now be considered as depending on two variables: q and β. In figure 3, they are represented as typically looking for a given value of β, belonging to [0, 1]  Therefore, the rational domain to search for SP optimal value is comprised between these two limits. For an arbitrary value q inside the rational domain, we have, equation (18): The performed numerical simulations (sampled in the next section) entitle us to state that SP has a variation with a point of maximum, S, reached when SP opt q q  .
Finding an optimal value for q when β is kept constant means an optimization of programming action only. If also taking into account the scheduling, then we have to look what happens to the values of SP opt q and SP max when β is varying. It is possible that there is an optimal value β opt , corresponding to which SP max is maximum. Hereby we define the solution of SP optimization problem by the couple ( SP opt q , β opt ).  As it can be observed, the optimization problem is consistent in terms of energy-optimal programming. The extreme values EC min , PO max and SP max correspond to different values for q: 12; 7.2; respective 8.4 [processes/hour] in heating case, and 12; 7.2; respective 9 [processes/hour] in cooling case. The percent of losses (referred to the optimal point), which appear if working with process intensity different from the optimal ones, define the efficiency of energy-optimal programming & scheduling. Their values, calculated after each optimization criterion, are presented in figure 5.  One can easily notice that the losses can reach very significant levels if the action of energyoptimal programming is not considered. For example, SP can be up to 40% smaller if the process intensity is too low, or even worse, up to 100% smaller if the process intensity is too high relative to the optimal one (both situations in the heating case).

Conclusion & perspectives
By starting from one of the main requirements of the sustainable development conceptthe energy efficiency increase, this paper presents a method for modeling and energy-optimal programming & scheduling the manufacturing operations. It lays on a holistic approach of the energy consumption, including components neglected up to now, despite being significant.
In the mentioned purpose, the concept of energy-based optimization was firstly introduced. Then, the optimization problem was modeled by choosing as optimization criteria the energy consumption, the profit per operation and the specific profit, and as optimization variablesthe operation working duration and the scheduling effectiveness coefficient. Detailed expressions for the corresponding objective functions were deducted and the main aspects of the optimization problem were discussed, at conceptual level.
The feasibility and the efficiency of energy-optimal programming & scheduling of the manufacturing operations were both tested by running a numerical simulation in the case of a turning operation consisting in a single process. The results are convincing and they show an important potential of improving the manufacturing operations efficiency by applying the energy-optimal programming & scheduling.
Regarding the considered simplifying hypothesis (e.g. see relation (6)), they are suitable to machining operations case, but it should be noticed that they can be generalized to most of the other types of manufacturing operations (welding, forming, casting etc.).
The presented approach of energy-based optimization may seem affected by limitations, and it really is, but the limits can be pushed out in future works, by addressing to an extended optimization area (operations consisting in more than one process, groups of operations), to multi-criteria optimization or to generalized optimization algorithms.