A Novel Multiobjective Game IDEA Cross-Efficiency Method Based on Boolean Possibility Degree for Ranking Biomass Materials With Interval Data

The concept of processing biomass materials into charcoal briquettes is a viable solution for every developing nation’s energy crisis. However, the important properties of each biomass material must first be considered to find suitable biomass materials for processing into charcoal briquettes. Sometimes these qualities are measured with imprecise values, making it exceedingly challenging to rank biomass materials’ decision-making units (DMUs). This problem is one of the interval data envelopment analysis (IDEA) ranking issues that make it difficult to calculate and rank all DMUs. In this paper, the concepts of the Game IDEA cross-efficiency method and Boolean possibility degree were utilized to solve the IDEA ranking problems. Unlike existing IDEA ranking models, a new multi-objective Game IDEA cross-efficiency (MO-G-IDEA-CE) method was used to obtain the Game interval cross-efficiency (GICE) scores of each DMU simultaneously. After that, the Boolean possibility degree was used to transform GICE scores into crisp values for ranking all DMUs. Three numerical examples, including a simple numerical example of China’s primary schools and seven biomass materials problems, are provided to demonstrate and validate the effectiveness of the proposed model. For the case study of seven biomass materials, after the Spearman correlation test, the correlation coefficients ( $r_{s}$ ) for the proposed method and Wang’s method, and Wu et al.’s method are calculated as $r_{s} =1.000$ and 0.964, respectively. In addition, it is worth noting that the proposed MO-G-IDEA-CE method has a very high correlation with the other ranking methods for all three numerical examples.

for enrichment of evaluations (PROMETHEE) and Elimi-97 nation and choice expressing reality (ELECTRE); (V) Util-98 ity/Evaluate methods such as Multi-attribute utility theory 99 (MAUT) and Multi-attribute value theory (MAVT); and (VI) 100 others such as Quality function development (QFD). DEA 101 is a distance-based approach that is widely used in solving 102 MADM problems. In the decision matrix, criteria can be 103 viewed as inputs and outputs for DEA, and alternatives can 104 be considered decision-making units (DMUs). The DEA can 105 be regarded as one of the MCDM tools because it can be 106 used to generate optimal weights of each criterion for ranking 107 alternatives/DMUs. Recently, the DEA is still being used as 108 an MCDM tool in various fields, such as the case of irrigation 109 management [9], the supplier's selection [10], and the solar 110 PV power plant site selection [11]. According to the relevant 111 literature review [12], despite the wide range of applications 112 of the DEA concept in renewable energy applications, there 113 has been no research using the multi-objective Game interval 114 data envelopment analysis (MO-G-IDEA-CE) approach for 115 evaluating charcoal briquettes. 116 DEA is a popular mathematical method used to measure 117 the performance of a set of DMUs with multiple inputs and 118 outputs. The relative efficiency of each DMU can be obtained 119 by calculating the ratio of the weighted sum of outputs to the 120 weighted sum of inputs. If a DMU has a relative efficiency 121 score of 1, it is defined as efficient. Otherwise, it is specified 122 as inefficient. Over the past several decades, various forms of 123 DEAs, the Charnes-Cooper-Rhodes (CCR) model [13] and 124 Banker-Charnes-Cooper (BCC) model [14], have been used 125 in a wide range of fields, such as banking [15], engineering 126 [16], education [17], agriculture [18], and corporate adminis-127 tration [19]. The main advantages of DEA are that it does not 128 require any possible assumptions related to the structure of 129 the production function, and the values of inputs and outputs 130 can have different measurement units [20], [21], [22]. The tra-131 ditional DEAs can estimate the relative efficiencies of DMUs 132 with precise values of inputs and outputs. If the values of the 133 inputs or outputs of DMUs are imprecise, such as interval 134 data, the existing DEAs fail to measure the performance of 135 the DMUs. Hence, many researchers [23], [24], [25] have 136 offered various Interval Data Envelopment Analysis models 137 (IDEA models) to solve this weak point. Cooper et al. [26] 138 first offered the IDEA model to measure the performance 139 of a set of DMUs with inaccurate data. Subsequently, this 140 theoretical approach has contributed to further development 141 by a group of scholars. Despotis and Smirlis [27] converted 142 the DEA-CCR model into the IDEA -CCR model to solve 143 DMUs with interval data, and outcomes were obtained as 144 the lower and upper values of efficiency scores. 145 Entani et al. [28] offered a pair of IDEA models, called 146 the optimistic IDEA and pessimistic IDEA models, to solve 147 IDEA ranking problems for DMUs with interval data of 148 inputs and outputs. However, Wang et al. [29] noted that 149 Despotis and Smirlis' model [27] employed two different 150 production frontiers to calculate the efficiencies of DMUs, 151 which may result in the incomparability of DMU efficiencies. 152 VOLUME 10,2022 To address this issue, Wang et al. [29]  The main contributions of this research are in the following 216 ways: 217 1) Based on the idea of Wang et al. [29], we formulated a 218 new multi-objective interval data envelopment analysis CCR 219 model (MO-IDEA-CCR) for measuring the interval efficien-220 cies of DMUs with interval data.   4) We apply the proposed method to a real case of the fuel 232 briquette problem; this will be extremely valuable for study 233 in this field in most, especially farming, nations.

234
The rest of this paper is structured as follows: In Section II, 235 we first offer existing IDEA-CCR models, IDEA cross-236 efficiency method, and IDEA cross-efficiency method based 237 on secondary goals. Next, Section III provides a new solution 238 for measuring and ranking DMUs with interval data. Then 239 three examples, a simple numerical example, China's primary 240 schools, and a biomass materials problem, are provided to 241 illustrate the idea proposed in Section IV. Finally, Section V 242 is the conclusion.

II. BACKGROUND
In Equations (1) and (2), DMU k is to be measured. Let v ik 262 and u rk be the weights of the input i and output r, respectively.

263
Then, E l kk and E u kk are the lower and upper efficiencies for 264 each DMU k , respectively. In the above two models, it is clear 265 that DMU k can be defined as an efficient DMU if its optimal 266 solution is E u kk = 1, or it is inefficient if E u kk < 1.

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After solving the IDEA-CCR models in Equation (1) and

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Equation (2), let u l * rd and u u * rd be the lower and upper bound of 270 the optimal output weights for a specific DMU k , respectively.

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If v l * id and v u * id are the lower and upper bound of optimal 272 input weights for a particular DMU k , respectively, then the 273 small cross-efficiency scores of each DMU j peer-evaluated 274 by DMU k , are provided by As a result, the average cross-efficiency (ACE l ) score of 277 DMU j is defined as Similarly, the values of large cross-efficiency can be 280 defined as

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As a result, from Sexton et al. [38], the ACE u of DMU j can 283 be defined as To enhance its efficiency ratio, the DMU under assessment 293 considers the inputs and outputs of a few favorable DMUs, 294 while ignoring the rest. In addition, optimal weights deter-295 mined with models (1) and (2) are not typically unique. 296 As a result, the calculating software may give varying 297 ideal weights, rendering the cross-efficiency scores arbitrary. 298 To address this deficiency, an interval cross-efficiency eval-299 uation method is employed. DEA's cross-efficiency process 300 uses peer evaluation rather than self-evaluation. It can define 301 the cross-efficiency ratings of DMUs based on their inter-302 val [29]. Some choices of weights in the traditional cross-303 efficiency approach may result in a lower cross-efficiency for 304 some DMUs and a higher cross-efficiency for others. To alle-305 viate the ambiguity, a secondary goal function is introduced. 306 Model (7), proposed by Wu et al. [34], can calculate the 307 values of small cross-efficiency for interval data. 308 Similarly, the values of large cross-efficiency can be cal-314 culated using model (8). After solving Equation (7) to Equation (8) (1) and Equation (2), with added constraints of scores, E l kk , E u kk , as shown in Equation (9).
In this section, we formulate the multi-objective Game 347 IDEA cross-efficiency method (MO-G-IDEA) for ranking 348 all DMUs with interval data. Based on the concept of 349 a traditional Game cross-efficiency method presented by 350 Liang et al. [36], the MO-G-IDEA method can be defined 351 as α l j , α u j is called the Game interval cross-efficiency (GICE) 364 score of DMU j (j = 1, 2, . . . , n). Based on Equation (10), the 365 iteration algorithm leading to Nash-equilibrium is:

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Step I: To obtain the GICE scores, the MO-IDEA-CCR 367 model in Equation (9) must be computed first. For each 368

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Step    into six steps: Step I: Collection of biomass materials. 419 Agricultural biomass is collected, sorted, diced into smaller 420 pieces, and then dried in the sun.
Step II: Biomass car-421 bonization. The collected biomass materials are burned in 422 an oil drum. After burning, the carbonized biomass mate-423 rials must be collected and weighed.
Step III: Preparation 424 of binder. The binding substance strengthens the charcoal 425 briquettes. For every 10 kilograms of complete carbonized 426 charcoal powder, combine 0.5 to 0.6 kilograms of starch or 427 cassava flour with 5 to 10 liters of water to create a binder. 428 Step IV: Mixing. Ensure that the binder is uniformly dis-429 tributed throughout the carbonized charcoal's particles. It will 430 enhance the adhesion of the charcoal and produce uniform 431 briquettes.
Step V: Briquetting. A briquetting machine is 432 used to turn the charcoal mixture into charcoal briquettes. 433 To manufacture briquettes of the same size, pour the mixture 434 immediately into the briquetting machine.
Step VI: Drying. 435 Each charcoal briquette is air-dried outdoors. For the bri-436 quette quality test, important properties (moisture content, 437 ash content, heating value, and fixed carbon) are analyzed 438 using ASTM D3173, ASTM D3174, ASTM D5865, and 439 ASTM D3172, in that order. These properties can be con-440 sidered inputs and outputs of each charcoal briquette/DMU 441 in terms of DEA. The selection of suitable biomass materials 442 from agricultural products for processing into fuel briquettes 443 is a complicated decision-making problem due to the multiple 444 interval qualities that must be considered simultaneously. 445 Consequently, the proposed approach is utilized to evaluate 446 each biomass material.

448
A. THE SIMPLE NUMERICAL EXAMPLE 449 Wu et al. [34] proposed a simple numerical example. There 450 are six DMUs with two inputs and two outputs. The data set 451 of the simple numerical example is shown in Table 2.

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The calculation procedure is as follows. Firstly, by solv-453 ing Equation (9) (MO-IDEA-CCR model), the interval CCR 454 scores are obtained as in the last column of Table 2. After 455 that, the optimal weights of inputs and outputs are utilized to 456 calculate the interval ACE scores of each DMU, according to 457 Equations (3) to (6). As a result, the interval ACE scores for 458 each DMU are achieved, as shown in Table 3.     (11), was used to generate the possibility degree matrix, 471 P = (p ij ) 6×6 . For example, the possibility degree score when

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The calculation procedure is the same as in Section A. 520 Firstly, by solving Equation (9), the interval CCR scores are 521 obtained as in the last column of Table 8. After that, the 522 interval ACE scores of each DMU are obtained according to 523 Equations (3) to (6). As a result, the interval ACE scores for 524 all DMUs are achieved as in the second column of Table 9. 525 VOLUME 10, 2022 After obtaining the ACE scores of each DMU, the interval 526 ACE score of arbitrary strategy is set as the initial solution for   Table 9 shows that the final GICE scores for all DMUs  Table 10.  After obtaining Table 10, the possibility degree matrix 538 (P) was transformed into the Boolean matrix (Q). If the 539 possibility degree score of DMU i vs DMU j (p ij ) ≥ 0.50, 540 the Boolean value of DMU i vs DMU j (q ij ) = 1. Otherwise, 541 q ij = 0. Details of Q are shown in Table 11. 542 VOLUME 10, 2022     After that, the ranking vector (λ i ) was calculated.   [35]. In the Wang et al. method [35], 548 the distance model based on entropy and TOPSIS was used  all DMUs generated by the proposed method attain constant 562 GICE scores, which means the optimal solutions are the Nash 563 equilibrium points, as demonstrated in [36].

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Thailand is an agricultural-based economy with various agri-566 cultural residue resources that can be used for manufacturing 567 charcoal briquettes. In developing countries, biomass from 568 farm residues can be transformed into charcoal briquettes to 569 VOLUME 10, 2022   Table 13.

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The properties of the seven biomass charcoal briquettes 582 are shown in Table 14, including seven agricultural residues 583 (DMUs) with interval data of inputs and outputs. Let moisture 584 content (%) and ash content (%) be input 1 (⊗x 1 ) and input 2 585 (⊗x 2 ), respectively. The heating value (kcal/kg) and fixed 586 carbon (%) are output 1 (⊗y 1 ) and output 2 (⊗y 2 ), 587  The calculation procedure is the same as in Section A.

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By solving the MO-IDEA-CCR model; the interval CCR 593 scores were obtained as in the last column of Table 14.        Table 19.  DMUs generated by the proposed method attain constant 625 GICE scores, which means the optimal solutions are the Nash 626 equilibrium points, as demonstrated elsewhere [36].

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In developing countries, agricultural residuals can be used to 629 make biomass charcoal briquettes for cooking and heating. 630 This idea is one good idea to solve the energy shortage 631 problem of almost every agricultural country. However, the 632 important properties of each biomass material must first 633 be considered to find suitable biomass for processing into 634 charcoal briquettes. Sometimes these qualities are measured 635 with imprecise values, making it exceedingly challenging 636 to rank biomass materials (DMUs). To solve this problem, 637 this paper offers the new MO-G-IDEA-CE method based 638 on the Boolean possibility degree to tackle the IDEA rank-639 ing problems, including seven biomass materials with inter-640 val properties and a simple numerical example. Unlike the 641 existing IDEA models, the proposed models can be used to 642 generate the lower and upper bounds of interval efficiencies 643 for all DMUs simultaneously. The optimal weights of inputs 644 and outputs from the proposed models satisfy the entire 645 IDEA, but the optimal weights of inputs and outputs from 646 the existing models do not. Through three examples, we find 647 that the proposed method has a very high correlation with 648 the other methods and provides a new direction for IDEA 649 ranking problems based on the ideas of the traditional Game 650 cross-efficiency method and the Boolean possibility degree. 651 In particular, for the case study of seven biomass materials, 652 after the Spearman correlation test, the correlation coeffi-653 cients (r s ) for the proposed method and Wang's method, and 654 Wu et al.'s method are calculated as r s = 1.000 and 0.964, 655 respectively.

656
Although three numerical examples have illustrated our 657 method's advantages, potential, and applications, the limi-658 tation of the proposed method is that using it for solving 659 more significant IDEA problems or other IDEA problems 660 equilibrium point, or it may take a more significant number of 662 calculation iterations. However, for future work, we believe 663 the proposed method can be extended or adapted to tackle 664 other complicated IDEA problems in real-world situations.

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In addition, it is hard to develop the proposed method with the 666 fuzzy cross-efficiency evaluation method to measure DMUs 667 with fuzzy or missing data, but this direction is worth further