Energy-aware integration of process planning and scheduling of advanced machining workshop

Abstract

Increasing energy consumption of manufacturing industry demands novel approaches to achieve energy conservation and emission reduction. Most of the previous research efforts in this area focused more on analyzing manufacturing energy consumption of a process or that of a machine tool with less concern on the system level of advanced machining workshop, especially a flexible manufacturing system. In this article, a new energy-saving approach of flexible manufacturing system is put forward based on energy evaluation model for integration of process planning and scheduling problem in flexible manufacturing system (flexible manufacturing system-integration of process planning and scheduling). First, complying with feature precedence and other technological requirements, flexible manufacturing system -integration of process planning and scheduling is mapped as an asymmetric traveling salesman problem of which operations are provinces and candidate operations are cities belonging to different provinces. To evaluate the performance of each solution of the asymmetric traveling salesman problem, energy consumption evaluation criteria for flexible manufacturing system-integration of process planning and scheduling are established and three energy efficiency indicators are also provided to perform further analysis on manufacturing energy consumption, that is, part energy efficiency, machine tool energy efficiency and feasible solution energy efficiency. Then, a mutation-combined ant colony optimization algorithm is proposed to solve the flexible manufacturing system-integration of process planning and scheduling which combined roulette and mutation selection methods to pick out the next candidate operation. The pheromone trails associated with edges are released by the so-far-best ant or the iteration-best ant probabilistically to both keep the search directed and avoid converging to the local best. Finally, a case study of flexible manufacturing system in advanced machining workshop is employed to demonstrate the feasibility and applicability of this approach in three different scenarios and compared with the “process planning then scheduling” approach; energy consumption obtained by the proposed method drops 10.7%.

Introduction

Manufacturing industry has been forced to reconsider the importance of saving energy due to the current energy pricing policy. The industrial sector is, indeed, responsible for over half of the world’s total energy consumption (EC) and 36% of CO₂ emissions,^1,2 and its EC has almost doubled over the last 60 years. ³ Both the European Community and the US government have been devoted into reducing primary EC. ⁴ Therefore, novel approaches are needed for considering energy-saving aspects of manufacturing industry, and it is promising that many benefits could be gained, such as direct costs reduction, regulation respect and positive product image. ⁵ According to the current standards (I.S. 393, ANSI/MSE2000-2008, EN16001, ISO 50001/20140/14955), it is a fact that energy conservation in machining workshop is more and more emphasized.

Moreover, for metal removal processes, actual EC of removing the material is, at most, 14.8% of the total energy required in manufacturing, while the rest 85.2% of energy is independent of whether or not a part is being processed. This significant amount of energy use is required for entire time since the machine is powered on. ⁶ It was also testified that huge amounts of energy is consumed during machines’ idle state, which implies that besides centering on energy reducing of individual operation or machine, attention should also be drawn to system-level improvement for energy conservation, such as process planning and scheduling. ⁷ However, most of the researchers considered process planning or scheduling separately for EC reduction, and effective energy-saving approaches for integration of process planning and scheduling problem are lacking, especially in a flexible manufacturing system (FMS) which has the flexibility to react in case of changes, whether predicted or unpredicted. In order to address the issue, a new energy-saving approach of FMS is put forward based on energy evaluation model for integration of process planning and scheduling problem in FMS (FMS-IPPS). Complying with feature precedence and other technological requirements, the FMS-IPPS is mapped as a unique asymmetric traveling salesman problem (ATSP) of which operations are provinces. The problem of FMS-IPPS is defined and converted to an ATSP of finding the shortest route to visit each node (operation) once. Then, in order to avoid locking into a local minimum, a roulette and mutation-combined selection method is employed to increase the diversity of ant colony optimization (ACO) algorithm.

Literature review

In order to reduce EC of manufacturing processes, a significant number of research studied the EC modeling or monitoring method which can be roughly viewed under two perspectives, namely, “process” level and “plant” level. ⁸ At the process level, the issue of EC is first presented by De Filippi et al. ⁹ In their study, the operating data, involved in various operations, were collected from 10 different numerically controlled machine tools (MTs), and their analysis results showed that the energy required of MTs during machining is significantly greater than the theoretical energy required in chip formation. Draganescu et al. ¹⁰ constructed an experimental data-based statistic model using response surface methodology to estimate machine tool efficiency and EC in cutting process. A new on-line energy efficiency monitoring approach was proposed without using any torque sensor or dynamometer to improve the environmental performance of machining systems. ¹¹ Kara and Li ¹² presented an empirical model to characterize the relationship between EC and process variables for material removal processes to enable industry to develop potential energy-saving strategies during product design and process planning stages. Considering EC, resource utilization, waste generation and their collective effect on equivalent carbon dioxide emission, quantitative analysis modeling of the entire grinding process is developed against the roughing, finishing and spark-out stages of the process. ¹³ Then, Yingjie ¹⁴ presented an overview of the state of the art in energy-efficient techniques in the domain of discrete part manufacturing, focusing on the techniques including energy assessment model for machining process and the energy efficiency analysis and evaluation for MTs, important components and machining systems. At the plant level, Seow and Rahimifard ¹⁵ adopted a novel approach to modeling energy flows within a manufacturing system based on a “product” viewpoint and utilized the EC data at “plant” and “process” levels to provide a breakdown of energy used during production. A modeling method of task-oriented EC for machining manufacturing system was proposed to explore the potential on energy-saving in production management. ¹⁶ Overall, many researchers mainly studied EC modeling of MTs or a plant, which provides a theoretical basis for energy conservation of FMS-IPPS.

At the system level of machining workshop, since process planning and scheduling are the two major governing operations, both of them are responsible for efficient allocation and utilization of manufacturing resources, and their relationships with EC of a plant have been studied by many researchers. Bruzzone et al. ⁵ proposed an approach based on a mixed integer programming model to modify a reference schedule generated by an advanced planning and scheduling system to account for EC. Anderberg et al. ¹⁷ investigated the role of process planning as an enabler for cost-efficient and environmentally benign computer numerical control (CNC) machining using specific energy as the principal indicator of energy-efficient machining. A systematic approach was developed for milling process planning and scheduling optimization, which consists of a process stage and a system stage, augmented with intelligent mechanisms for enhancing the adaptability and responsiveness to job dynamics in machining shopfloors. ¹⁸

Moreover, Dai et al. ¹⁹ proposed an energy-efficient model for flexible flow shop scheduling and adopted an improved generitic-simulated annealing algorithm to make a significant grade-off between the makespan and the total EC to implement a feasible scheduling. Based on a carbon footprint model of multi-job processing, Zhang et al. ²⁰ presented a model of low-carbon scheduling of the flexible job shop and put forward three carbon efficiency indicators to estimate the carbon emission of parts and MTs. Through reviewing the literature on energy-efficient machining systems, Peng and Xu ⁷ pointed that the complexity of machining systems, such as diverse nature of materials, sophisticated machine center and various machining operations, makes it more difficult to reduce EC. However, most of these methods considered process planning or scheduling separately for EC reduction, and this hierarchical treatment may neglect their clearly existing relevance. Although process planning can enhance productivity of manufacturing systems and scheduling can optimize process sequences, it has been proved that the separated process planning and scheduling systems might be able to significantly improve productivity of manufacturing system largely. ²¹ Dai et al. ²² proposed an energy-aware mathematical model for job shops that integrates process planning and scheduling, but they neglected the buffers’ energy consumption (BE) and job-conveying energy consumption (JCE). Therefore, this article reports our research of the energy-aware IPPS for generating feasible and optimal solutions.

In order to solve the IPPS, researchers have proposed various integration models and algorithms over the last decade. A recent review of IPPS showed that the common feature of recent research was making full use of the intersection of process planning and scheduling and then improving the performance and the flexibility of the process planning system. ²³ Since IPPS problem is a non-deterministic polynomial hard (NP-hard) problem, various artificial intelligence–based approaches have been developed to facilitate the optimization process. Chan et al. ²⁴ put forward an IPPS model encapsulating the salient features of outsourcing strategy and proposed an artificial immune system–based algorithm embedded with the fuzzy logic controller (AIS-FLC) to solve the complex problem. Guo et al. ²⁵ developed a unified representation model for IPPS and proposed a modified particle swarm optimization (PSO) algorithm to optimize the IPPS problem, which has been enhanced with new operators to improve its performance. ACO, first introduced by Dorigo et al. ²⁶ in the early 1990s, is a population-based, cooperative search metaphor inspired by the foraging behavior of real ants and has been widely applied to combinatorial optimization problems such as traveling salesman problem (TSP), quadratic assignment problem (QAP) and job shop scheduling problem (JSP). ²⁷ Kumar et al. ²⁸ used the ACO technique to solve the scheduling problem for FMS and applied a graph-based representation technique with nodes and arcs representing operation and transfer from one stage of processing to the other. In order to cope with the difficulty of scheduling semiconductor wafer fabrication system, a decomposition-based classified ant colony optimization (D-CACO) method was proposed which comprised decomposition procedure and classified ACO algorithm. ²⁹ Moreover, Liu et al. ³⁰ carried out the process planning optimization of hole-making operations using the ACO, and the optimized results showed that the optimal auxiliary time is significantly less than the intuitive auxiliary time. In summary, ACO algorithm has been used to solve various optimization problems and has proved to be an effective algorithm especially for process planning and scheduling problem. However, as the pheromone accumulates, a global optimum may not be reached because the search process can get stuck in a local minimum. ³¹ To avoid locking into local minima, Yang et al. ³² introduced a mutation process and a local searching technique into the ACO method to solve the generalized TSP. Yu et al. ³³ proposed an improved ACO, which possessed a new strategy to update the increased pheromone, called ant-weight strategy, and a mutation operation to solve the vehicle routing problem. Changdar et al. ³⁴ presented a novel ACO algorithm to solve binary knapsack problem and performed crossover and mutation processes between the solutions generated by ants. Analogously, based on the original ACO, a roulette and mutation-combined selection method is employed to increase the diversity of ant colony.

Based on the present research, this article presents a new energy-saving approach of FMS based on energy evaluation model for in FMS-IPPS, and the remaining sections of this article are organized as follows: in section “Problem statements,” the problem of FMS-IPPS is defined and converted to an ATSP of finding the shortest route to visit each node (operation) once. Section “EC evaluation criteria for FMS-IPPS” presents the evaluation criteria of EC and three energy efficiency indicators are provided, that is, part energy efficiency (PAEE), machine tool energy efficiency (MTEE) and IPPS solution energy efficiency. Then a mutation-combined ACO algorithm for IPPS is described in section “Mutation-combined ACO algorithm for IPPS.” Section “Case study” is a case study of advanced machining workshop with two parts and four MTs. Finally, in section “Conclusion,” some conclusions are stated.

Problem statements

Definition and assumptions of FMS-IPPS

The definition of FMS-IPPS is described as follows: given a set of parts, which are to be processed on several machines with alternative process plans, manufacturing resources and other technological constraints, then selecting a suitable process plan and corresponding manufacturing resources and sequencing the operations at the same time so as to determine a schedule in which the technological constraints among operations can be satisfied and the corresponding objectives can be achieved.

FMS-IPPS can be formulated as follows. There is a set $\mathit{P}$ of K parts ( $\mathit{P}::=\{P_1,\dots ,P_k,\dots ,P_K\}$ ) that will be processed on M machines. The set of machines is $\mathit{MT}::=\{M{T}_1,\dots ,M{T}_m,\dots ,M{T}_M\}$ . Each job $P_k$ consists of ${\mathit{N}}_O^k$ operations $\mathit{O}{\mathit{P}}^k::=\{{\mathrm{OP}}_1^k,\dots ,{\mathrm{OP}}_i^k,\dots ,{\mathrm{OP}}_{N_O^k}^k\}$ .

There are precedence constraints between operations in one part, which are usually classified into six types: (1) fixture interaction, (2) tool interaction, (3) datum interaction, (4) thin-wall interaction, (5) material-removal interaction and (6) fixed order of machining operations. ³⁵ These constraints will be considered for each job to ensure that each solution is feasible, which will be stated in section “Constraint and state matrix.”

Furthermore, some assumptions of FMS-IPPS are made as follows:

1. Every operation of each part should be processed by one and only one machine.

2. One machine can process at most one operation at a time.

3. All machines are available at $t=0$ .

4. There is no precedence relationship between operations of different jobs, but there are precedence relationships between operations of one job in some occasion.

5. Preemption is not allowed for processing each job, that is, once an operation is started, it must be finished without interruption.

6. For the same operation, the process time and energy consumed might be different when using different process plans, such as different types of machines.

7. The time and energy used for conveying jobs for different distances are taken into consideration. The time and energy consumed by MTs while waiting idly or changing cutting tools (CTs) are also taken into account.

Mathematical description of candidate operations

At the process planning phase, the manufacturing features of each part are first recognized by analyzing the geometrical and topological information of the part. The description of a feature consists of its type, position, dimensions, tolerance and surface finish. Also, each feature is mapped to one or several operations according to its geometrical and technological requirements and available manufacturing resources. ³⁶ If there are K parts machining in the job shop, for the job $P_k$ with $N_O^k$ operations, there are a set of candidate MTs, a set of CTs) and that of tool approach directions (TAD), ³⁵ as shown in Figure 1. A candidate operation (CO) is constructed by different combinations of MTs, CTs and TADs. Now, ${\mathit{CO}}_i^k$ represents the CO set for ${\mathrm{OP}}_i^k$ ( $i=1,2,\dots ,N_O^k$ ) which is defined as

$${\mathit{CO}}_i^k::=\{{\mathrm{CO}}_{i,1}^k,\dots ,{\mathrm{CO}}_{i,j}^k,\dots ,{\mathrm{CO}}_{i,N_{\mathrm{co}}^{k,i}}^k\}$$

(1)

where ${\mathrm{CO}}_{i,j}^k$ is the $j\mathrm{th}\mathrm{CO}$ for ${\mathrm{OP}}_i^k$ of $P_k$ , and there are $N_{\mathrm{co}}^{k,i}$ COs available for each ${\mathrm{OP}}_i^k$ . Some important notations are described in Table 1.

Figure 1.

Illustration of tool approach directions.

Table 1.

Some important notations.

Notation	Definition	Equations
$N_O$	Total number of operations of all parts	$N_O=\sum _{k=1}^K{N}_O^k$ (2)
$N_{\mathrm{CO}}^k$	Total number of COs of job $P_k$	$N_{\mathrm{CO}}^k=\sum _{i=1}^{N_O^k}{N}_{\mathrm{co}}^{k,i}$ (3)
$N_{\mathrm{CO}}$	Total number of COs of all operations of all parts	$N_{\mathrm{CO}}=\sum _{k=1}^K{N}_{\mathrm{CO}}^k$ (4)

CO: candidate operation.

Since TADs are determined by different set-ups (SUs) of the part to some degree, a CO is the combination of MT, CT and SU

$${\mathrm{CO}}_{i,j}^k=\{{\mathrm{MT}}_{i,j}^k,{\mathrm{CT}}_{i,j}^k,{\mathrm{SU}}_{i,j}^k\}$$

(5)

where ${\mathrm{MT}}_{i,j}^k,{\mathrm{CT}}_{i,j}^k,{\mathrm{SU}}_{i,j}^k$ are the MT, CT and SU strategy chosen by ${\mathrm{CO}}_{i,j}^k$ , respectively.

A feasible solution consists of not only the selection of ${\mathrm{CO}}_{i,j}^k$ for each operation but also the sequences of operations. Therefore, a feasible solution is a sequential list of COs. By combining and sorting dissimilar COs, the EC of machining would be significantly different, which will be described in section “EC modeling for FMS-IPPS” in detail.

Directed graph modeling of FMS-IPPS problem

Since FMS-IPPS problem shares similarities with the ATSP problem, it is converted to ATSP problem to find the optimal solution. To match FMS-IPPS problem with ATSP, a to-be-machined operation is regarded as a city, while the weight and distance reflect the EC of each operation and that of transportation, respectively. However, every operation owns several available COs, so the weight and distance of different COs vary. Therefore, the ATSP problem needs some modifications to be compatible with FMS-IPPS problem. Here, each operation is regarded as a “province” which owns several cities representing its COs. The salesman’s mission is to visit every province (operation) once, that is, he could visit any one city (CO) in this province, while the distance is still between the two cities and every city has its own weight. The construction graph used here contains provinces representing operations plus two virtual provinces: a source and a destination, and FMS-IPPS problem actually could be mapped to a directed graph with nonnegative weights

$$G=(V,C,E,W)$$

(6)

where $V$ contains $N_{\mathrm{CO}}$ nodes (cities) which are divided into $N_O$ groups (provinces), C is the set of nodes’ weights representing the EC of machining processes and E is the set of edges between COs. It needs to be specified that only cities belonging to two different groups have a directed weighted edge. W is the weight set of edges, which denotes the EC caused by changing MTs, CTs and SUs, such as the tool changing EC, the transporting EC and idle EC, which will be described in the next section. As a result, the objective of finding the most energy-saving route is turned to the aim of finding a tour that could visit these nodes with the lowest cost.

Also, geometrical and technological precedence constraints in the manufacturing process could not be violated, that is, if one operation must be processed before another one, the edge between them is directed. Even if there is no precedence between the two operations, there will be two direct edges connecting them with different weights. Thus, FMS-IPPS problem is a constraint-based ATSP problem.

In order to illustrate the FMS-IPPS vividly, an example with three jobs is shown in Figure 2. A feasible solution to FMS-IPPS must comply with precedence constraints: ${\mathrm{OP}}_1^1>{\mathrm{OP}}_2^1,{\mathrm{OP}}_1^1>{\mathrm{OP}}_3^1,{\mathrm{OP}}_1^2>{\mathrm{OP}}_2^2,{\mathrm{OP}}_1^3>{\mathrm{OP}}_3^3,{\mathrm{OP}}_2^3>{\mathrm{OP}}_3^3$ . Next, for each operation, there might be several COs available, with unique and proper combinations of MTs, CTs and TADs. For instance, ${\mathrm{OP}}_1^1$ has two COs: ${\mathrm{CO}}_{11}^1$ and ${\mathrm{CO}}_{12}^1$ . A feasible visiting route $\mathrm{FS}:{\mathrm{CO}}_{11}^3\to {\mathrm{CO}}_{12}^1\to {\mathrm{CO}}_{32}^1\to {\mathrm{CO}}_{11}^2\to {\mathrm{CO}}_{21}^1\to {\mathrm{CO}}_{21}^3\to {\mathrm{CO}}_{22}^2\to {\mathrm{CO}}_{31}^3$ , starting and ending at virtual spots, is generated by choosing the operations and the specified CO one by one under precedence constraints, as shown in Figure 2.

Figure 2.

Workflow of generating a feasible solution for IPPS.

EC evaluation criteria for FMS-IPPS

EC modeling for FMS-IPPS

As previously mentioned, FMS-IPPS covers multi-job processing, and the inputs are raw materials or semi-finished products, while the outputs include end products and effluent. During the multi-job machining processes, many activities will generate EC, such as machining processes, job-conveying and buffers’ usage. To calculate EC of a workshop accurately, machining processes of a part are drawn based on life cycle assessment, which is shown in Figure 3. The EC of a processing system mainly comes from three parts, namely, processing energy consumption (PE), BE and JCE.

Figure 3.

Energy consumption of machining processes.

PE

According to the spindle power profile of a machine, a machining process mainly contains five states, that is, the startup state, idle state, cutting state, tool changing state and the shutdown state. ¹¹ Since the research emphasis in this article is the cutting EC, and the EC of the startup state and shutdown state is excluded, EC of machining processes shown in this article may be underestimated.

With regard to EC of the cutting state, an empirical relationship between specific EC (SEC) and material removal rate (MRR) for an MT is adopted, ¹² as shown in equation (7)

$$\mathrm{SEC}=C_0+\frac{C_1}{\mathrm{MRR}}$$

(7)

where C₀ and C₁ are the machine-specific coefficients and are related to the workpiece material, tool geometrics, spindle drive characteristics and MT, and usually obtained through the experiments.

Then, the EC of the cutting state can be calculated by multiplying the removal volume of a part by SEC

$$\mathrm{PE}\_\mathrm{cutting}(M{T}_i{)}_j=V_j*\mathrm{SE}{C}_j$$

(8)

where $\mathrm{PE}\_\mathrm{cutting}(M{T}_i{)}_j$ stands for the EC of the jth cutting state of MT_i, and V_j and SEC_j represent its the removal volume and SEC, respectively.

For the idle state, its EC takes a dramatic portion of the total EC, caused by MTs’ waiting for the next job or changing fixtures

$$\mathrm{PE}\_\mathrm{idle}(M{T}_i)=T\_\mathrm{idl}{e}_i*P\_\mathrm{idl}{e}_i$$

(9)

where $\mathrm{PE}\_\mathrm{idle}(M{T}_i)$ stands for the total idle EC of MT_i, and $T\_\mathrm{idl}{e}_i$ and $P\_\mathrm{idl}{e}_i$ are the idle time and idle power of MT_i, respectively.

As for the EC of tool changing state, the average EC of tool changing is used to simplify the calculation process

$$\mathrm{PE}\_\mathrm{toolch}(M{T}_i)=N\_\mathrm{toolc}{h}_i*E\_\mathrm{toolc}{h}_i$$

(10)

where $\mathrm{PE}\_\mathrm{toolch}(M{T}_i)$ stands for the total EC of the tool changing state of $M{T}_i$ , and $N\_\mathrm{tooc}{h}_i$ and $E\_\mathrm{toolc}{h}_i$ represent the number of times and average EC of tool changing, respectively.

Then, the total EC of $M{T}_i$ covers EC of all the above states

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{PE}(M{T}_i)=\mathrm{PE}\_\mathrm{cutting}(M{T}_i)}\\ +\mathrm{PE}\_\mathrm{toolch}(M{T}_i)+\mathrm{PE}\_\mathrm{idle}(M{T}_i)\hfill \end{array}$$

(11)

Except the above direct EC, there are also some indirect EC during machining processes, like EC of CT ware. Since huge energy has been consumed during manufacturing of a CT, each use of CTs will generate EC indirectly due to its limited lifetime

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{CTE}(C{T}_i)=\frac{t\_\mathrm{too}{l}_i}{T\_\mathrm{too}{l}_i(N\_g{r}_i+1)}(E\_\mathrm{toolpro}{d}_i+N\_g{r}_i*E\_g{r}_i)}\end{array}$$

(12)

where $\mathrm{CTE}(C{T}_i)$ is the total EC of CT_i; $t\_\mathrm{too}{l}_i$ , $T\_\mathrm{too}{l}_i$ and $E\_\mathrm{toolpro}{d}_i$ denote its usage time, lifetime and EC of production, respectively, and $N\_g{r}_i$ and $E\_g{r}_i$ represent its grinding time and average grinding EC, respectively.

BE

Since a part will be kept in a buffer temporarily before the next process, buffers also will generate EC which can be calculated through the time of storage

$$\mathrm{BE}(B{F}_i)=t\_\mathrm{buffe}{r}_i*E\_\mathrm{buffe}{r}_i$$

(13)

where $\mathrm{BE}(B{F}_i)$ stands for the EC of BF_i, and $t\_\mathrm{buffe}{r}_i$ and $E\_\mathrm{buffe}{r}_i$ denote its usage time and EC per unit time, respectively.

JCE

JCE is the energy consumed by transportation equipment. Especially, in FMS, automated machines (lathe, mill, drill, etc.) are served by an automated material handling system, such as automated conveyors and automated guided vehicles, so energy consumed by transportation shares a considerable partition of the total EC

$$\mathrm{JCE}(P_i)=E\_\mathrm{conve}{y}_i*L_i$$

(14)

where $\mathrm{JCE}(P_i)$ is the transportation EC of P_i, and $E\_\mathrm{conve}{y}_i$ and L_i stand for the SEC and the transportation distance, respectively.

Finally, the total EC of the FMS can be expressed as follows

$$\mathrm{TE}=\sum _{i=1}^M\mathrm{PE}(M{T}_i)+\mathrm{CTE}+\mathrm{BE}+\mathrm{JCE}$$

(15)

Energy efficiency indicators of parts and MTs

Although many activities will generate EC as previously mentioned, only EC of the cutting state will add the value of a part, and other types of EC just play supplementary roles. To evaluate the energy efficiency of different parts, MTs and IPPS solutions, three energy efficiency indicators are proposed, that is, PAEE, MTEE and feasible solution energy efficiency (FSEE).

To compare the energy efficiency of different parts, it is necessary to have a parameter, which does not depend on the total EC of a part. PAEE is such a parameter, which can be defined as the ratio of EC of the cutting state of a part to the total EC of that part. And the PAEE could be used to compare the energy efficiency of different parts to find the least efficient part

$$\mathrm{PAEE}=\frac{\mathrm{PE}\_\mathrm{cutting}(P_i)}{\mathrm{PE}\_\mathrm{cutting}(P_i)+\mathrm{CTE}(P_i)+\mathrm{BE}(P_i)+\mathrm{JCE}(P_i)}$$

(16)

To evaluate the energy efficiency of an MT, MTEE is defined as the ratio of the EC of cutting state of a machine to the total EC of the machine in a scheduling plan

$$\mathrm{MTEE}=\frac{\mathrm{PE}\_\mathrm{cutting}\left(M_i\right)}{\mathrm{PE}\left(M_i\right)}$$

(17)

To evaluate the energy efficiency of a feasible IPPS solution, FSEE is defined as the ratio of the EC of cutting state to the total EC in a scheduling plan

$$\mathrm{FSEE}=\frac{\mathrm{PE}\_\mathrm{cutting}}{\mathrm{TE}}$$

(18)

Mutation-combined ACO algorithm for IPPS

Based on the original ACO, a roulette and mutation-combined selection method is employed to increase the diversity of ant colony. In order to settle the constraints in the problem, a constraint and state matrix CSM is built first. Figure 4 shows the flowchart of the proposed mutation-combined ACO algorithm. The main components of the algorithm are summarized below.

Figure 4.

Flowchart of the mutation-combined ACO algorithm.

Constraint and state matrix

With the purpose of describing precedence constraints in the problem, a constraint matrix CM with $N_O^k\times N_O^k$ elements is defined for P_k

$${\mathrm{cm}}_{i,j}^k=\{\begin{array}{c}1,\mathrm{if}i\ne j\mathrm{and}{\mathrm{OP}}_i^k\mathrm{has}\mathrm{to}\mathrm{be}\mathrm{executed}\mathrm{before}{\mathrm{OP}}_j^k\hfill \\ \multicolumn{1}{c}{0,\mathrm{otherwise}}\end{array}$$

(19)

Since an ant at each construction step chooses the next city only among those it has not visited yet, a state vector sv is needed to record whether an operation has been visited or not for each job

$${\mathrm{sv}}_i^k=\{\begin{array}{cc}0\hfill & \mathrm{if}{\mathrm{OP}}_i^k\mathrm{has}\mathrm{not}\mathrm{been}\mathrm{visited}\hfill \\ \multicolumn{1}{c}{-1,}& \mathrm{otherwise}\hfill \end{array}$$

(20)

The CM and sv can be combined as a matrix CSM^k to simplify the model, as shown in equation (21). For each iteration, the initial CSM^k is assigned to each ant, and after processing ${\mathrm{OP}}_i^k$ , the ith row of the matrix is zeroed, representing eliminating precedence constraints of ${\mathrm{OP}}_i^k$ for other operations, and the element ${\mathrm{csm}}_{i,i}^k$ is set to −1, implying that ${\mathrm{OP}}_i^k$ has already been visited

$${\mathrm{csm}}_{i,j}^k=\{\begin{array}{cc}{\mathrm{cm}}_{i,j}^k,\hfill & \mathrm{if}i\ne j\hfill \\ \multicolumn{1}{c}{{\mathrm{sv}}_i^k,}& \mathrm{if}i=j\hfill \end{array}$$

(21)

Finally, the CSM can be expressed as in equation (22), which contains $\mathbf{CS}{\mathbf{M}}^{\mathrm{k}}(k\in [1,K])$ as diagonal elements. In addition, as stated before, two additional virtual nodes are added which represent the starting node and the terminal node. Since the starting node should be the first to be visited, so elements in the first row should be set to 1, then the elements in the first column should be set to 0. Analogically, the terminal node is the last one to visit, so elements in the last column should be set to 1 and the elements in the last row should be set to 0

$$\mathbf{CSM}=\left[\begin{array}{c}\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\hfill \\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\begin{array}{c}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{\mathrm{CS}{M}^1}\\ \multicolumn{1}{c}{}\end{array}\hfill \\ \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{0}\end{array}\begin{array}{c}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{\mathrm{CS}{M}^2}\end{array}\hfill \\ \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{0}\end{array}\begin{array}{c}\begin{array}{c}\cdots \hfill \\ \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}\end{array}\hfill \\ \multicolumn{1}{c}{\ddots }\\ \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{\cdots }\end{array}\begin{array}{c}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}\end{array}\hfill \\ \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{\mathrm{CS}{M}^K}\\ \multicolumn{1}{c}{0}\end{array}\begin{array}{c}\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{1}\end{array}\hfill \\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{0}\end{array}\right]$$

(22)

Roulette and mutation-combined selection for solution construction

During the computing process of ant r, if ${\mathrm{CO}}_{i,j}^k$ is just selected as the current operation, then the following rule will be used to judge whether ${\mathrm{CO}}_{i\prime ,j\prime }^{k\prime }$ can be selected in the following computing steps. First, let ${\mathit{J}}_{i,j,k}^r$ be the list of COs which remains to be selected and could be selected according to the current values in CSM. Since every ant makes its choice from ${\mathit{J}}_{i,j,k}^r$ , the generated solutions are always feasible. The probability of each element in ${\mathit{J}}_{i,j,k}^r$ is calculated as ³⁷

$$p_{i,j,i^{\prime },j^{\prime }}^{k,k^{\prime }}=\frac{{[{\tau }_{i,j,i^{\prime },j^{\prime }}^{k,k^{\prime }}]}^{\alpha }\cdot {[{\eta }_{i,j,i^{\prime },j^{\prime }}^{k,k^{\prime }}]}^{\beta }}{{\displaystyle \sum _{i^{″}{j}^{″}{k}^{″}\in {}_{i,j,k}^r}{[{\tau }_{ij{i}^{″}{j}^{″}}^{(k)(k^{″})}]}^{\alpha }\cdot {[{\eta }_{i,j,i^{″},j^{″}}^{k,k^{″}}]}^{\beta }}}ifC{O}_{i^{″},j^{″}}^{k^{″}}\in J_{i,j,k}^r$$

(23)

where $\alpha$ and $\beta$ are the parameters which weigh the relative importance of learned pheromone and heuristic values. ${\tau }_{i,j,i\prime ,j\prime }^{k,k\prime }$ and ${\eta }_{i,j,i\prime ,j\prime }^{k,k\prime }$ represent the pheromone trail and heuristic information, respectively, which will be introduced in the next section.

After calculating probabilities of all the available COs, a roulette and mutation-combined selection method is employed to pick out the CO:

1. First, a preliminary CO is selected by roulette wheel selection.

2. Then, to avoid converging to a local optimum, the mutation idea is introduced from the genetic algorithm (GA). After a preliminary CO is selected, it will perform the mutation process according to the given mutation probability P_mu. The mutation process is similar to single-point mutation, and it alters the current selection from its initial CO and stochastically chooses one CO from ${\mathit{J}}_{i,j,k}^r$ to guarantee that the generated result is always feasible. Hence, the diversity is kept and the algorithm can come to better solution.

Updating of pheromone trail and heuristic information

Pheromone ${\tau }_{i,j,i\prime ,j\prime }^{k,k\prime }$ , which is a positive real value attached to the edge $e_{i,j,i\prime ,j\prime }^{k,k\prime }$ between ${\mathrm{CO}}_{i,j}^k$ and ${\mathrm{CO}}_{i\prime ,j\prime }^{k\prime }$ , will be updated by the rule of the MAX–MIN ant system (MMAS) in this article ³⁸

$${\tau }_{i,j,i\prime ,j\prime }^{k,k\prime }=(1-\rho )·{\tau }_{i,j,i\prime ,j\prime }^{k,k\prime }+\Delta {\tau }_{i,j,i\prime ,j\prime }^{k,k\prime }$$

(24)

where parameter $\rho$ is the evaporation rate of the pheromone on the path, and $\Delta {\tau }_{i,j,i\prime ,j\prime }^{k,k\prime }=w/C^{\mathrm{best}}$ in which w is a parameter and could modify the influence of the best length $C^{\mathrm{best}}$ . The ant that is allowed to add pheromone may be either the best-so-far, in which case $\Delta {\tau }_{i,j,i\prime ,j\prime }^{k,k\prime }=w/C^{\mathrm{bs}}$ , or the iteration-best, in which case $\Delta {\tau }_{i,j,i\prime ,j\prime }^{k,k\prime }=w/C^{\mathrm{ib}}$ , where C^bs and C^ib are the length of the best-so-far tour and the iteration-best one, respectively.

In general, both the iteration-best and the best-so-far update rules are used in an alternate way. When pheromone updates are always performed by the best-so-far ant, the search focuses very quickly around the best-so-far tour, whereas when it is the iteration-best ant that updates pheromones, then the number of arcs that receive pheromone is larger and the search is less directed. The probability for choosing the best-so-far ant to update the pheromone is noted as P_bs.

In addition, representing a priori information about the problem, ${\eta }_{i,j,i\prime ,j\prime }^{k,k\prime }$ is a heuristic value associated with edge weight between ${\mathrm{CO}}_{i,j}^k$ and ${\mathrm{CO}}_{i\prime ,j\prime }^{k\prime }$ . Conventionally, ${\eta }_{i,j,i\prime ,j\prime }^{k,k\prime }$ is the inverse of the sum of the cost of the current operation and that of the next operation to evaluate the goodness of this potential selection. In our case, the cost is the sum of PE and cutting tool energy consumption (CTE). Moreover, according to the precedence constraints, some operations could be carried out only when a certain operation is finished, and there should be more chances for the operation with higher urgency degree to be chosen as the next operation. So allowing for the total number of precedence constraints, ${\eta }_{i,j,i\prime ,j\prime }^{k,k\prime }$ could be computed as follows

$${\eta }_{i,j,i\prime ,j\prime }^{k,k\prime }=\frac{\sum _{j=1}^{N_O^k}{\mathrm{csm}}_{i,j}^k}{(\mathrm{PE}({\mathrm{CO}}_{i,j}^k)+\mathrm{CTE}({\mathrm{CO}}_{i,j}^k))+(\mathrm{PE}({\mathrm{CO}}_{i\prime ,j\prime }^{k\prime })+\mathrm{CTE}({\mathrm{CO}}_{i\prime ,j\prime }^{k\prime }))}$$

(25)

Case study

Case description

To verify the rationality of the proposed model and algorithm, two prismatic parts modified on the basis of Zhang et al. ³⁹ and Li et al. ⁴⁰ are adopted, as shown in Figures 5 and 6. Part 1 has 14 manufacturing features and 18 operations, while Part 2 owns 14 manufacturing features and 20 operations. The related information of machining resources and the features, and the operations of the two parts are shown in Tables 2–4 correspondingly.

Figure 5.

All the features of Part 1.

Figure 6.

All the features of Part 2.

Table 2.

Machining resources and their energy-related data.

Machines	Type	Idle power (W)	Average operating power (W)	Tool-change energy (W min)
MT 1	CNC drill press	90	153	50
MT 2	Vertical CNC milling	150	255	90
MT 3	Vertical CNC milling center	200	320	35
MT 4	CNC boring machine	180	306	80
Tools	Type	Average tool life (min)		Tool-manufacturing EC (kW min)
T 1	Drill1 ϕ2	60		6
T 2	Drill2 ϕ4	60		6
T 3	Drill3 ϕ6	90		9
T 4	Drill4 ϕ6	90		6
T 5	Mill1 ϕ5	180		12
T 6	Mill2 ϕ8	180		15
T 7	Mill3 ϕ10	180		18
T 8	Tapping tool M2	60		9
T 9	Reaming ϕ8	180		15
T 10	Boring tool $\varphi 12-\varphi 18$	180		18
T 11	Chamfer tool ${60}^{\circ }$	180		12
T 12	Slot cutter ϕ5	120		9

CNC: computer numerical control; EC: energy consumption.

Table 3.

Operations and candidate machining resources of Part 1.

Feature	Operation (Oper id)	MT candidates	TAD	CT candidates (runtime/min)	Precedence constraints (processed before)
F 1	Drilling (OP1)	MT 1, MT2, MT3	+Z, −Z	T 2(2)	OP 2
F 2	Milling (OP2)	MT 2, MT3	−Z	T 11(4)
F 3	Milling (OP3)	MT 2, MT3	−Y, −Z	T 2(13, 14), T6(10, 10)	OP 2
F 4	Milling (OP4)	MT 2, MT3	−Y	T 5(6), T6(6)	OP 3, OP11, OP13_OP18
F 5	Milling (OP5)	MT 2, MT3	−Y	T 5(10), T6(9), T7(8)	OP 3, OP4, OP11, OP13_OP18
F 6	Drilling (OP6)	MT 1, MT2, MT3	−Z	T 1(3)
	Tapping (OP7)	MT 1, MT2, MT3		T 8(3)
F 7	Drilling (OP8)	MT 1, MT2, MT3	+Z, −Z	T 2(2), T3(2), T4(3)	OP 6, OP7, OP9, OP10
	Reaming (OP9)	MT 1, MT2, MT3		T 9(2)	OP 6, OP7, OP10
	Boring (OP10)	MT 3, MT4		T 10(2)	OP 6, OP7
F 8	Drilling (OP11)	MT 1, MT2, MT3	−Z	T 2(2)
F 9	Milling (OP12)	MT 2, MT3	−X	T 5(15), T6(13)	OP 2, OP3, OP6, OP13_OP18
F 10	Drilling (OP13)	MT 1, MT2, MT3	+Z, −Z	T 2(2)
F 11	Milling (OP14)	MT 2, MT3	+Y, −Z	T 6(16), T7(14)
F 12	Milling (OP15)	MT 2, MT3	+Y	T 5(5), T12(5)
F 13	Drilling (OP16)	MT 1, MT2, MT3	+Y	T 1(2)	OP 17
	Tapping (OP17)	MT 1, MT2, MT3		T 8(2)
F 14	Milling (OP18)	MT 2, MT3	+Y	T 5(16), T6(14), T7(12)	OP 1, OP2, OP8_OP10, OP13_OP18

MT: machine tool; TAD: tool approach direction; CT: cutting tool.

Table 4.

Operations and candidate machining resources of Part 2.

Feature	Operation (Oper_id)	MT candidates	TAD	CT candidates (runtime/min)	Precedence constraints (processed before)
F 1	Milling (OP1)	MT 2, MT3	+Y	T 5(13), T6(11), T7(9)	OP 2_OP20
F 2	Milling (OP2)	MT 2, MT3	−Z	T 5(14), T6(12), T7(10)	OP 12_OP16
F 3	Milling (OP3)	MT 2, MT3	+X	T 5(10)
F 4	Drilling (OP4)	MT 1, MT2, MT3	+Z, −Z	T 2(4)
F 5	Milling (OP5)	MT 2, MT3	+X, −Z	T 5(13, 14), T6(10, 10)	OP 4, OP7
F 6	Milling (OP6)	MT 2, MT3	+Y	T 6(12), T7(10)	OP 12, OP13, OP14
F 7	Milling (OP7)	MT 2, MT3	−A	T 6(12), T7(10)	OP 8, OP9, OP10
F 8	Drilling (OP8)	MT 1, MT2, MT3	−A	T 2(3), T3(3), T4(4)	OP 9, OP10
	Reaming (OP9)	MT 1, MT2, MT3		T 9(5)	OP 10
	Boring (OP10)	MT 3, MT4		T 10(5)
F 9	Milling (OP11)	MT 2, MT3	−A	T 6(11), T7(9)	OP 12, OP13, OP14
F 10	Drilling (OP12)	MT 1, MT2, MT3	−Z	T 2(4), T3(4), T4(6)	OP 13_OP16, OP19, OP20
	Reaming (OP13)	MT 1, MT2, MT3		T 9(6)	OP 14, OP15, OP16
	Boring (OP14)	MT 3, MT4		T 10(6)	OP 15, OP16
F 11	Drilling (OP15)	MT 1, MT2, MT3	−Z	T 1(7)	OP 16
	Tapping (OP16)	MT 1, MT2, MT3		T 8(7)
F 12	Milling (OP17)	MT 2, MT3	−X	T 5(10), T6(8)
F 13	Milling (OP18)	MT 2, MT3	−X, −Z	T 5(16), T6(14), T7(12)	OP 4, OP17
F 14	Reaming (OP19)	MT 1, MT2, MT3	+Y	T 9(3)	OP 20
	Boring (OP20)	MT 3, MT4		T 10(3)

MT: machine tool; TAD: tool approach direction; CT: cutting tool.

Parameter setting of the ACO algorithm

For the proposed ACO approach, the main parameters include the number of ants Q; the parameters $w,\alpha ,\beta$ ; the evaporation rate of the pheromone $\rho$ and mutation probability $P_{\mathrm{mu}}$ . Different combination of these parameters has significant influence on the astringency, the efficiency and the result. For the update of pheromone, the probability $P_{\mathrm{bs}}$ determines how greedy the search is. Therefore, the experiments with orthogonal arrays (OAs) are involved to discover the most robust and nearly optimal combination of parameters, as shown in Table 5. The iterative time is set as 1000 and in order to find the most robust set of parameters, each run is repeated 10 times to obtain average values ( $T{E}^{*}$ ). The orthogonal table $\mathrm{OA}(32,4^7)$ and its results, which contains 32 runs of experiments, are listed in Table 6.

Table 5.

Factor levels of $\mathrm{OA}(32,4^7)$ .

Factors	A (Q)	B (w)	C (α)	D (β)	E (ρ)	F (Pmu)	G (Pbs)
Level 1	50	50	1	1	0.1	0	0
Level 2	100	100	2	2	0.15	0.05	0.35
Level 3	200	200	3	3	0.2	0.1	0.65
Level 4	250	400	4	4	0.25	0.2	1

Table 6.

Orthogonal table and results of $\mathrm{OA}(32,4^7)$ .

Run	Factors
	A	B	C	D	E	F	G	$T{E}^{*}$
1	50	50	1	1	0.1	0	0	115917.1
2	50	100	2	2	0.15	0.05	0.35	104245.5
3	50	200	3	3	0.2	0.1	0.65	105454.8
4	50	400	4	4	0.25	0.2	1	110564.3
5	100	50	1	2	0.15	0.1	0.65	97678.33
6	100	100	2	1	0.1	0.2	1	103075.2
7	100	200	3	4	0.25	0	0	130857.5
8	100	400	4	3	0.2	0.05	0.35	99376.87
9	200	50	2	3	0.25	0	0.35	121614.3
10	200	100	1	4	0.2	0.05	0	96886.53
11	200	200	4	1	0.15	0.1	1	96599.6
12	200	400	3	2	0.1	0.2	0.65	102486.5
13	250	50	2	4	0.2	0.1	1	99384.07
14	250	100	1	3	0.25	0.2	0.65	100749.6
15	250	200	4	2	0.1	0	0.35	126981.7
16	250	400	3	1	0.15	0.05	0	97200.93
17	50	50	4	1	0.25	0.05	0.65	104834.7
18	50	100	3	2	0.2	0	1	132517.4
19	50	200	2	3	0.15	0.2	0	115772.2
20	50	400	1	4	0.1	0.1	0.35	104,843
21	100	50	4	2	0.2	0.2	0	102620.7
22	100	100	3	1	0.25	0.1	0.35	110443.7
23	100	200	2	4	0.1	0.05	0.65	106449.4
24	100	400	1	3	0.15	0	1	101275.7
25	200	50	3	3	0.1	0.05	1	113921.4
26	200	100	4	4	0.15	0	0.65	100353.3
27	200	200	1	1	0.2	0.2	0.35	120259.4
28	200	400	2	2	0.25	0.1	0	100389.1
29	250	50	3	4	0.15	0.2	0.35	98479.8
30	250	100	4	3	0.1	0.1	0	101881.9
31	250	200	1	2	0.25	0.05	1	96,777
32	250	400	2	1	0.2	0	0.65	96608.27
R	6930.8	4048.9	6553.8	3509.8	2824.8	22272.9	1740.5

From Table 6, it can be observed that the mutation probability P_mu has a great influence on the final result. Using mutation ( $P_{\mathrm{mu}}=0.05$ ), the diversity is kept and the algorithm can come to a better solution. However, if it is set too high ( $P_{\mathrm{mu}}=0.2$ ), the search will turn into a primitive random search without achieving the lowest or near-lowest EC. Besides, it is also shown that P_bs has an influence on how greedy the search is. With a proper probability ( $P_{\mathrm{bs}}=0.65$ ), the algorithm could reach the lowest result. From the results, when $P_{\mathrm{mu}}=0.05$ , Q = 250, α = 1, β = 1, w = 400, ρ = 0.15 and $P_{\mathrm{bs}}=0.65$ , the algorithm can achieve a better performance, and one of the best IPPS solutions and its corresponding Gannt chart are shown in Table 7 and Figure 7, respectively.

Table 7.

Near-best IPPS solution for Scenario 1.

No.	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
Job	2	2	1	2	2	1	1	1	2	2	1	2	2	1	2	1	1	1	2
Oper id	1	6	12	3	5	5	4	3	2	18	18	7	11	14	17	15	16	17	12
MT id	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	1	1	1
TAD	+Y	+Y	−X	+X	+X	−Y	−Y	−Y	−Z	−Z	+Y	−A	−A	+Y	−X	+Y	+Y	+Y	−Z
CT id	7	7	6	5	6	6	6	6	7	7	7	7	7	7	6	5	1	8	3
No.	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38
Job	1	1	1	2	2	1	1	1	2	2	2	1	2	2	1	2	2	1	2
Oper id	13	1	11	13	14	8	9	10	15	16	4	7	8	9	6	19	20	2	10
MT id	1	1	1	1	4	1	1	4	1	1	1	1	1	1	1	1	3	3	3
TAD	−Z	−Z	−Z	−Z	−Z	−Z	−Z	−Z	−Z	−Z	−Z	−Z	−A	−A	−Z	+Y	+Y	−Z	−A
CT id	2	2	2	9	10	2	9	10	1	8	2	8	2	9	1	9	10	11	10
TE = 84,267, PE = 55,350, TE = 22,367, BE = 3350, JCE = 3200

MT: machine tool; TAD: tool approach direction; CT: cutting tool; PE: processing energy consumption; BE: buffers’ energy consumption; JCE: job-conveying energy consumption; TE: total energy consumption.

Figure 7.

Gannt chart of one of the best solutions for Scenario 1.

Experiment results and discussions

Computation results under different conditions

Since a machining workshop will encounter different situations at different times, in order to verify the practicability of the proposed model and algorithm, three experimental scenarios are set as following and the simulation under each scenario is repeated 10 times:

Scenario 1

All machines and tools are available, and the four different kinds of EC in equation (15) are all taken into account. One of the best results for this scenario is listed in Table 7: the minimum EC is 84,267 W min. It is obvious that the sum EC of $\mathrm{PE}$ and $\mathrm{CTE}$ reaches to 77,717 W min, accounting for 92.2% of the total EC, so it is more effective for the workshop to reduce EC of $\mathrm{PE}$ and $\mathrm{CTE}$ . From the Gannt chart in Figure 7, $M{T}_1$ and $M{T}_2$ are employed with high frequency, while $M{T}_3$ and $M{T}_4$ only machine five processes including $P_2$ and $P_{10}$ of Part 1 and $P_{10}$ , $P_{14}$ and $P_{20}$ of Part 2 due to the processing constraints.

Furthermore, the proposed algorithm is compared with the original ACO and GA optimization approaches in terms of optimum results and convergence, as shown in Figure 8. In GA, the operators are inspired by a natural evolution process, which includes selection, crossover and mutation to improve the populations gradually. The proposed mutation-combined ACO is a hybrid algorithm which consists of mutation operator in GA and ACO. Hence, an enhanced ability of finding an optimum solution can be obtained using the mutation-combined ACO. From the results in Figure 8, it can be clearly seen that GA has a strong convergence but spends more time. The original ACO expends a short time to find an optimal solution but converges too early. Different from these two algorithms, the proposed mutation-combined ACO algorithm can reach a better result after 624 iterations, which means that it is an effective method for the FMS-IPPS problem. However, it can also be clearly seen that the proposed algorithm needs to spend plenty of time to converge, especially for complex problems. This may be related to the mutation process because the mutation process will increase the calculation amount of the algorithm, although it can expand the diversity of the population. This disadvantage of the proposed algorithm needs to be considered in the future research.

Figure 8.

Comparisons of the three optimization approaches for energy conservation.

Scenario 2

All machines and tools are available, but only PE is considered and JCE and BE are ignored. This scenario simulates the conventional optimization method of EC during manufacturing, which merely considers the EC of an operation and its manufacturing resources without taking EC for transportation and buffers into consideration. One of the best results for this scenario is listed in Table 8, and the minimum EC is 77,040 W min. According to this plan, the parts change MTs for 21 times. The EC seems lower, but if we add $\mathrm{JCE}$ and $\mathrm{BE}$ needed for this solution, the result will be TE = 120,989 W min, which is much more than the best solution shown in Table 7. So the conventional method did not obtain the optimal solution due to the defects of the model.

Table 8.

Best result for Scenario 2.

No.	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
Job	2	2	2	2	1	1	2	2	1	1	1	1	2	2	2	2	2	1	2
Oper id	1	6	2	11	5	12	5	12	4	18	16	1	18	17	4	7	8	3	9
MT id	2	2	2	2	2	2	2	1	2	2	1	1	2	2	1	2	1	2	1
TAD	+Y	+Y	−Z	−A	−Y	−X	+X	−Z	−Y	+Y	+Y	+Z	−Z	−X	+Z	−A	−A	−Y	−A
CT id	7	7	7	7	7	6	6	2	5	7	1	2	7	6	2	7	3	6	9
No.	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38
Job	2	2	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	1	1
Oper id	10	3	17	2	8	9	10	6	7	14	13	13	14	19	15	16	20	11	15
MT id	4	2	1	2	1	1	4	1	1	2	1	1	4	1	1	1	4	1	2
TAD	−A	+X	+Y	−Z	+Z	−Z	−Z	−Z	−Z	−Z	+Z	−Z	−Z	+Y	−Z	−Z	+Y	−Z	+Y
CT id	10	5	8	11	2	9	10	1	8	7	2	9	10	9	1	8	10	2	5
$\mathrm{TE}=\{\begin{array}{c}77,040,\mathrm{if}\mathrm{JCE}\mathrm{and}\mathrm{BE}\mathrm{are}\mathrm{excluded}\hfill \\ \multicolumn{1}{c}{120,989,\mathrm{if}\mathrm{JCE}\mathrm{and}\mathrm{BE}\mathrm{are}\mathrm{included}}\end{array}$ , PE = 54,723, CTE = 22,316, BE = 35,550, JCE = 8400

MT: machine tool; TAD: tool approach direction; CT: cutting tool; JCE: job-conveying energy consumption; BE: buffers’ energy consumption; PE: processing energy consumption; TE: total energy consumption; CTE: cutting tool energy consumption.

Scenario 3

• Assume that $M{T}_3$ and $T_7$ are not available for some reason. This scenario mainly contraposes emergencies during processing. One of the optimal solutions is listed in Table 9. The total EC is 94,400 W min, which is a little more than that for Scenario 1, and the EC of buffers and job-conveying increase obviously, while the PE and CT EC have changed little.

Table 9.

Best result for Scenario 3.

No.	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
Job	2	2	1	2	2	1	2	1	2	2	1	2	2	2	1	1	2	1	1
Oper id	1	6	12	11	5	5	3	4	18	2	3	7	8	12	1	2	17	18	8
MT id	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
TAD	+Y	+Y	−X	−A	+X	−Y	+X	−Y	−Z	−Z	−Y	−A	−A	−Z	−Z	−Z	−X	+Y	−Z
CT id	6	6	6	6	6	6	5	5	6	6	6	6	2	2	2	11	5	5	3
No.	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38
Job	1	2	1	2	2	2	1	1	2	2	2	1	1	2	1	1	1	2	1
Oper id	15	13	14	19	9	10	9	10	20	14	15	16	17	16	7	13	11	4	6
MT id	2	2	2	2	2	4	2	4	4	4	1	1	1	1	1	1	1	1	1
TAD	+Y	−Z	−Z	+Y	−A	−A	+Z	+Z	+Y	−Z	−Z	+Y	+Y	−Z	−Z	−Z	−Z	−Z	−Z
CT id	12	9	6	9	9	10	9	10	10	10	1	1	8	8	8	2	2	2	1
TE = 94,400, PE = 62,832, CTE = 22,108, BE = 9460, JCE = 1600

MT: machine tool; TAD: tool approach direction; CT: cutting tool; PE: processing energy consumption; BE: buffers’ energy consumption; JCE: job-conveying energy consumption.

In order to perform further analysis on manufacturing EC, different energy efficiency indicators under different conditions are compared in Table 10. It can be clearly seen that $\mathrm{PE}$ contributes to the largest portion of $\mathrm{TE}$ . By choosing different sets of manufacturing resources, it could vary significantly, as well as $\mathrm{JCE}$ , which will decrease through reducing changing times of MTs. As for $\mathrm{BE}$ , it fluctuates strongly as the scheduling plan changes. Moreover, Scenario 1 has the highest FSEE. The PAEEs of the parts of Scenario 1 and Scenario 3 are nearly equal, while the energy efficiency of Part 2 of Scenario 2 is a little better than that of Part 1. In addition, the MTEE of $M_4$ of Scenario 1 is the highest, while that of $M_1$ is only 0.87, which means some actions need to be taken to decrease its EC of the idle state and tool changing state.

Table 10.

Energy efficiency comparisons of different scenarios.

Indicator	Scenario 1	Scenario 2	Scenario 3
TE	84,267	120,989	94,400
PE	55,350	54,723	62,832
CTE	22,367	22,316	22,108
BE	3350	35,550	9460
JCE	3200	8400	1600
FSEE	0.657	0.452	0.655
$\mathrm{PAE}{E}_1$	0.685	0.634	0.724
$\mathrm{PAE}{E}_2$	0.684	0.645	0.728
$\mathrm{MTE}{E}_1$	0.870	0.402	0.948
$\mathrm{MTE}{E}_2$	0.957	0.866	0.896
$\mathrm{MTE}{E}_3$	0.950	–	–
$\mathrm{MTE}{E}_4$	0.995	0.24	0.773
Makespan	258	304	308

PE: processing energy consumption; BE: buffers’ energy consumption; JCE: job-conveying energy consumption; FSEE: feasible solution energy efficiency; PAEE: part energy efficiency; MTEE: machine tool energy efficiency.

Comparisons and discussions

As stated above, for the “process planning then scheduling” approach, the process plan is usually determined before scheduling with the assumption that all manufacturing resources are available. Specifically, the procedure of the process planning then scheduling method has two steps: (1) for each part, using the algorithm to get the most energy-saving process plan by assigning MTs, CTs and SUs for each operation and (2) then using the algorithm a second time to scheduling (sequencing) the determined process plan. However, this hierarchical method neglects their relevance clearly existing between them. To compare it with the IPPS method, the results of the “process planning then scheduling” method are obtained, as listed in Table 11. It is obvious that the lowest EC obtained by the “process planning then scheduling” method is 93,249 W min, which increases 10.7% compared with that of Scenario 1. Furthermore, the EC of buffers increases obviously from 3350 to 9130 W min, which means that the waiting time of the parts is much longer than the IPPS solution.

Table 11.

Result of the “process planning then scheduling” method.

No.	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
Job	2	2	1	1	2	1	2	2	2	2	2	1	1	2	1	1	2	2	2
Oper id	1	6	5	4	5	1	11	2	12	18	17	12	18	3	14	15	19	13	4
MT id	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
TAD	+Y	+Y	−Y	−Y	+X	+Z	−A	−Z	−Z	−Z	−X	−X	+Y	+X	+Y	+Y	+Y	−Z	−Z
CT id	7	7	7	5	6	2	7	7	3	7	5	6	7	5	7	5	9	9	2
No.	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38
Job	2	2	1	1	2	2	2	1	2	1	2	2	2	2	1	1	2	2	1
Oper id	19	20	6	1	4	13	14	2	17	14	7	8	9	10	13	11	15	16	7
MT id	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3
TAD	+Y	+Y	−Z	−Z	−Z	−Z	−Z	−Z	−X	+Y	−A	−A	−A	−A	−Z	−Z	−Z	−Z	−Z
CT id	9	10	1	2	2	9	10	11	6	6	6	3	9	10	2	2	1	8	8
TE = 93,249, PE = 60,002, CTE = 22,117, BE = 9130, JCE = 2000

MT: machine tool; TAD: tool approach direction; CT: cutting tool; PE: processing energy consumption; BE: buffers’ energy consumption; JCE: job-conveying energy consumption.

In addition, it must be pointed out that there are still several weaknesses which need to be studied in the future work:

1. The EC modeling for FMS-IPPS is only suitable for normal manufacturing processes, but the real manufacturing process is complex, which includes machining tool breakdown, the changing of orders and so on. So the proposed models need to be improved to solve problems of practical production through considering the EC fluctuation caused by temporary changes in production.

2. The input data about EC were obtained from the shop floor, which includes the idle power, average operating power and tool changing energy of MTs, manufacturing EC of cutting tools. However, the EC results shown in Tables 7–9 are all theoretical results, and it is difficult to obtain the actual processing results due to the high complexity of manufacturing processes. Therefore, an EC monitoring platform of shop floor needs to be established to collect the real data about EC, which will be carried out in the future work.

Conclusion

A new energy-saving approach of FMS has been proposed based on energy evaluation model for FMS-IPPS in the article. First, complying with feature precedence and other technological requirements, FMS-IPPS is mapped as a unique ATSP of which operations are provinces and COs are cities belonging to different provinces. To evaluate the performance of each solution of the ATSP, EC evaluation criteria are established, and three energy efficiency indicators are also provided to perform further analysis on manufacturing EC, that is, PAEE, MTEE and FSEE. Then, a mutation-combined ACO algorithm is proposed to solve the FMS-IPPS problem which combined roulette and mutation selection methods to pick out the next CO after calculating probabilities of all the available COs. The pheromone trails associated with edges are released by the so-far-best ant or the iteration-best ant probabilistically to both keep the search directed and avoid converging to the local best. Finally, a case which contains two parts, four MTs and more than 10 CTs is studied to demonstrate the feasibility and applicability of the proposed model in three different scenarios. Compared with the proposed method, EC obtained by the process planning then scheduling approach increases 10.7%.

Future research in this area will include the extension of the FMS-IPPS model to other more complex aspects, such as dynamic IPPS and batch IPPS for energy conservation. On the other hand, the ACO algorithm will be reinforced by other algorithms to generate better solutions and more algorithms will be used to be compared to obtain much better solutions, such as GA, PSO, honey-bee mating optimization or teaching learning–based optimization algorithm.