# Sustainable Energy Consumption Model for Textile Industry Using Fully Intuitionistic Fuzzy Optimization Approach

Authors: Sajida Kousar; Urooj Shafqat; Nasreen Kausar; Dragan Pamucar; Yeliz Karaca; Mohammed Abdullah Salman
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

Consumption of renewable energy is on the rise because new technologies have made it cheaper and easier to meet the needs of a long-term energy source. In the present study, the idea of optimal usage of sustainable energy is discussed, taking into consideration the environmental and economic conditions that exist in Pakistan’s textile manufacturing industry. By taking into account the regional potential for the application of renewable energy resources, solar energy generators are taken into consideration, and a fully intuitionistic fuzzy (FIF) textile energy model is constructed. Using the FIF model to determine the optimal distribution of solar energy units resulted in a tolerable number of unused energy units. These units may be returned to the central power supply station, which would save both money and energy.


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## Body

## 1. Introduction

In the textile industry, electricity and thermal energy usage is the most common, as generated at industrial level through nonrenewable energy resources (petroleum, hydrocarbon gas liquids, natural gas, coal, etc.) or obtained directly through the government according to industrial policies. For example, in Pakistan, coal, nuclear, natural gas, hydroelectric, wind, and solar generators are the major electricity developing sources. Specifically, in Pakistan, the two crucial electricity producers are WAPDA (water and power development authority) that generates hydroelectricity and PEPCO (Pakistan electric power company) where distribution companies (DISCOs) under PEPCO work to maintain the path between electricity producers to consumer end by purchasing and selling it to representative area distribution companies. Except for one (K-electric), all these companies are owned by the government, and hence, the government provides different policies to different consuming sectors. In Pakistan, the energy crisis has been a never-ending situation from last two decades due to imbalance production and consumption ratio. At this point of energy crisis, the government of Pakistan and APTMA (All Pakistan Textile Mills Association) are on continuous conflict regarding the availability and favourable rates of energy. Working of textile industry is based on multiple stages like spinning, weaving, rewinding, dying, and so on, which are further subdivided into several other stages (see Figure1).Figure 1
Textile industry processes.All those stages require a huge amount of energy for their processing. Proper energy management planning can help industries to survive in natural crises or any other unpredictable conditions. Only if a single factor like energy optimization is focused, firstly, it will help industries to minimize their investing cost; secondly, the buyer will be able to get the product at a low cost. Thirdly, the saved energy will be used by other sectors like domestic sectors to fulfill their energy need and most importantly for the sake of healthy environmental and economic conditions, energy saving is a significant step. In the present time, when COVID-19 or other natural and economic disasters are happening, it is almost impossible to make an accurate and successful management plan. To ensure better energy management for textile industries under all these circumstances, developing an energy optimization model that can work in uncertain situations and generate the best optimal output is needed. To reduce the energy cost, one way is to reconsider the source of energy and replace it with the best convenient sustainable resource. Another way is to allocate energy units optimally. For the first way, considering the situation of Pakistan, it is best for textile industries to generate their own electricity and thermal energy through the Solar System. For the second way, it is best to modify the linear programming (LP) model for the textile industry according to the fuzzy environment. For this purpose, the conversion of the objective function, constraint equations, demand, and supply into fuzzy number is needed. Precisely that fuzzy number can overcome the uncertainty regarding material availability, working hours, financial and social barriers, labour, money, and space.


## 2. Literature Review

Although in Pakistan, textile industries cover 46% of the whole manufacturing sector and are a macro contributor with 8.5% of Pakistan’s gross domestic product (GDP), it is also a huge environmental pollution contributor. In this setting, textile industries have a significant share in water and air pollution. The amount of clean water utilized by the textile industrial sector is substantially more as compared to the agricultural one [1], and for the production of one kg fabric output at the wet processing stage, the amount of water needed is almost 80–150 liters in addition to other chemicals [2]. Not only that, the report of 2015 stated that textile sector solely accounted for 1.2 billion tons of carbon dioxide [3], while according to the present behaviour of this industrial sector, 26% of carbon emission and 0.3 billion tons of crude oil consumption by the year 2025 is estimated by the study as well [3]. This is because the textile industry’s fuel consumption is not only related to the production level but also during the transfer of textile products through different transportation means to different areas [4] taking part in carbon emission. All these aspects clear the reason for being called textile industry as the most polluting industry. In Faisalabad, Pakistan, a chain of textile industries is located around the road connecting twin districts Khuranwala and Jaranwala as shown in Figure 2.Figure 2
Industrial chain of Khuranwala.Due to industrial water waste and carbon emission, the Khuranwala/Jaranwala road of 25–28 km long is full of unbearable smell and smog. The residents and daily travellers nearby face exposure to this unhealthy environment and suffer from breathing and eye diseases. By controlling abrupt energy consumption and ways connected to pollution either directly (industrial waste) or indirectly, like nonrecyclable fabric formation, excess of textile products, and so on, can help to lessen down pollution. Several studies have been conducted to reuse industrial water waste. Nadeem et al. [5] discussed the recycling and treatment potential of textile waste water by using membrane technology. According to Dehghani and Yoo [6], the organic components and approximate temperature of the textile industry’s waste water are sufficient enough for the production of biofuel that can be used as a cheap source for creating thermal energy.Efficient consumption of energy except proper planning is not possible in the region due to the complicated nature of the textile manufacturing sector. Spinning and weaving stages fully depend on electricity, while sizing, dying, and rewinding requires thermal energy to heat water or to dry the fabric [7, 8]. Studies have shown that, globally one trillion KWh of electricity is consumed per year to produce 60 billion kg fabric [9]. Usage of renewable energy resources like solar energy, geothermal energy, and landfill gas is helpful in the reduction of textile production costs. It lowers the total energy per unit usage that automatically degrade the environmental pollution factor as well. In Atlanta, seven textile manufacturing plants only use renewable electricity, and 89% of this electricity is renewable source [10].Specifically, in Pakistan, the existence of energy self-sufficient textile industries is rare. The All Pakistan Textile Mills Association (APTMA) Punjab Chairman also insisted on adopting renewable solar energy resources to fulfill the need for electricity with affordable cost for the maintenance of industrial sustainability. According to their reporting, Pakistan textile industry is ready to shift to solar or hybrid energy generating systems [11]. This shift will not only reduce the cost but also reduce the carbon emission. In developing countries like Pakistan, where energy production is not proportional to the need, the cheap production and optimal allocation of energy is required. The usage of renewable energy resources can also be of great help for the industrial sector. Considering solar energy as a means for electricity is the best suitable option. The temperature and weather conditions of Pakistan as shown in Figure 3 can accelerate the outcome of the solar energy system.Figure 3
Temperature map of Pakistan (Pakistan Meteorological Department [12]).The advancement of solar energy system makes electricity production much more efficient. Using this kind of system is considered as sustainable and ecological investment. Textile industry extensively uses electricity, so the optimal utilization of energy at each stage is regarded as an important initial target.The conventional and most widely used method with reliable results for optimization is linear programming (LP) created by Kantorovich [13]. The only drawback of LP is its nonflexibility regarding nature. Natural scenarios are full of ambiguity, while classical optimization based on LP does not cover these uncertainties. After the revelation of fuzzy sets by Zadeh [14], LP also started to be modified. Zimmerman [15] used the concept of fuzziness in LP. He invented the technique to find a solution for fuzzy multiobjective LP. The fuzziness in this technique is due to the presence of fuzzy optimization conditions in it. Atanassov [16] improved the concept of fuzziness more by introducing a new generalization of a fuzzy set named intuitionistic fuzzy set, in which both the membership grade and nonmembership grade of element from decision set are necessary. This generalization created more optimization techniques [17]. Angelov [18] made the intuitionistic fuzzy LP technique. Subsequently, several researches has appeared on intuitionistic fuzzy linear programming (IFLP). Hussain and Kumar [19] worked on intuitionistic fuzzy transportation problem (IFTP). Ebrahimnejad and Verdegay [20] worked on a fully intuitionistic transportation problem (TP).Several optimization models for sustainable production and consumption of renewable energy are constructed using bilevel programming [21], mixed integer linear programming [22], simulation-optimization modelling [23], linear programming [24–26], and goal programming [27] approach. Nematian and Farzi [28] considered the case study of energy recovered from urban solid waste in Iran and developed energy and environmental management model using fuzzy LP approach. Zhou et al. [29] used type-2 fuzzy chance-constrained fractional integrated modelling method for the management of energy system subjected to uncertainties and risks. Kouaissah and Hocine [30] and Hashemizadeh and Ju [31] introduced sustainable and renewable energy portfolios using a fuzzy interval goal programming technique. Khan et al. [32] examined and assessed the optimal cost system of electricity generation for the socio-economic sustainability of India by developing a sustainable and flexible electricity generation model using flexible fuzzy goal programming. Sustainable production in the textile industry depends upon the search of sustainable ways of energy management. Emeç and Akkaya [33] developed a fuzzy optimal renewable energy model (F-OREM) to solve the energy problem involving fuzzy parameters. Abbas et al. [34] systematically analyzed the potential of cotton crop waste to synergize industrial energy systems by integrating strategic and tactical decision models into an integrated model. The results indicate that cotton crop waste is a conducive and convenient source of sustainable energy supply for the textile industry. Techato et al. [35] presented a systematic review of the optimization models used in the textile industry, mostly established for cost minimization in logistics and production and optimized mainly with linear programming, integer programming, Markov chains, genetic algorithms, and multi-objective programming. The effectiveness of the present study in contrast with conventional LP is further discussed in upcoming sections where a fully intuitionistic fuzzy energy optimization model for Pakistan’s textile industry is formed that results in optimal energy allocation that is further presented. For sustainability, the idea of replacing typical energy resources with solar energy generators along with its investing and working cost is also to be considered in reference to the existing self-sufficient textile industry of Pakistan.


## 3. Preliminaries

### 3.1. Intuitionistic Fuzzy Set

The intuitionistic fuzzy set was presented by Atanassov [16]; the degree of nonmembership and the degree of membership were expressed by the two characteristic functions. An intuitionistic fuzzy set (IFS) N in Q can be described as an element of the following form N=s,ζNs,ηAs|s∈Q where the functions ηN:Q⟶0,1 and ζN:Q⟶0,1 represents the degree of nonmembership and the degree of membership of the component s∈Q, respectively.


### 3.2. Triangular Intuitionistic Fuzzy Number

It is based on membership and nonmembership functions [36], considering Ql⋎˜=q1,q2,q3;q´1,q2,q´3 such that q´1≤q1≤q2≤q3≤q´3 and its membership function is defined as(1)μQl⋎˜=x−q2q2−q1q1≤x<q21,x=q2q3−xq3−q2q2<x≤q30,otherwise.The nonmembership function is as follows:(2)νQl⋎˜=q2−xq2−q´1q´1≤x<q20,x=q2x−q2q´3−q2q2<x≤q´31,otherwise.With restriction0≤μQl⋎˜x+νQl⋎˜x≤1. If q1, q2, q3, q´1, and q´3 are nonnegative, then Ql⋎˜ will be a nonnegative intuitionistic fuzzy number. The geometrical interpretation of Triangular intuitionistic fuzzy number is presented in Figure 4.Figure 4
Graphical representation of a triangular intuitionistic fuzzy number.


### 3.3. Arithmetic Operations

LetQl⋎˜=q1,q2,q3;q´1,q2,q´3 and Pl⋎˜=p1,p2,p3;p´1,p2,p´3 are two triangular intuitionistic fuzzy numbers, then the algebraic operations between them are defined as follows:(1)
Ql⋎˜⊕Pl⋎˜=q1+p1,q2+p2,q3+p3;q´1+p´1,q2+p2,q´3+p´3(2)
Ql⋎˜⊖Pl⋎˜=q1−p3,q2−p2,q3−p1;q´1−p´3,q2−p2,q´3−p´1(3)
κQl⋎˜=κq1,κq2,κq3;κq´1,κq2,κq´3;forκ≥0(4)
κQl⋎˜=κq3,κq2,αq1;κq´3,κq2,κq´1;forκ<0(5)
Ql⋎˜⊛Pl⋎˜=r1,r2,r3;r´1,r2,r´3Here,(3)r1=minq1p1,q1p3,q3p1,q3p3r3=maxq1p1,q1p3,q3p1,q3p3r2=q2p2=r´2r´1=minq´1p´1,q´1p´3,q´3p´1,q´3p´3r´3=maxq´1p´1,q´1p´3,q´3p´1,q´3p´3.In particular, ifQl⋎˜ and Pl⋎˜ are two nonnegative triangular intuitionistic fuzzy numbers, then their product Ql⋎˜⊛Pl⋎˜ will be q1p1,q2p2,q3p3;q´1p´1,q2p2,q´3p´3.For the conversion of triangular intuitionistic fuzzy number into crisp following accuracy function [20] is used.(4)IQl⋎˜=18q1+4q2+q3+q´1+q´3.


## 3.1. Intuitionistic Fuzzy Set

The intuitionistic fuzzy set was presented by Atanassov [16]; the degree of nonmembership and the degree of membership were expressed by the two characteristic functions. An intuitionistic fuzzy set (IFS) N in Q can be described as an element of the following form N=s,ζNs,ηAs|s∈Q where the functions ηN:Q⟶0,1 and ζN:Q⟶0,1 represents the degree of nonmembership and the degree of membership of the component s∈Q, respectively.


## 3.2. Triangular Intuitionistic Fuzzy Number

It is based on membership and nonmembership functions [36], considering Ql⋎˜=q1,q2,q3;q´1,q2,q´3 such that q´1≤q1≤q2≤q3≤q´3 and its membership function is defined as(1)μQl⋎˜=x−q2q2−q1q1≤x<q21,x=q2q3−xq3−q2q2<x≤q30,otherwise.The nonmembership function is as follows:(2)νQl⋎˜=q2−xq2−q´1q´1≤x<q20,x=q2x−q2q´3−q2q2<x≤q´31,otherwise.With restriction0≤μQl⋎˜x+νQl⋎˜x≤1. If q1, q2, q3, q´1, and q´3 are nonnegative, then Ql⋎˜ will be a nonnegative intuitionistic fuzzy number. The geometrical interpretation of Triangular intuitionistic fuzzy number is presented in Figure 4.Figure 4
Graphical representation of a triangular intuitionistic fuzzy number.


## 3.3. Arithmetic Operations

LetQl⋎˜=q1,q2,q3;q´1,q2,q´3 and Pl⋎˜=p1,p2,p3;p´1,p2,p´3 are two triangular intuitionistic fuzzy numbers, then the algebraic operations between them are defined as follows:(1)
Ql⋎˜⊕Pl⋎˜=q1+p1,q2+p2,q3+p3;q´1+p´1,q2+p2,q´3+p´3(2)
Ql⋎˜⊖Pl⋎˜=q1−p3,q2−p2,q3−p1;q´1−p´3,q2−p2,q´3−p´1(3)
κQl⋎˜=κq1,κq2,κq3;κq´1,κq2,κq´3;forκ≥0(4)
κQl⋎˜=κq3,κq2,αq1;κq´3,κq2,κq´1;forκ<0(5)
Ql⋎˜⊛Pl⋎˜=r1,r2,r3;r´1,r2,r´3Here,(3)r1=minq1p1,q1p3,q3p1,q3p3r3=maxq1p1,q1p3,q3p1,q3p3r2=q2p2=r´2r´1=minq´1p´1,q´1p´3,q´3p´1,q´3p´3r´3=maxq´1p´1,q´1p´3,q´3p´1,q´3p´3.In particular, ifQl⋎˜ and Pl⋎˜ are two nonnegative triangular intuitionistic fuzzy numbers, then their product Ql⋎˜⊛Pl⋎˜ will be q1p1,q2p2,q3p3;q´1p´1,q2p2,q´3p´3.For the conversion of triangular intuitionistic fuzzy number into crisp following accuracy function [20] is used.(4)IQl⋎˜=18q1+4q2+q3+q´1+q´3.


## 4. Fully Intuitionistic Fuzzy Linear Programming Model

Mathematically, a general triangular fuzzy intuitionistic energy optimization model is expressed as follows:(5)minα⋎˜=∑j=1mCj⋎˜⊛xj⋎˜minα⋎˜=∑j=1mcj,1xj,1,∑j=1mcj,2xj,2,∑j=1mcj,3xj,3;∑j=1mc´j,1x´j,1,∑j=1mcj,2xj,2,∑j=1mc´j,3x´j,3.Subjected to triangular intuitionistic fuzzy energy optimization constraints,(6)s21⋎˜x1⋎˜+s2j⋎˜xj⋎˜+⋯+s2m⋎˜xm⋎˜≤=≥D2⋎˜⋮≤=≥⋮sn1⋎˜x1⋎˜+snj⋎˜xj⋎˜+⋯+snm⋎˜xm⋎˜≤=≥Dn⋎˜s11⋎˜x1⋎˜+s1j⋎˜xj⋎˜+⋯+s1m⋎˜xm⋎˜≤=≥D1⋎.˜Here, xj⋎˜≥0⋎˜. The values of xj⋎˜ will optimize the intuitionistic fuzzy energy cost objective function where all the xj⋎˜ will satisfy the nonnegativity condition and constraint equations. In the above objective function Cj⋎˜=cj,1,cj,2,cj,3;c´j,1,cj,2,c´j,3j=1,2,3,… represent the triangular intuitionistic fuzzy cost coefficients and sij are the technological coefficients representing the amount of i th resource consuming at rate of xj⋎˜ per unit where(7)Di⋎˜=dj,1,dj,2,dj,3;d´j,1,dj,2,d´j,3refers to the total availability of the i th triangular intuitionistic fuzzy resource. For the conversion of intuitionistic fuzzy energy optimization model into linear programming problem, the accuracy function is used as defined in (4). The following equation presents the defuzzified objective function.(8)min∑j=1mcj,1xj,1,cj,2xj,2,cj,3xj,3;c´j,1x´j,1,cj,2xj,2,c´j,3x´j,3=min18∑j=1mcj,1xj,1+4cj,2xj,2+cj,3xj,3+c´j,1x´j,1+c´j,3x´j,3..The above defined set of triangular intuitionistic constraints (6) can be rewritten as follows:(9)∑j=1msijxj,1≤=≥di,1.∑j=1msijxj,2≤=≥di,2∑j=1msijxj,3≤=≥di,3∑j=1msijx´j,1≤=≥d´i,1∑j=1msijx´j,3≤=≥d´i,3The optimal solution ofxj,1, xj,2, xj,3, x´j,1, and x´j,3 will be obtained by solving objective function (7) under constraints of (4).


## 5. Energy Optimization Model for Textile Industry

The textile industry is usually based on five stages which are further subdivided into several others. The main five stages are spinning, sizing, rewinding, weaving, and dyeing. Each stage’s productxi′s forward lessens quantity to the next stage, that is, input<output always. For example, 97% of stage 2 product is further processed for next weaving stage, and remaining 3% goes to the rewinding stage whose product again becomes the input of stage 2. Considering minimum waste up to 7% for weaving stage, the material processed further is 93% of total x3, at the last stage, the wastage is approximately 4%. Considering a standard five stages textile model, the per month demand of stage x1,x4, and x5 products 400 units, 600 units, and 20,000 units. Per month total working hours are 720 hours [37]. The electricity cost per unit is 20.62 PKR/kWh, fuel oil is 85.68 PKR/litre, and LPG cost at the rate of 19.4103 PKR/litre [38, 39]. Energy cost for each stage is presented in Table 1.Table 1
Energy cost in PKR.
Energy
Spinning
Sizing
Weaving
Rewinding
Dyeing
Electricity
51.55
5.155
41.24
15.465
20.62
LPG
0
19.0103
0
0
135.8721
Furnace fuel
0
0
0
0
21.40
Total energy cost
51.55
24.5653
41.24
15.465
177.892The formulation of above model in intutitionistic fuzzy environment provided triangular intuitionistic fuzzy cost coefficients that represent the cost spent during production period as follows:(10)Cj⋎˜=C1⋎˜=46.55,51.55,56.55;41.55,51.55,61.55,C2⋎˜=19.5653,24.5653,29.5653;14.5653,24.5653,34.5653,C3⋎=36.24,41.24,46.24;31.24,41.24,51.24,C4⋎=10.465,15.465,20.465;5.465,15.465,25.465,C5⋎=172.892,177.892,182.892;167.892,177.892,187.892˜˜˜.The monthly triangular intuitionistic fuzzy production demand in kg for three products and total availability of working hours in a month are stated as follows:(11)Di⋎˜=D1⋎=350,400,450;300,400,500˜.D2⋎˜=550,600,650;500,600,700,D3⋎=19950,20000,20050;19900,20000,20100,D4⋎=670,720,770;620,720,820˜˜Mathematically, intuitionistic fuzzy energy optimization is framed as follows:(12)minα⋎˜=∑j=1mCj⋎˜⊛xj⋎˜=46.55,51.55,56.55;41.55,51.55,61.55⊛x1⋎˜⊕19.5653,24.5653,29.5653;14.5653,24.5653,34.5653⊛x2⋎˜⊕36.24,41.24,46.24;31.24,41.24,51.24⊛x3⋎˜⊕10.465,15.465,20.465;5.465,15.465,25.465⊛x4⋎˜⊕172.892,177.892,182.892;167.892,177.892,187.892⊛x5⋎˜.Subject to(13)x1⋎˜⊖x2⋎˜≥350,400,450;300,400,500,0.03x2⋎˜⊖x3⋎˜⊕0.07x4⋎˜=0,0,0;0,0,0,0.97x2⋎˜⊖x4⋎˜=0,0,0;0,0,0,0.93x4⋎˜⊖x5⋎˜≥550,600,650;500,600,700,0.96x5⋎˜≥19950,20000,20050;19900,20000,20100,0.007x1⋎˜⊖0.007x2⋎˜⊕0.013x4⋎˜⊕0.0062x5⋎˜≤670,720,770;620,720,820,xi⋎˜≥0,0,0;0,0,0,i=1,2,3,4,5,According to the above intuitionistic model, this formulated intuitionistic fuzzy energy optimization model is defuzzified as follows:(14)min=1846.55x1,1+4×51.55x1,2+56.55x1,3+41.55x´1,1+61.55x´1,3+1819.5653x2,1+4×24.5653x2,2+29.5653x2,3+14.5653x´2,1+34.5653x´2,3+1836.24x3,1+4×41.24x3,2+46.24x3,3+31.24x´3,1+51.24x´3,3+1810.465x4,1+4×15.465x4,2+20.465x4,3+5.465x´4,1+25.465x´4,3+18174.892x5,1+4×177.892x5,2+182.892x5,3+167.892x´5,1+187.892x´5,3.Subject to(15)xi,1≥0,xi,2≥0,xi,3≥0,x´i,1≤0,x´i,3≤0wherei=1,2,3,4,50.007x1,1−0.007x2,3+0.013x4,1+0.0062x5,1≤670,0.007x1,2−0.007x2,2+0.013x4,2+0.0062x5,2≤720,0.007x1,3−0.007x2,1+0.013x4,3+0.0062x5,3≤770,0.007x´1,1−0.007x´2,3+0.013x´4,1+0.0062x´5,1≤620,0.007x´1,2−0.007x´2,2+0.013x´4,2+0.0062x´5,2≤820,x1,1−x2,3≥350,x1,2−x2,2≥400,x1,3−x2,1≥450,0.03x2,1−x3,3+0.07x4,1=0,0.03x2,2−x3,2+0.07x4,2=0,0.03x2,3−x3,1+0.07x4,3=0,0.03x´2,1−x´3,3+0.07x´4,1=0,0.03x´2,3−x´3,1+0.07x´4,3=0,x´1,1−x´2,3≥300,x´1,3−x´2,1≥500,0.97x2,1−x4,3=0,0.97x2,2−x4,2=0,0.97x2,3−x4,1=0,0.97x´2,1−x´4,3=0,0.97x´2,3−x´4,1=0,0.93x4,1−x5,3≥550,0.93x4,2−x5,2≥600,0.93x4,3−x5,1≥650,0.93x´4,1−x´5,3≥500,0.93x´4,3−x´5,1≥700,0.96x5,1≥19950,0.96x5,2≥20000,0.96x5,3≥20050,0.96x´5,1≥19900,0.96x´5,3≥20100.The optimal solution of (13) is(16)x1⋎˜=24111.7,24159.4,24207.1;24064,24159.4,24254.8,x2⋎˜=23757.1,23759.4,23761.7;23754.8,23759.4,23764,x3⋎˜=874.161,874.107,874.054;874.125,874.107,874,x4⋎˜=23048.8,23046.6,23044.4;23051.1,23046.6,23024.1,x5⋎˜=20781.3,20833.3,20885.4;20729.2,20833.3,20937.5.By substituting the values ofxi⋎˜’s in objective function (10), the minimal triangular intuitionistic fuzzy energy cost is calculated as follows:(17)minα⋎˜=∑j=15Cj⋎˜⊛xj⋎˜=5463813.64,5927615.1,6403249.54;4979403.76,5927615.1,7880260.7μQl⋎˜α⋎˜=x−5927615.15927615.1−5463813.645463813.64≤x<5927615.11,x=5927615.16403249.54−x6403249.54−5463813.645927615.1<x≤6403249.540,otherwise,νQl⋎˜α⋎˜=5927615.1−x5927615.1−4979403.764979403.76≤x<5927615.10,x=5927615.1x−5927615.17880260.7−5927615.15927615.1<x≤7880260.71,otherwise.


### 5.1. Postoptimal Analysis

Table2 elaborates the flexibility of the presented model in intuitionistic fuzzy environment. Within the provided range for each variable, the model remains optimal and feasible. Since xiγ=xi,1,xi,2,xi,3;xi,1′,xi,2,xi,3′; therefore, the allowable change in xi,j will vast the range for xiγ as well. This change will not impact the feasibility of the given model.Table 2
Optimal feasible range.
Variable
Final value
Reduced cost
Objective
Allowable increase
Allowable decrease
Range
x
1,1
24111.6857
0
5.819
1E + 30
5.819
5.819
x
1,2
24159.37627
0
25.775
1E + 30
25.775
25.775
x
1,3
24207.06684
0
7.069
1E + 30
7.069
7.069
x
1,1
′
24063.99512
0
5.194
1E + 30
5.194
5.194
x
1,3
′
24254.75742
0
7.694
1E + 30
7.694
7.694
x
2,1
23757.06684
0
2.46
1E + 30
12.2144187
12.2144187
x
2,2
23759.37627
0
12.283
1E + 30
46.3171348
46.3171348
x
2,3
23761.6857
0
3.696
1E + 30
10.9589062
10.9589062
x
2,1
′
23754.75742
0
1.821
1E + 30
12.82117495
12.82117495
x
2,3
′
23763.99512
0
4.321
1E + 30
10.33814995
10.33814995
x
3,1
874.1610548
0
4.53
1E + 30
365.2968733
365.2968733
x
3,2
874.107453
0
20.62
1E + 30
1258.959902
1258.959902
x
3,3
874.0538512
0
5.78
1E + 30
407.14729
407.14729
x
3,1
′
874.2146565
0
3.905
1E + 30
344.6049983
344.6049983
x
3,3
′
874.0002494
0
6.405
1E + 30
427.3724983
427.3724983
x
4,1
23048.83513
0
1.308
1E + 30
11.29784144
11.29784144
x
4,2
23046.59498
0
7.7325
1E + 30
47.74962351
47.74962351
x
4,3
23044.35484
0
2.558
1E + 30
12.59218423
12.59218423
x
4,1
′
23051.07527
0
0.683
1E + 30
10.65788655
10.65788655
x
4,3
′
23042.1147
0
3.183
1E + 30
13.21770613
13.21770613
x
5,1
20781.25
0
21.8615
1E + 30
35.40148304
35.40148304
x
5,2
20833.33333
0
88.946
1E + 30
140.2896812
140.2896812
x
5,3
20885.41667
0
22.8615
1E + 30
35.00971661
35.00971661
x
5,1
′
20729.16667
0
20.9865
1E + 30
35.19908724
35.19908724
x
5,3
′
20937.5
0
23.4865
1E + 30
34.94659306
34.94659306As shown in Table3, the optimal results changed accordingly within the limits without impacting feasibility. The limit report of this model provided lower limit and upper limit for each decision variable along with its corresponding objective values. As the main objective is to minimize the energy cost, so here the optimal result is totally based on the lower limits to obtain minimum output. Whereas the upper limit is providing the range for feasible solution. Table 4 presents the sensitivity report for the constraint equation. As there is not always perfect availability of resources and time so for each situation making new model is not possible. The following report allows us to predict the flexibility regarding the optimal and feasible results we could have obtained if the availability of resources fluctuate between the allowable increase and decrease.Table 3
Limit report.
Variable
Final value
Lower limit
Objective value
Upper limit
Objective value
x
1,1
24111.6857
24111.6857
5933823.792
58264.74189
6132560.426
x
1,2
24159.37627
24159.37627
5933823.792
65363.31892
6995855.414
x
1,3
24207.06684
24207.06684
5933823.792
72461.89595
6274937.179
x
1,1
′
24063.99512
24063.99512
5933823.792
51166.16486
6074592.462
x
1,3
′
24254.75742
24254.75742
5933823.792
79560.47298
6359345.968
x
2,1
23757.06684
23757.06684
5933823.792
23757.06684
5933823.792
x
2,2
23759.37627
23759.37627
5933823.792
23759.37627
5933823.792
x
2,3
23761.6857
23761.6857
5933823.792
23761.6857
5933823.792
x
2,1
′
23754.75742
23754.75742
5933823.792
23754.75742
5933823.792
x
2,3
′
23763.99512
23763.99512
5933823.792
23763.99512
5933823.792
x
3,1
874.1610548
874.1610548
5933823.792
874.1610548
5933823.792
x
3,2
874.107453
874.107453
5933823.792
874.107453
5933823.792
x
3,3
874.0538512
874.0538512
5933823.792
874.0538512
5933823.792
x
3,1
′
874.2146565
874.2146565
5933823.792
874.2146565
5933823.792
x
3,2
′
874.0002494
874.0002494
5933823.792
874.0002494
5933823.792
x
4,1
23048.83513
23048.83513
5933823.792
23048.83513
5933823.792
x
4,2
23046.59498
23046.59498
5933823.792
23046.59498
5933823.792
x
4,3
23044.35484
23044.35484
5933823.792
23044.35484
5933823.792
x
4,1
′
23051.07527
23051.07527
5933823.792
23051.07527
5933823.792
x
4,3
′
23042.1147
23042.1147
5933823.792
23042.1147
5933823.792
x
5,1
20781.25
20781.25
5933823.792
20781.25
5933823.792
x
5,2
20833.33333
20833.33333
5933823.792
20833.33333
5933823.792
x
5,3
20885.41667
20885.41667
5933823.792
20885.41667
5933823.792
x
5,1
′
20729.16667
20729.16667
5933823.792
20729.16667
5933823.792
x
5,3
′
20937.5
20937.5
5933823.792
20937.5
5933823.792Table 4
Sensitivity report.
Constraint LHS
Final value
Shadow price
Constraint RHS
Allowable increase
Allowable decrease
1
430.9286066
0
670
1E + 30
239.0713934
2
431.5724014
0
720
1E + 30
288.4275986
3
432.2161962
0
770
1E + 30
337.7838038
4
430.2848118
0
620
1E + 30
189.7151882
5
432.859991
0
820
1E + 30
387.140009
6
350
5.819
350
34153.0562
24111.6857
7
400
25.775
400
41203.94265
24159.37627
8
450
7.069
450
48254.82911
24207.06684
9
300
5.194
300
27102.16974
24063.99512
10
500
7.694
500
55305.71557
24254.75742
11
−1.7053E-13
−5.78
0
874.0538512
1E + 30
12
−1.13687E-13
−20.62
0
874.107453
1E + 30
13
3.97904E-13
−4.53
0
874.1610548
1E + 30
14
−2.84217E-13
−6.405
0
874.0002494
1E + 30
15
2.27374E-13
−3.905
0
874.2146565
1E + 30
16
−7.27596E-12
10.00247423
0
1E + 30
23044.35484
17
−3.63798E-12
39.87278351
0
1E + 30
23046.59498
18
−7.27596E-12
9.949381443
0
1E + 30
23048.83513
19
−3.63798E-12
10.00737113
0
1E + 30
23042.1147
20
−3.63798E-12
9.930051546
0
1E + 30
23051.07527
21
19950
36.87654483
19950
23197.95231
19950
22
20000
146.1350846
20000
13722.05904
20000
23
20050
36.4684548
20050
16418.68769
20050
24
19900
36.66571588
19900
26587.58462
19900
25
20100
36.4027011
20100
13029.05538
20100
26
550
12.14821661
550
17102.79968
21435.41667
27
600
51.34368119
600
20633.66667
21433.33333
28
650
13.53998304
650
24164.53365
21431.25
29
500
11.46009306
500
13571.93269
21437.5
30
700
14.21258724
700
27695.40064
21429.16667
31
24111.6857
0
0
24111.6857
1E + 30
32
24159.37627
0
0
24159.37627
1E + 30
33
24207.06684
0
0
24207.06684
1E + 30
34
24063.99512
0
0
24063.99512
1E + 30
35
24254.75742
0
0
24254.75742
1E + 30
36
23757.06684
0
0
23757.06684
1E + 30
37
23759.37627
0
0
23759.37627
1E + 30
38
23761.6857
0
0
23761.6857
1E + 30
39
23754.75742
0
0
23754.75742
1E + 30
40
23763.99512
0
0
23763.99512
1E + 30
41
874.1610548
0
0
874.1610548
1E + 30
42
874.107453
0
0
874.107453
1E + 30
43
874.0538512
0
0
874.0538512
1E + 30
44
874.2146565
0
0
874.2146565
1E + 30
45
874.0002494
0
0
874.0002494
1E + 30
46
23048.83513
0
0
23048.83513
1E + 30
47
23046.59498
0
0
23046.59498
1E + 30
48
23044.35484
0
0
23044.35484
1E + 30
49
23051.07527
0
0
23051.07527
1E + 30
50
23042.1147
0
0
23042.1147
1E + 30
51
20781.25
0
0
20781.25
1E + 30
52
20833.33333
0
0
20833.33333
1E + 30
53
20885.41667
0
0
20885.41667
1E + 30
54
20729.16667
0
0
20729.16667
1E + 30
55
20937.5
0
0
20937.5
1E + 30


### 5.2. Comparison

The solution of linear optimization problem under consideration is carried out by already existing linear programming (LP) and fuzzy linear programming techniques (FLP). The objective function value for the proposed method is less than LP and FLP techniques. The results are presented in Table5.Table 5
Comparison of optimization techniques.
Variable
LP
FLP
FIF LP
x
1
24195.38
(24126.743, 24159.376, 24192.008)
(24111.69, 24159.38, 24207.07; 24064, 24159.38, 24254.76)
x
2
23759.38
(23736.743, 23759.376, 23782.008)
(23757.07, 23759.38, 223759.38; 23754.76,23759.38, 23764)
x
3
2326.043
(2323.872, 2326.0429, 2328.2586)
(874.1611, 874.1075, 874.0539; 874.2147, 874.1075, 874.0002)
x
4
23046.59
(23024.64, 23046.59, 23068.55)
(23048.84, 23046.59, 23044.35; 23051.08, 23046.59, 23042.11)
x
5
20833.33
(20822.92, 20833.33, 20843.75)
(20781.25, 20833.33333, 20885.42; 20729.15, 20833.33333, 20937.5)
Objective function
5987499
5980612.1
5927615.1From Figure5, the degree of acceptance (rejection) of energy cost per month increases (decreases) from 5463813.64 to 5927615.1 and decrease (increases) from 5927615.1 to 6403249.54, while 5927615.1 is considered as required value where the level of acceptance is fully satisfied and degree of rejection is fully zero. The optimal solution of linear programming and fuzzy linear programming whose resulting cost are PKR 5987499 per month and 5980612.1 per month, while for proposed method it is 5927615.1 per month. This is evident that the fully intuitionistic fuzzy optimization model is more effectively minimize the energy cost as compared to linear programming approach.Figure 5
Graphical representation of LP, FLP, and FIF LP.


## 5.1. Postoptimal Analysis

Table2 elaborates the flexibility of the presented model in intuitionistic fuzzy environment. Within the provided range for each variable, the model remains optimal and feasible. Since xiγ=xi,1,xi,2,xi,3;xi,1′,xi,2,xi,3′; therefore, the allowable change in xi,j will vast the range for xiγ as well. This change will not impact the feasibility of the given model.Table 2
Optimal feasible range.
Variable
Final value
Reduced cost
Objective
Allowable increase
Allowable decrease
Range
x
1,1
24111.6857
0
5.819
1E + 30
5.819
5.819
x
1,2
24159.37627
0
25.775
1E + 30
25.775
25.775
x
1,3
24207.06684
0
7.069
1E + 30
7.069
7.069
x
1,1
′
24063.99512
0
5.194
1E + 30
5.194
5.194
x
1,3
′
24254.75742
0
7.694
1E + 30
7.694
7.694
x
2,1
23757.06684
0
2.46
1E + 30
12.2144187
12.2144187
x
2,2
23759.37627
0
12.283
1E + 30
46.3171348
46.3171348
x
2,3
23761.6857
0
3.696
1E + 30
10.9589062
10.9589062
x
2,1
′
23754.75742
0
1.821
1E + 30
12.82117495
12.82117495
x
2,3
′
23763.99512
0
4.321
1E + 30
10.33814995
10.33814995
x
3,1
874.1610548
0
4.53
1E + 30
365.2968733
365.2968733
x
3,2
874.107453
0
20.62
1E + 30
1258.959902
1258.959902
x
3,3
874.0538512
0
5.78
1E + 30
407.14729
407.14729
x
3,1
′
874.2146565
0
3.905
1E + 30
344.6049983
344.6049983
x
3,3
′
874.0002494
0
6.405
1E + 30
427.3724983
427.3724983
x
4,1
23048.83513
0
1.308
1E + 30
11.29784144
11.29784144
x
4,2
23046.59498
0
7.7325
1E + 30
47.74962351
47.74962351
x
4,3
23044.35484
0
2.558
1E + 30
12.59218423
12.59218423
x
4,1
′
23051.07527
0
0.683
1E + 30
10.65788655
10.65788655
x
4,3
′
23042.1147
0
3.183
1E + 30
13.21770613
13.21770613
x
5,1
20781.25
0
21.8615
1E + 30
35.40148304
35.40148304
x
5,2
20833.33333
0
88.946
1E + 30
140.2896812
140.2896812
x
5,3
20885.41667
0
22.8615
1E + 30
35.00971661
35.00971661
x
5,1
′
20729.16667
0
20.9865
1E + 30
35.19908724
35.19908724
x
5,3
′
20937.5
0
23.4865
1E + 30
34.94659306
34.94659306As shown in Table3, the optimal results changed accordingly within the limits without impacting feasibility. The limit report of this model provided lower limit and upper limit for each decision variable along with its corresponding objective values. As the main objective is to minimize the energy cost, so here the optimal result is totally based on the lower limits to obtain minimum output. Whereas the upper limit is providing the range for feasible solution. Table 4 presents the sensitivity report for the constraint equation. As there is not always perfect availability of resources and time so for each situation making new model is not possible. The following report allows us to predict the flexibility regarding the optimal and feasible results we could have obtained if the availability of resources fluctuate between the allowable increase and decrease.Table 3
Limit report.
Variable
Final value
Lower limit
Objective value
Upper limit
Objective value
x
1,1
24111.6857
24111.6857
5933823.792
58264.74189
6132560.426
x
1,2
24159.37627
24159.37627
5933823.792
65363.31892
6995855.414
x
1,3
24207.06684
24207.06684
5933823.792
72461.89595
6274937.179
x
1,1
′
24063.99512
24063.99512
5933823.792
51166.16486
6074592.462
x
1,3
′
24254.75742
24254.75742
5933823.792
79560.47298
6359345.968
x
2,1
23757.06684
23757.06684
5933823.792
23757.06684
5933823.792
x
2,2
23759.37627
23759.37627
5933823.792
23759.37627
5933823.792
x
2,3
23761.6857
23761.6857
5933823.792
23761.6857
5933823.792
x
2,1
′
23754.75742
23754.75742
5933823.792
23754.75742
5933823.792
x
2,3
′
23763.99512
23763.99512
5933823.792
23763.99512
5933823.792
x
3,1
874.1610548
874.1610548
5933823.792
874.1610548
5933823.792
x
3,2
874.107453
874.107453
5933823.792
874.107453
5933823.792
x
3,3
874.0538512
874.0538512
5933823.792
874.0538512
5933823.792
x
3,1
′
874.2146565
874.2146565
5933823.792
874.2146565
5933823.792
x
3,2
′
874.0002494
874.0002494
5933823.792
874.0002494
5933823.792
x
4,1
23048.83513
23048.83513
5933823.792
23048.83513
5933823.792
x
4,2
23046.59498
23046.59498
5933823.792
23046.59498
5933823.792
x
4,3
23044.35484
23044.35484
5933823.792
23044.35484
5933823.792
x
4,1
′
23051.07527
23051.07527
5933823.792
23051.07527
5933823.792
x
4,3
′
23042.1147
23042.1147
5933823.792
23042.1147
5933823.792
x
5,1
20781.25
20781.25
5933823.792
20781.25
5933823.792
x
5,2
20833.33333
20833.33333
5933823.792
20833.33333
5933823.792
x
5,3
20885.41667
20885.41667
5933823.792
20885.41667
5933823.792
x
5,1
′
20729.16667
20729.16667
5933823.792
20729.16667
5933823.792
x
5,3
′
20937.5
20937.5
5933823.792
20937.5
5933823.792Table 4
Sensitivity report.
Constraint LHS
Final value
Shadow price
Constraint RHS
Allowable increase
Allowable decrease
1
430.9286066
0
670
1E + 30
239.0713934
2
431.5724014
0
720
1E + 30
288.4275986
3
432.2161962
0
770
1E + 30
337.7838038
4
430.2848118
0
620
1E + 30
189.7151882
5
432.859991
0
820
1E + 30
387.140009
6
350
5.819
350
34153.0562
24111.6857
7
400
25.775
400
41203.94265
24159.37627
8
450
7.069
450
48254.82911
24207.06684
9
300
5.194
300
27102.16974
24063.99512
10
500
7.694
500
55305.71557
24254.75742
11
−1.7053E-13
−5.78
0
874.0538512
1E + 30
12
−1.13687E-13
−20.62
0
874.107453
1E + 30
13
3.97904E-13
−4.53
0
874.1610548
1E + 30
14
−2.84217E-13
−6.405
0
874.0002494
1E + 30
15
2.27374E-13
−3.905
0
874.2146565
1E + 30
16
−7.27596E-12
10.00247423
0
1E + 30
23044.35484
17
−3.63798E-12
39.87278351
0
1E + 30
23046.59498
18
−7.27596E-12
9.949381443
0
1E + 30
23048.83513
19
−3.63798E-12
10.00737113
0
1E + 30
23042.1147
20
−3.63798E-12
9.930051546
0
1E + 30
23051.07527
21
19950
36.87654483
19950
23197.95231
19950
22
20000
146.1350846
20000
13722.05904
20000
23
20050
36.4684548
20050
16418.68769
20050
24
19900
36.66571588
19900
26587.58462
19900
25
20100
36.4027011
20100
13029.05538
20100
26
550
12.14821661
550
17102.79968
21435.41667
27
600
51.34368119
600
20633.66667
21433.33333
28
650
13.53998304
650
24164.53365
21431.25
29
500
11.46009306
500
13571.93269
21437.5
30
700
14.21258724
700
27695.40064
21429.16667
31
24111.6857
0
0
24111.6857
1E + 30
32
24159.37627
0
0
24159.37627
1E + 30
33
24207.06684
0
0
24207.06684
1E + 30
34
24063.99512
0
0
24063.99512
1E + 30
35
24254.75742
0
0
24254.75742
1E + 30
36
23757.06684
0
0
23757.06684
1E + 30
37
23759.37627
0
0
23759.37627
1E + 30
38
23761.6857
0
0
23761.6857
1E + 30
39
23754.75742
0
0
23754.75742
1E + 30
40
23763.99512
0
0
23763.99512
1E + 30
41
874.1610548
0
0
874.1610548
1E + 30
42
874.107453
0
0
874.107453
1E + 30
43
874.0538512
0
0
874.0538512
1E + 30
44
874.2146565
0
0
874.2146565
1E + 30
45
874.0002494
0
0
874.0002494
1E + 30
46
23048.83513
0
0
23048.83513
1E + 30
47
23046.59498
0
0
23046.59498
1E + 30
48
23044.35484
0
0
23044.35484
1E + 30
49
23051.07527
0
0
23051.07527
1E + 30
50
23042.1147
0
0
23042.1147
1E + 30
51
20781.25
0
0
20781.25
1E + 30
52
20833.33333
0
0
20833.33333
1E + 30
53
20885.41667
0
0
20885.41667
1E + 30
54
20729.16667
0
0
20729.16667
1E + 30
55
20937.5
0
0
20937.5
1E + 30


## 5.2. Comparison

The solution of linear optimization problem under consideration is carried out by already existing linear programming (LP) and fuzzy linear programming techniques (FLP). The objective function value for the proposed method is less than LP and FLP techniques. The results are presented in Table5.Table 5
Comparison of optimization techniques.
Variable
LP
FLP
FIF LP
x
1
24195.38
(24126.743, 24159.376, 24192.008)
(24111.69, 24159.38, 24207.07; 24064, 24159.38, 24254.76)
x
2
23759.38
(23736.743, 23759.376, 23782.008)
(23757.07, 23759.38, 223759.38; 23754.76,23759.38, 23764)
x
3
2326.043
(2323.872, 2326.0429, 2328.2586)
(874.1611, 874.1075, 874.0539; 874.2147, 874.1075, 874.0002)
x
4
23046.59
(23024.64, 23046.59, 23068.55)
(23048.84, 23046.59, 23044.35; 23051.08, 23046.59, 23042.11)
x
5
20833.33
(20822.92, 20833.33, 20843.75)
(20781.25, 20833.33333, 20885.42; 20729.15, 20833.33333, 20937.5)
Objective function
5987499
5980612.1
5927615.1From Figure5, the degree of acceptance (rejection) of energy cost per month increases (decreases) from 5463813.64 to 5927615.1 and decrease (increases) from 5927615.1 to 6403249.54, while 5927615.1 is considered as required value where the level of acceptance is fully satisfied and degree of rejection is fully zero. The optimal solution of linear programming and fuzzy linear programming whose resulting cost are PKR 5987499 per month and 5980612.1 per month, while for proposed method it is 5927615.1 per month. This is evident that the fully intuitionistic fuzzy optimization model is more effectively minimize the energy cost as compared to linear programming approach.Figure 5
Graphical representation of LP, FLP, and FIF LP.


## 6. Conclusion

Since Pakistan has a huge textile industrial sector and is now facing so many uncertainties due to the unpredicted policy shift 2020–25, therefore economically improving this sector will help every person belonging to the textile industry’s hierarchy from consumers, labourers to stakeholder, and government. In Pakistan, government suggests industries to have a local supply of gas, which will create an explosion in production costs. According to the executive director of all Pakistan mills associations (APTAMA), the textile industry will face a 50% increase in the production cost [40]. The best option for Pakistan’s industrial sector is to become self-sufficient as soon as possible. For this purpose, they need to change their habit of being dependent on the government for their energy resources. It is only possible to shift their energy modes from nonrenewable to renewable ones for sustainable energy production. According to regional temperature conditions of Pakistan as shown in Figure 2, solar energy production refines option. Since Pakistan’s Kohinoor Textile Mills (KTML) in Rawalpindi, Pakistan, converted to a 6 Mw solar power plant with Reon energy making it affordable, sustainable, and more competitive. The implementation of this solar plant resulted in cost reduction and carbon footprint reduction. Because of its success, further three projects are also covered by KTML from 2017 to 2019, and their captive solar plant is saving almost 30% energy as well [41]. Therefore in Punjab, Pakistan, where there is a hub of textile industries should take a step forward towards the solar power plants.The application of this solar plant project approximately requires 65 to 80 million in Pakistan. This project will contain a one Mw system that can produce 5000 units per day if minimum five hours of sunlight is considered. Monthly electricity unit production from this system will approximate between 150000 kWh and 180000 kWh. If LP solution is considered, the electricity unit per month consumed by all stages are 109108.6563 kWh and from FIFLP 106204.7783 kWh. In both cases, monthly electricity unit production through a solar plant is greater than need. The remaining electricity units can be sold out to the government or nearby industries that will accelerate to overcome the investing cost and the amount obtained from sold electricity can be further used for maintenance purpose as well. Decision making through FIFLP provide more flexible and optimal outputs. The less units industry will consume the more will be available for sale and soon the invested cost will overcome and providing profit. The conversion of few Pakistan’s textile industry on solar power plants will help their neighboring industries, because the electricity cost per unit in solar generators is cheaper than other. That is, the government provide electricity at the rate of 20.62 Rs/Kwh while solar generating electricity cost is between 8 and 14 Rs/kWh.The idea of intuitionistic fuzzy model helps in sufficient allocation of energy units that provides less wastage of energy. The implementation of this modelling can definitely lower down the production cost. For pilot study the implementation of sustainable energy optimization model on one of the industry from that region mentioned in Figure1 can provide fruitful results along with the optimizing production cost of the neighboring industries as well.



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*Source: 5724825-2022-08-17.xml*