Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 9803497 | 19 pages | https://doi.org/10.1155/2019/9803497

A Low-Carbon Design Method Integrating Structure Design and Injection Process Design for Injection Molding Machines

Academic Editor: Akhil Garg
Received22 Mar 2019
Accepted19 Aug 2019
Published17 Sep 2019

Abstract

In the past decades, environmental problems are widely concerned and solved. However, as most solutions, they are methods of “end-of-pipe” treatment which are inefficient and of high cost. Low-carbon design (LCD) is a novel way to solve the problem of pollution emissions at source. Injection molding machine (IMM) as important manufacturing equipment has been widely used in many industries. In the pursuit of high-quality plastic products, the environmental qualities of IMM are often neglected. To achieve low carbon of IMM at source, a LCD method is proposed combining the structure design and injection process design for IMM. At first, LCD decision variables are determined based on interval number theory. Subsequently, the IMM structural carbon emissions and injection molding process carbon emissions are calculated, respectively. Based on this architecture, the carbon emission mathematical model is constructed. To solve the multiobjective optimization problem, the improved strength Pareto evolutionary algorithm based on epsilon dominance (E-SPEA-II) is used, and the design result schemes are sorted using the multiattribute decision-making method for intervals. Finally, the validity of this method is demonstrated by an IMM injection component-integrated low carbon design (ILCD) example.

1. Introduction

Injection molding machine (IMM) is the main equipment for plastic production, which maintains a complex system integrating mechanical, electrical, and hydraulic properties [1]. The production of plastic products is 7.7 × 107 tons in 2016 in China. The annual growth of the plastic-processing industry is 8%. The annual growth rate is in the forefront of the light industry, 1.21% higher than others in the light industry and 2% higher than others in the Chinese national industry [2]. However, as a kind of high-energy consumption equipment, the whole life cycle of an IMM is accompanied by a large number of environmental impacts. The ecological environment properties of plastic products have been considered as Chinese national VOC (volatile organic compounds) emissions and PM 2.5 index. Carbon emission is one of the most important green sustainable development indicators among these ecological environment properties.

A schematic of a typical inline reciprocating screw IMM with hydraulic drive discussed in this paper is shown in Figure 1. This IMM is a mechatronics equipment which consists of injection molding components (3, 4, 10, 13), feeding components (5), clamping components (1, 2, 14, 15), heating components (12), and hydraulic components (6, 7, 8, 9, 11).

Among the existed LCD researches, scholars focus on two major fields, product structure LCD and product process LCD. Plenty of research studies have been conducted in the study field of product structure LCD [4, 5]. Song and Lee [6] divided the product into several structure parts to analyze the greenhouse gas (GHG) emissions separately. They developed a design system which allowed quick calculation of the GHG emissions of a product that helped product structure low-carbon design. Zhang et al. [7] considered the product as the organic combination of connection units. By analyzing the parts’ connection characteristics, they obtained the carbon emissions of each product connection unit. Lu et al. [8] proposed a selection method based on the characteristic of carbon emissions for low-carbon structure design. The total carbon emissions distribution of every product part can be achieved by the proposed method. As typical mechatronics equipment, IMM is widely used in many applications. The carbon emissions of the IMM structure grow with increase of the complexity of the IMM structure [911]. A number of scholars focus on improving the IMM structure LCD method. Agazzi et al. [12] proposed a new design method to determine the structure of IMM cooling runner in the three-dimensional model. The optimal fluid temperature distribution along cooling is determined. Li et al. [13] used the method of configuration space to optimize the layout design of IMM cooling channel. This method overcomes the limitation of invariable topological structure in the previous design. Both methods mentioned above helped to decrease the carbon emissions of the IMM cooling system.

In the study field of product process LCD, Tan et al. [14] constructed a multiobjective cutting fluid decision-making model considering the quality, cost, and environmental impact for machine tools process planning. It is verified by an example of cutting fluid selection. Sivapirakasam et al. [15] proposed a method hybridizing Taguchi and fuzzy TOPSIS to solve multiresponse parameter optimization problems in green manufacturing. Jiang and Zhang [16] established a machine tool processing selection model for low carbon emissions. A vector projection method is developed in their study to justify machine tool processing alternatives. In addition, injection molding process is a typical reciprocating process. Every step of an injection molding process cycle has been accompanied by carbon emissions [1719]. Scholars have been studied for years to find the efficient way to reduce the injection molding process carbon emissions. Meekers et al. [20] proposed a method to optimize the injection molding process in order to reduce energy consumption as well as ensuring the part quality. They found that the cooling time had the most sensitive impact on energy consumption. To verify the effect on energy consumption of optimized process parameters, Tranter et al. [21] generated a novel experimental schedule. The result showed that the cooling time incurred the largest variation in energy consumption during the cycle time injection molding process.

LCD problems usually can be considered as the optimization problems. There are two major solutions to the optimization problems. They are evolution algorithm and machine-learning algorithm. Wang et al. [22] proposed a new automatic niching technique based on the affinity propagation clustering and designed a novel niching differential evolution algorithm. It helped to solve the multimodal optimization problems. Liu et al. [23] proposed an adaptive sorting-based environmental selection strategy. An adaptive promising subpopulation sorting-based environmental selection strategy is provided for problems which may have irregular Pareto fronts. Chen et al. [24] proposed an automated neural network search method in optimizing the power output of cleaner energy production systems.

Previous researches have made some achievements in LCD of products. However, the product structure design [25, 26] and the product manufacturing process design [27, 28] are separated in the LCD process. In the structural design of IMMs, there is several uncertainties in the injection process plan because that the injection molding process has not yet been carried out. In the process of injection molding process design, the structure of IMMs has been determined and it could not be modified according to the requirements of low-carbon design. The separation makes the LCD of IMMs lack the overall situation, which affects the accuracy of LCD results. Therefore, an integrated low-carbon design (ILCD) method is proposed combining the structure design and injection process design for IMMs.

This paper proposes a low-carbon design method integrating structure design and injection process design for IMMs. The framework of the new low-carbon design method is introduced in Section 2. Then, a carbon emissions calculation model is proposed in Section 3 in detail. Section 4 presents an optimization method to solve the carbon emissions problem. It helps to find the most proper scheme for IMMs ILCD. Section 5 describes a case study of DSH280/750TW IMM injection component low-carbon design problem. And, the discussion of the results is shown in Section 6. Finally, Section 7 concludes this paper.

2. Framework of the ILCD Method for IMM

The framework of the ILCD method for IMM, which is shown in Figure 2, is illustrated in this section. At first, the decision variables of IMM LCD problem are determined based on the LCD requirement. Considering the uncertainty of ILCD, the related parameters are interval expressed for further dominance comparing. Based on that, the total carbon emissions are divided into structural carbon emissions and injection molding process carbon emissions and, respectively, calculated. In structural carbon emissions calculation part, the raw material carbon emissions are calculated using the Kriging model to fit the unknown structural carbon emissions response; at the meantime, the carbon emissions when manufacturing the IMM are calculated based on the energy consumption model during machine tools manufacturing. On the other side, in injection molding process carbon emissions calculation part, it is divided into four subprocess calculations, of which the energy consumptions are calculated first. Subsequently, the energy consumptions are converted to carbon emissions. In addition, it will be meaningless if the quality of injection products is not guaranteed in ILCD; thus, the plasticizing capacity is considered as the extra objective in integrated carbon emissions calculation. Above all, the carbon emissions mathematical model can be developed as a three-objective optimization problem, which can be optimized by an improved evolutionary algorithm. Due to the design parameters being interval expressed, the optimal results of Pareto frontier are also the interval numbers. A dynamic multiattribute decision-making method is used to sort the optimal populations. According to the sorting result, ILCD optimal scheme is obtained.

3. ILCD Carbon Emissions Calculation Model

3.1. LCD Decision Variables

In order to develop a carbon emissions calculation model for the integrated LCD, some design parameters should be determined as LCD decision variables. In the IMM life cycle, different LCD decision variables from each stage are stored in set , which is shown as follows:where presents the decision variables set at kth stage of the life cycle of IMM and u is the amount of decision variables at kth stage. For example, at the material stage, one of the decision variables is {nonreturn valve (45CrMo) 15 kg, heating bands (stainless steel) 5 kg, and injection ram (HT200) 16.5 kg}. Then, at the manufacturing stage, these raw materials are manufactured with different processes.

In the very beginning of the IMM scheme design stage, the process parameters are uncertain. Therefore, intervals are used to express these uncertain design parameters. The ternary interval A is represented by two numbers, which is expressed as follows:where represents the lower limit of and represents the upper limit of .

Definition 1. If  =  and  = , the distance between A and B can be recorded as d(A, B):

Definition 2. If  =  and  = , is the maximum interval of A and B, recorded as , where and .
As is the maximum interval of A and B, a conclusion can be drawn that and .

Definition 3 [29, 30]. If  =  and  = , is the maximum interval of A and B, the relative dominance coefficient of can be expressed as follows:Analogously, the relative dominance coefficient of can be expressed as follows:When two intervals are compared, the one with higher relative dominance is usually chosen by decision makers. According to that, the relative dominance is used to compare the dominant relationship between two intervals. ILCD is conducted as a multiobject problem (MOP):where x is the decision variable with u dimensions in decision space . is the th objective function with interval parameter and is the th component of the interval parameter vector . can also be expressed as an interval .

Definition 4. For , if , and that makes , then dominates , expressed as .

Definition 5. For , if that makes , then is called the Pareto optimum solution.

3.2. IMM Structural Carbon Emissions Calculation

IMM structural carbon emissions consist of two parts, the carbon emissions of IMM raw material and the carbon emissions of manufacturing an IMM process. Sun [31] proposed a product carbon emissions calculation method which also divides the product carbon emissions into two parts:where represents the IMM structural carbon emissions consumption; the unit is ()kg. represents the carbon emissions consumption of the IMM raw material; the unit is ()kg. represents the carbon emissions consumption during process manufacturing the IMM; the unit is ()kg. represents the quantity consumption of the ith kind of material for the jth component; the unit is kg. represents the carbon emissions factor of the ith kind of material; the unit is ()kg/kg. represents the amount of material types, while represents the amount of components. represents the time consumption of the kth process for manufacturing the jth component; the unit is h. represents the carbon emissions factor of the kth process for manufacturing the jth component; the unit is ()kg/h. represents the amount of manufacturing processes.

However, at the beginning of IMM LCD, plenty of structural parameters are uncertain. The quantity consumption of IMM is difficult to be calculated. In order to overcome this problem, the Kriging approximate model is used for fitting the carbon emissions consumption of IMM raw material (). Because of its strong ability to remove noise and high optimization efficiency, the Kriging approximate model is widely used in describing nonlinear problems that are difficult to obtain analytical expressions [32, 33]. Suppose that can be expressed as follows:where represents the decision variables set of IMM LCD problem. represents the estimated carbon emissions consumption of the IMM raw material. is the undetermined coefficient of the estimated function. represents a random error function which can be expressed in a variety of forms. In this paper, is expressed as the Gauss distribution in terms of expectation 0 and variance . The covariance of the can be expressed as follows:where R represents the correlation matrix. R(, ) represents the correlation function of any two samples. p, q = 1, 2, …, , where represents the amount of data in the samples. R(, ) can be expressed in a variety of forms. For the consistency, R(, ) is expressed as Gauss function:where represents the sth decision variable and the amount of decision variables is k. According to the Kriging model theory, the estimated response value of the unknown point x can be expressed as follows:where represents the estimated value, which can be expressed as follows:where K represents the column vector of response value of sample data. represents the unit column vector. represents the correlation vector between the sample point and the prediction point, which can be expressed as follows:

The estimated value of variance can be expressed as follows: can be obtained by maximum likelihood estimation as

Then, initial samples were selected and finite element analysis was used to obtain injection equipment volume. The Latin hypercube sampling method is used to randomly select the sample points in the global design space. In order to ensure the fitting accuracy of the overall output response, the correlation coefficient is calculated. The closer the correlation coefficient is to 1, the higher the global accuracy of the model is:where represents the output response value of the ith sample point obtained from the finite element analysis software. The amount of sample points is . represents the response value obtained from the Kriging model. is the average value of the sample points. In addition, the relative maximum of absolute error is calculated. The closer the relative maximum of absolute error is to 0, the higher the local accuracy of the model is:where represents the standard deviation of sample points. Only if and meet the requirement of precision, the iterations are complete and the final fitting function is obtained. On the other hand, is firstly calculated as energy consumption. The components and parts of IMM are manufactured by using computerized numerical control machine tools (CNCMT). The energy consumption model during machine tools manufacturing has already been built [34, 35]. The total energy consumption in manufacturing an IMM is divided into five parts:where represents the total energy consumption in manufacturing an IMM. represents the energy consumption when the CNCMT is standby. represents the energy consumption during the CNCMT spindle rotation. represents the energy consumption during the CNCMT X-, Y-, and Z-axis feeding. represents the energy consumption during the CNCMT tool change. represents the energy consumption during the CNCMT A, B, C-axis rotation. The relationship between energy consumption and carbon emissions can be mathematically expressed aswhere represents the consumption of the kind of energy and represents the carbon emissions coefficient of the kind of energy, . represents the total time of the kind of energy consumption during single injection molding cycle, h. There are u kinds of energy counted when the carbon emissions of are calculated.

Above all, the IMM structural carbon emissions can be calculated as follows:

3.3. Carbon Emissions Calculation during Injection Molding Process

Injection molding is a repetitive process in which melted polymer (plastic) is injected into mold cavities, where it is held under certain pressure until it is removed into a solid state. Therefore, the whole injection molding process can be regarded as the cycle of multiple same processes, which is shown in Figure 3. It takes a 60-seconds injection molding cycle process as an example.

The carbon emissions during injection molding process are firstly calculated as energy consumption. The heating mode of IMM is the main factor affecting energy consumption in injection molding process. There are several heating modes used in IMM among which the electromagnetic induction heating is the most wildly used heating mode. Owing to that, the energy consumption is calculated as it is electromagnetic induction heating IMM. The energy consumption during a whole injection molding process, , which consists of four parts as shown in Figure 4 in grey can be calculated as follows:where represents the energy consumption during the plasticizing process, represents the energy consumption during the packing and holding process, represents the energy consumption during the cooling process, and represents the energy consumption during the ejection process. can be calculated as four individual parts as follows:where represents the energy consumption of heating bands, which can be calculated as

represents the energy consumption of the heating extruder barrel, which can be calculated as

represents the energy consumption of the heating injection screw, which can be calculated as

represents the energy consumption of the hydraulic motor, which can be calculated as

The schematic of a typical metering-type screw is shown in Figure 4 along with the main screw design parameters.

Based on the experience, the packing pressure is 75% of the injection pressure [36] so that the energy consumption during the packing and holding process, , can be calculated as follows:where represents the change ratio in unit volume of the polymer for a given decrease in temperature; the unit is . The coefficient of the cooling performance is regarded as the theoretical maximum [37]:

It is studied that the energy consumption during ejection process is 25% of the whole process [38] so that the energy consumption during the ejection process, , can be calculated as follows:

Above all, the energy consumption during an injection process cycle can be calculated as

On this basis, the carbon emissions in a single cycle of injection molding can be calculated as similar as equation (19):

Combining equations (30) and (31), the carbon emissions in a single cycle of injection molding can be expressed as

3.4. Carbon Emissions Calculation Extra Objective

The basic design purpose of an IMM is to increase the injection molding performance. It will be practically significant to analysis injection molding equipment carbon emission only if the quality of injection products is guaranteed. Plasticizing capacity is an important index to evaluate the quality of plasticizing products in injection molding equipment. If the plasticizing capability is too low in relation to the shot size required, the chances are that the injected plastic will not be completely melted. Therefore, the conveying capacity of the screw metering section is used to calculate the plasticizing capacity of the screw as the carbon emissions calculation extra objective, which can be express as follows:

3.5. Carbon Emissions Mathematical Model

Based on the above mentioned, equation (6) is rewritten as equation (34). In the carbon emissions mathematical model, is taken as the decision variables set; are taken as the three optimization objectives. It is worth mentioning that as an extra objective is required to obtain the maximum value in ILCD process. However, for the convenience of multiobjective decision at later stage, is set as . Besides, in many certain IMM LCD cases, there is an association between those known design parameters. These variables are limited as constraint conditions in the carbon emissions mathematical model:

4. Optimization Method

4.1. Optimization Based on E-SPEA-II

Zitzler et al. [39] proposed the improved strength Pareto evolutionary algorithm (SPEA-II). The advantage of this algorithm is that the less parameter artificially set, the more efficient optimization performance, the higher calculation speed. In addition, this algorithm can obtain the uniform distribution of the Pareto front. However, it still lacks in the maintenance of the target space and convergence speed. In order to overcome these problems, we introduce a novel SPEA-II powered by epsilon dominance which is used by Saxena et al. [40] implemented within the framework of NSGA-II. The improved algorithm flow is shown in Figure 5. The details of the algorithm is as follows:

Step 1. Initialize new population and an external set ; set the current generation as gen = 0, the maximum generation as , the crossover probability as , the mutation probability as ; set N as the number of the evolution population, .Step 2. Calculate the basic fitness values, , of individuals in current population and external set as

where represents the conjunctive symbol in mathematical logic, represents the dominant relationship, and is the pressure value of individual which represents the individual number of as follows:

where (x) represents the cardinality function of set x. Considering the influence of the constraint violation and the intensive rejection between individuals, the final fitness values can be calculated as

where represents the dispersion index of k-nearest neighbor algorithm [41].Step 3. Copy the nondominated individuals from current population and external set into next external set . If the number of individuals in is larger than N, archive truncation procedure to reduce the individuals number of , and if the number of individuals in is less than N, fill with the current nondominant individuals of and .Step 4. If , output the individuals in as Pareto optimal solution; otherwise, go to Step 5.Step 5. Perform binary tournament selection in to select the optimal fitness individuals filling into the mate pool.Step 6. Compare the individuals in and , select the optimal one into based on dominant relationship. If there is no dominant one between them, randomly choose one to crossover and mutate to generate new individuals in . Meanwhile, update : =  and go to Step 2.

4.2. Multiattribute Decision Making Method for Intervals

After optimization by evolution algorithm, the Pareto optimal solution set is obtained. However, there are three s’s in the ILCD problem, the multiattribute decision-making method should be used to weigh the importance of each objective. TOPSIS (technique for order preference by similarity to an ideal solution), proposed by Yoon and Hwang [42], is a widely used method for multiattribute decision. However, in this paper, the ILCD problem is based on interval numbers and the elements in the decision matrix are intervals. In order to solve the problems, a multiattribute decision-making method for interval numbers is proposed in this paper. After the carbon emissions model has been optimized, several optimization schemes are obtained as shown in matrix D:

In the decision matrix D, are the final design schemes optimized by the evolution algorithm, where pn is the number of populations. are the three optimization objectives. The interval represents the optimization objective of uth scheme. represents the weight of the optimization objective. The specific steps of A are as follows:Step 1. Normalize the decision matrix D to normalization decision matrix , where represents the interval which is normalized:Step 2. Considering the difference between the importance of the optimization objectives, the normalization decision matrix is weighted to obtain the weighted normalization decision matrix :where is the weight of the optimization objective.Step 3. Calculate the positive ideal solution and the negative ideal solution as follows:where U is associated with the benefit criteria, while V is associated with the cost criteria. The maximum and minimum of are calculated based on the relative dominance coefficient RD determined in definition 3. Take as an example. (represents the relative dominance that ) is obtained by comparing the elements from . Hence, the complementary matrix is constructed as , where , and :According to the order relationship of , the maximum and minimum of are obtained.Step 4. Calculate the distance between every scheme point and the positive ideal solution point (or the negative ideal solution point, respectively):Step 5. Calculate the relative closeness .Step 6. The ILCD schemes are sorted based on the relative closeness . In the ILCD carbon emissions optimization problem, the solution is to minimize all three objectives. So the smaller the relative closeness is, the better the ILCD scheme is.

5. A Case Study: Structure Optimization Design of the IMM Injection Component

In this section, a case study is conducted in an IMM injection component LCD process, which is provided by Zhejiang Sound Machine Manufacturing Co. Ltd. The model of the IMM is DSH280/750TW. The main design parameters of DSH280/750TW IMM injection component is presented in Table 1 as design requirement. The details of the 3D model are shown in Figure 6. According to design experiment and design requirement, eight parameters are chosen as ILCD decision variables. The ILCD decision variables are shown in Table 2 which consists of four structure design parameters and four injection process parameters.


Parameter

Value5025 : 14714283654815935045Open nozzle5
Unitmmcm3gg/scm3/sMPar/minmm


Parameter

Unitmmmm°mmKKss

After the ILCD requirement and decision variables are determined, the ILCD carbon emissions calculation model can be constructed. The Kriging model is used in DSH280/750TW injection component carbon emissions calculation. There are four structure design parameters chosen in Table 2 as decision variables which are . Twenty-six initial sample points are obtained based on design cases to fit the volume of the barrel and screw which is sampled in 3D modeling software. The Kriging model iterative process data of injection component carbon emissions are shown in Table 3. According to equations (8)–(17), the volume of the barrel and screw of IMM can be expressed as


Number of iterationsNumber of samples (%) (%)

0260.61520.4263
1410.74320.3412
2530.80140.2581
3600.85030.1387
4650.88210.0945
5680.91420.0695
6690.93090.0652
7700.93520.0623

The raw material of the barrel and screw of DSH280/750TW IMM is HT300 carbon steel. The density of HT300 carbon steel is 7300 kg/m3, and according to IPCC, the carbon emission of HT300 carbon steel per kilogram is 1.06 kg. So the carbon emissions consumption of IMM raw material can be expressed as

On the other hand, are firstly calculated as energy consumption. The injection screw 2D process sketch of DSH280/750TW is shown in Figure 7 which contains geometric parameter information and surface roughness information. And the machining process card of DSH280/750TW injection screw is shown in Table 4. As in this ILCD case, the decision variables have been already determined and the related process steps are extracted in Table 5, so that the energy consumption is calculated in each related process step. According to equation (19), the carbon emissions of manufacturing the DSH280/750TW injection component can be expressed as (the current situation of electric power structure in China of 2017 is shown in Table 6 along with the corresponding carbon emission coefficient):


Machining process cardProduct modelDSH280/750TW
Component nameInjection screw
Material38CrMoAlARough typeCastingDimension of rough
No.Work procedure namesProcess contentsEquipmentClamping

10LatheRough turning the cylinder of C620-1 horizontal lathe3-jaw chuck
20LatheFinish turning the cylinder of C620-1 horizontal lathe3-jaw chuck
30ModulationModulating to HB240-270
40LatheRough turning the cylinder of C620-1 horizontal lathe3-jaw chuck
50LatheFinish turning the cylinder of C620-1 horizontal lathe3-jaw chuck
60LatheFinish turning the cylinder of C620-1 horizontal lathe3-jaw chuck
70LatheFinish turning the cylinder of C620-1 horizontal lathe3-jaw chuck
80MillingRough milling the cylinder of X53T vertical milling machineTonger
90MillingFinish milling the cylinder of X53T vertical milling machineTonger
100Check


OperationFeature contentsProcessing parametersMachine toolEnergy consumption

Turning (No. 10, No. 11)Rough turning the cylinder of , , , C620-1 horizontal lathe [43]
Finish turning the cylinder of , , , C620-1 horizontal lathe [43]

Milling (No. 80, No. 90)Rough milling the cylinder of , , , X53T vertical milling machine [43]
Finish milling the cylinder of , , , X53T vertical milling machine [43]


Types of powerThermal powerHydroelectric powerWind powerNuclear powerPhotovoltaic power

Proportion (%)68.419.65.34.01.7
Carbon emission coefficient

After that, the IMM injection component carbon emissions and the carbon emissions during injection plasticizing process can be calculated, respectively. Finally, the integrated carbon emissions mathematical model with extra objective is built and equation (34) is rewritten as follows:

The main design parameters of DSH280/750TW IMM are shown in Table 7, in which the decision variables are in grey. Some of the design parameters are unknown, so they are expressed as interval numbers.


ParameterInterval valueUnit

70–100mm
mm
10–50kg
0.46kJ/(kg·K)
1–5L/r
0.5–0.9
10–50kg
0.46kJ/(kg·K)
350r/min
50mm
mm
°
mm
2.5–3.5mm
200–500mm
0.25–0.80kJ/(kg·K)
0.9kJ/(kg·K)
0.16w/(m·k)
50–160K
K
K
0.1–2.5mm
s
s
0.5–20MPa
1000–5000Pa·s
1000–5000Pa·s
20–120MPa

The initial range of the decision variables in ILCD problem is limited according to the design experiment as follows: 500–100 mm, 2.4–5.7 mm, 25–30°, 10–25 mm, 5–60 K, 150–250 K, 60–120 s, and 60–120 s. Equation (50) is optimized based on the method in Section 4. In order to reduce the unnecessary complexity of algorithm, the initial population is set at 50. The crossover probability is 0.9, and the mutation probability is 0.1, the distribution indexes of which are all 20. In this three-objective optimization problem, the generation is set at 100. The Pareto front of the ILCD results after 100 generations is shown in Figure 8. It is worth noting that the Pareto front is also an interval front. In Figure 8, the red points represent the upper limit of the optimized results and the blue points represent the lower limit of the optimized results. The detailed results after 100 generations are shown in Table 8. The number in Table 8 represents the serial number of the population. This means there are fifty optimized ILCD schemes that should be compared using the multiattribute decision-making method mentioned in Section 4.2.


No.

1<79.85, 102.89><4.72, 9.15><34.17, 49.36>
2<80.07, 102.28><5.09, 9.03><43.01, 59.96>
3<77.24, 94.35><5.28, 9.28><37.10, 58.11>
4<77.12, 98.17><4.51, 8.53><38.53, 53.68>
5<78.56, 99.38><4.88, 8.54><36.75, 52.49>
6<75.24, 94.50><5.02, 8.76><35.23, 50.78>
7<81.25, 103.96><4.33, 8.27><35.96, 51.29>
8<75.57, 94.81><5.19, 8.91><38.94, 54.21>
9<82.96, 105.19><4.29, 7.96><37.15, 53.10>
10<78.13, 100.67><5.08, 8.93><39.21, 54.98>
11<74.85, 102.50><4.47, 7.86><37.58, 53.92>
12<82.37, 101.14><5.68, 9.02><40.28, 55.97>
13<78.52, 98.57><5.17, 8.59><38.71, 56.42>
14<76.39, 94.96><4.37, 8.13><36.52, 51.37>
15<80.57, 100.02><4.93, 8.54><39.52, 56.36>
16<73.58, 90.36><4.58, 8.06><36.90, 58.32>
17<75.48, 99.42><4.63, 8.26><39.01, 58.23>
18<78.32, 96.89><5.35, 9.18><37.25, 52.19>
19<79.62, 97.82><5.26, 9.05><38.29, 56.28>
20<80.34, 98.89><5.15, 8.96><39.32, 57.15>
21<79.25, 98.17><5.08, 9.36><40.25, 59.68>
22<80.16, 99.78><5.19, 9.28><41.01, 57.58>
23<77.45, 97.82><4.92, 8.75><38.29, 53.85>
24<78.62, 96.85><5.07, 8.07><40.18, 55.58>
25<76.18, 94.28><4.75, 9.11><36.89, 49.98>
26<77.13, 93.26><5.12, 8.28><39.35, 51.04>
27<75.18, 98.25><4.78, 8.68><37.19, 50.47>
28<74.38, 96.27><4.85, 9.01><38.29, 56.31>
29<78.34, 93.28><4.93, 9.12><38.12, 55.07>
30<80.93, 100.98><5.08, 8.21><36.82, 53.39>
31<74.36, 93.28><4.66, 8.47><37.12, 52.11>
32<79.15, 101.25><5.42, 8.16><41.19, 53.61>
33<78.96, 100.41><5.06, 9.08><40.12, 54.98>
34<80.13, 103.98><5.18, 8.91><39.16, 56.78>
35<75.16, 102.15><4.93, 9.12><37.82, 57.12>
36<77.79, 99.58><4.98, 9.03><40.23, 55.19>
37<78.91, 102.25><5.08, 9.31><41.98, 59.71>
38<76.82, 100.98><5.33, 9.28><40.02, 58.46>
39<81.24, 104.78><4.83, 9.09><39.85, 58.92>
40<80.03, 102.64><4.71, 9.17><42.16, 57.21>
41<79.13, 101.18><5.19, 9.03><37.16, 59.18>
42<79.82, 103.87><5.27, 9.27><38.89, 56.71>
43<77.46, 100.13><4.99, 8.89><42.75, 58.76>
44<81.06, 101.98><4.92, 8.93><37.74, 57.99>
45<75.21, 93.51><4.58, 8.32><37.83, 52.62>
46<79.83, 98.56><5.17, 9.08><41.39, 60.04>
47<79.01, 97.15><4.79, 9.00><40.15, 58.27>
48<77.96, 98.18><5.29, 8.97><39.85, 57.65>
49<76.82, 99.47><5.31, 9.16><42.07, 59.42>
50<78.91, 102.46><4.98, 9.12><38.65, 59.16>

The results in Table 8 are firstly normalized and weighted according to the importance of each optimization objective. In this case, the weight vector is set as W =  = (0.4, 0.4, 0.2). Based on that, the positive ideal solution and the negative ideal solution are calculated:

According to (44)–(46), the relative closeness of each ILCD scheme is calculated. The detailed results of are shown in Table 9. The fifty Pareto optimal schemes are arranged from small to large according to . In this case, the solution is to minimize all three objectives. So the smaller the relative closeness is, the better the ILCD scheme is. In Table 10, the best ten schemes are shown with decision variables.


No.

10.01420.01640.5359
20.00790.01930.7099
30.01320.01600.5482
40.01710.01030.3765
50.01380.01280.4819
60.01640.01100.4018
70.01830.01500.4508
80.01390.01380.4982
90.01900.01650.4654
100.01000.01630.6189
110.01870.01280.4063
120.00540.02220.8034
130.01120.01500.5732
140.02130.00570.2126
150.01180.01480.5553
160.02260.00610.2123
170.01620.01170.4195
180.01130.01670.5952
190.01010.01660.6217
200.00980.01650.6279
210.01030.01730.6254
220.00830.01850.6909
230.01350.01270.4852
240.01430.01280.4726
250.01740.01070.3808
260.01660.01120.4034
270.01560.01130.4188
280.01510.01230.4496
290.01570.01250.4425
300.01230.01530.5534
310.01950.00700.2626
320.01020.01830.6423
330.00970.01670.6321
340.00750.01950.7216
350.01150.01670.5927
360.01110.01550.5825
370.0080