Application and Analysis of Methods for Selecting an Optimal Solution

Article pubs.acs.org/IECR

Application and Analysis of Methods for Selecting an Optimal Solution from the Pareto-Optimal Front obtained by Multiobjective Optimization Zhiyuan Wang and Gade Pandu Rangaiah* Department of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 117585 ABSTRACT: Process optimization often has two or more objectives which are conflicting. For such situations, multiobjective optimization (MOO) provides many optimal solutions, which are equally good from the perspective of the given objectives. These solutions, known as Paretooptimal front and as nondominated solutions, provide deeper insights into the trade-off among the objectives and many choices for implementation. In the past 20 years, researchers have applied MOO to numerous applications in chemical engineering. However, selection of an optimal solution from the set of nondominated solutions has not received much attention in the chemical engineering literature. In the present study, 10 methods for selecting an optimal solution from the Pareto-optimal front are carefully chosen and implemented in an MS Excel-based program. Then, they are applied to the selection of an optimal solution in many benchmark or mathematical problems and chemical engineering applications, and their effectiveness and similarities are analyzed. Results of analysis indicate that, among the 10 methods studied, technique for order of preference by similarity to ideal solution, gray relational analysis, and simple additive weighting are better for choosing one of the Pareto-optimal solutions. been on finding the nondominated solutions for the application under study. The next step of selecting one of the nondominated (i.e., Pareto-optimal) solutions did not receive much attention in chemical engineering until recently. Rangaiah et al.6 noted that 20 out of 65 studies reviewed have employed one or more selection methods for choosing one of the nondominated solutions, and most of these 20 studies are related to energy. Selection methods employed in these 20 studies include technique for order of preference by similarity to ideal solution (TOPSIS) in Shirazi et al.,7 linear programming technique for multidimensional analysis of preference (LINMAP) in Sanaye and Modarrespoor,8 faire un choix adequat (FUCA) in Ouattara et al.,9 preference ranking organization method for enrichment of evaluations II (PROMETHEE II) in Alsayed et al.,10 clustering and closest to the ideal point in Sharafi and Elmekkawy,11 and fuzzy membership function in Niknam et al.12 Of these, TOPSIS13 is the common selection method employed. Although methods for selecting a Pareto-optimal solution are yet to become common in chemical engineering, there is extensive literature in the area, known as multicriteria decision analysis (MCDA) and multicriteria/choice decision making (MCDM).9

1. INTRODUCTION Optimization of design, operation, and control of chemical processes often involves two or more objectives. Multiobjective optimization (MOO), also called multicriteria optimization, refers to finding decision variables which correspond to the optimum of two or more objectives. Objectives for a particular application are often conflicting, which means some compromise in one or more other objectives is required to achieve the optimum for one objective. Therefore, MOO provides many optimal solutions, known as nondominated solutions or Paretooptimal front, except when the objectives are not conflicting for which only one unique solution exists. By applying MOO and obtaining nondominated solutions, decision makers could gain deeper insight into the quantitative trade-off among objectives and then select one of the optimal solutions based on the results generated and their own judgment. MOO has found numerous applications in chemical engineering, particularly in the last two decades, motivated by the development of stochastic optimization methods for MOO. These applications have been reviewed by Bhaskar et al.,1 Masuduzzaman and Rangaiah,2 and Sharma and Rangaiah.3 Of these, the last two reviews cover more than 300 applications reported in chemical engineering journals from 2000 to 2012. There are also two monographs on MOO in chemical engineering.4,5 Our recent work6 identified 65 journal articles of interest to chemical engineers, published during January 2013 and February 2015. This and earlier reviews clearly show that the focus of the studies on MOO in chemical engineering has © XXXX American Chemical Society

Received: Revised: Accepted: Published: A

September 6, 2016 November 19, 2016 December 16, 2016 December 16, 2016 DOI: 10.1021/acs.iecr.6b03453 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

It has m rows (with one row for each solution) and n columns (with one column for each objective). The objective can be for maximization (referred to as a benefit criterion) or for minimization (referred to as a cost criterion). The objective matrix contains values of objective functions and not values of decision variables. Therefore, it should not be confused with decision variables in the optimization problem. Decision variables and their values are not required for choosing one of the Pareto-optimal solutions by the selection methods. However, in applications, decision variable values could be important in choosing an optimal solution depending on their relative feasibility. Common symbols used in the following subsections are f ij is the ith value of the jth objective in the objective matrix, Fij the value of f ij after normalization, vij the weighted value or rank of f ij or Fij, and wj the weightage of the jth objective. Note that normalization and weighting may not be the same in different selection methods although the same symbol is used. 2.1. Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). According to this technique, the chosen optimal solution should have the smallest Euclidean distance from the ideal (also known as positive ideal) solution and also the largest Euclidean distance from the negative-ideal solution. The ideal solution is a combination of the best value of each objective in the given optimal solutions. In contrast, negative-ideal solution is a combination of the worst value of each objective in the given optimal solutions. The algorithm given below is from Hwang and Yoon.13 Step 1. Construct normalized objective matrix with m rows and n columns by applying

The main objective of the present study is to apply different methods for selecting a nondominated solution to both mathematical (benchmark) problems and chemical engineering applications and to analyze their performance. For this, 10 methods are chosen considering their simplicity, ability to handle objectives with significantly different magnitude, and user input required. They are implemented in MS Excel using Visual Basic for Applications (VBA) programming. The methods are then applied to 12 mathematical problems and 13 MOO applications to chemical processes. Selection methods with similar concept are compared and discussed to obtain insights on their applicability. Application, analysis, and Excel program of this study will encourage selection and discussion of a Pareto-optimal solution as part of future studies on MOO applications in chemical engineering. The next section presents reasons for choosing the 10 selection methods followed by their principles and algorithm. Section 3 describes the program development and its verification on the examples available in the literature. Section 4 presents the application of the 10 selection methods to both mathematical and benchmark MOO problems and chemical engineering applications. Section 5 compares and discusses the selection methods with similar concepts. Section 6 discusses the significance of user inputs and proposes an approach to minimize the subjectivity of choosing a selection method and its inputs. Concluding remarks are given in the final section.

2. METHODS AND THEIR PRINCIPLES AND ALGORITHMS The recent literature on MOO and its applications in chemical engineering and related fields is reviewed, and 14 selection methods are identified. Note that each method is different in its basis, and consequently the chosen nondominated solution may not be the same by different selection methods. Hence, one may have to choose the selection method itself. One aim of this study is to provide guidelines for this. As outlined in the previous section, three characteristics are used to decide on the methods to implement in the Excel program, apply, and analyze in this study. First, less user input is preferred for convenience and for minimizing its effect on the selected Pareto-optimal solution. Second, the selected method should be able to deal with objectives having significantly different magnitudes, which essentially requires a normalization step. This is particularly important in applications which involve a variety of objectives. Lastly, the principle and algorithm of the method should be simple to understand by decision makers so that they can interpret the chosen solution by different methods. For this, each of the 14 selection methods is assessed for each of the three criteria using a 10-point scale, and then 10 methods with high total score are chosen for the present study. Principle and algorithm of the chosen 10 methods are presented in the following subsections. Of these 10 methods, 9 require weightage (as a fraction) for each of the objectives, as input from the user. Some methods require other inputs, which are mentioned as part of the principle. The GRA method is an exception and does not require any inputs including weightage. The terminology used in the following description of the chosen methods is as follows. Objective matrix (also known as decision matrix in MCDM/MCDA literature) is a matrix of n objectives and their values at m Pareto-optimal (or nondominated) solutions found by an optimization method.

Fij =

fij m

∑i = 1 fij2

(1)

Step 2. Construct weighted normalized objective matrix by multiplying each column with its weight, wj: vij = Fij × wj

(2)

Step 3. Determine the ideal solution, A+, and negative-ideal solution, A−, as follows. First, find the best value of each objective. For maximization objective, the best value is the largest value within the column of the objective matrix. In addition, for minimization objective, the best value is the smallest value in the column. Mathematically, these are given by A+ = {(Max i(vij)|j ∈ J ), (Mini(vij)|j ∈ J ′)|i ∈ 1, 2, ..., m} = {v1+ , v2+ , v3+ , ..., vj+ , ..., vn+}

(3)

where J is the set of maximization objectives and J′ is the set of minimization objectives, from the total set of {1, 2, 3, 4, ..., n}. Next, find the worst value of each objective, which is the smallest and largest value within the column of the objective matrix for maximization and minimization objective, respectively. These values constitute the negative-ideal solution given by A− = {(Mini(vij)|j ∈ J ), (Max i(vij)|j ∈ J ′)|i ∈ 1, 2, ..., m} = {v1− , v2− , v3− , ..., vj− , ..., vn−} B

(4)

DOI: 10.1021/acs.iecr.6b03453 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article


where S+ = Mini∈mSi, S− = Maxi∈mSi, R+ = Mini∈mRi, R− = Maxi∈mRi. Parameter γ with 0 ≤ γ ≤ 1 is the weight of maximum group utility, and 1 − γ is the weight of regret, both of a Pareto-optimal solution. In other words, γ > 0.5 represents a decision-making process that uses the strategy of emphasizing group utility, by consensus when γ = 0.5 or with veto when γ < 0.5. Step 4. Rank Pareto-optimal solutions, sorting by the value of Q i|i = 1, 2, ..., m in decreasing order. Propose a compromise solution, A(1) by the measure Mini∈mQ i if the following two conditions are satisfied. C1. Acceptable advantage: R(A(2)) − R(A(1)) ≥ 1/(m − 1), where A(1) and A(2) are the solutions in the first and second positions, respectively, in the ranking list by Qi. C2. Acceptable stability in decision making: solution A(1) must also be the best ranked by Si and Ri|i = 1, 2, ..., m. If one of the above conditions is not satisfied, then a set of compromise solutions is proposed as follows. Solutions A(1) and A(2) are both recommended if condition C2 is not satisfied, and solutions A(1), A(2), A(3), ..., A(d) are all recommended if condition C1 is not satisfied. Here, A(d) is determined by the relation R(A(d)) − R(A(1)) < 1/(d − 1) for maximum d (i.e., positions of these alternatives are close). 2.4. Faire Un Choix Adéquat (FUCA). For each of the objectives, rank 1 is assigned to the best value, and rank m is assigned to the worst value. Then, for each solution in the Pareto-optimal front, a weighted summation is computed. The selected solution is that having the smallest overall rank value.16 Step 1. For each of the objectives, rank 1 is assigned to the best value, and rank m is assigned to the worst value. If the optimization purpose is maximization, then the best value would be the largest value within the column; otherwise, the best value would be the smallest value in the column. This is the same as the best and worst values defined in TOPSIS. Step 2. A weighted summation for each optimal solution, i, is computed.

Step 4. Calculate the Euclidean distance between each solution and the ideal and negative-ideal solution: Distance to positive ideal n

∑ (vij − vj+)2

Si + =

i = 1, 2, 3, ..., m (5)

j=1

Distance to negative ideal n

∑ (vij − v−j )2

Si − =

i = 1, 2, 3, ..., m (6)

j=1

Step 5. Calculate the closeness of each optimal solution: Ci =

Si − Si − + Si +

(7)

When Si− = 0, Ci = 0 and solution i is the closest to the negative ideal. When Si+ = 0, Ci = 1 and i is the closest to the positive ideal. The optimal solution having the largest Ci is the recommended solution. 2.2. Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP). LINMAP shares a principle similar with that of TOPSIS, except that the latter involves the calculation of both distances to positive and negative ideal points whereas the former considers only the distance to the positive ideal. LINMAP selects the optimum solution based on the distances of Pareto optimal solutions from the ideal solution. The chosen solution should have the shortest Euclidean distance from the ideal solution.14 Steps 1−4 in LINMAP are the same as those in TOPSIS except that the distance to the negative ideal is not calculated. The recommended solution is that having the largest Si+. 2.3. Viekriterijumsko Kompromisno Rangiranje (VIKOR). In this method, the chosen solution has the smallest selection value, which is determined by the following steps.15 The physical meaning of the selected solution recommended by VIKOR is similar to that of LINMAP, both representing the shortest distance to the positive ideal solution, but with different mathematics. Besides weightage of each objective, VIKOR requires a parameter, γ, in step 3 below. In the Excel program developed, γ = 0.5 was used as the default value; this can be changed, if required. Step 1. For each objective j = 1, 2, 3, ..., n, determine the best and worst values given by F+j = Maxi∈m f ij and F−j = Mini∈m f ij for maximization objectives and F+j = Mini∈m f ij and F−j = Maxi∈m f ij for minimization objectives. Step 2. Compute Si (i.e., sum of weighted fractional distances of each solution from the best value) and Ri (i.e., maximum of weighted fractional distances of each solution from the best value) for each solution, i = 1, 2, 3, ..., m, given by ⎛ F j+ − f ⎞ ij ⎟ Si = ∑ wj⎜⎜ + F − F j− ⎟⎠ j=1 ⎝ j

n

vi =

(11)

j=1

where rij is the rank of solution i for objective j. The solution with the smallest vi is the recommended optimal solution. 2.5. Gray Relational Analysis (GRA). In this algorithm, gray relational coefficient (GRC) is used to describe the similarity between each candidate network (i.e., objective values of each optimal solution) and the best reference network (i.e., an ideal network formed by choosing the best value of each objective). GRA is usually implemented in three steps: data normalization, defining the ideal sequence, and computing GRC.15 As stated earlier, this method does not require weights or any other input from the user. Step 1. Normalization of objective values of Pareto-optimal solutions is performed according to the two situations: largerthe-better for maximization and smaller-the-better for minimization, as follows respectively

n

⎡ ⎛ F + − f ⎞⎤ j ij ⎟⎥ R i = Max j ∈ n⎢wj⎜⎜ + ⎢ F j − F j− ⎟⎥ ⎠⎦ ⎣ ⎝

∑ (rij × wj)

(8)

Fij =

(9)

fij − mini ∈ m fij max i ∈ m fij − mini ∈ m fij

(12)

Step 3. Compute Qi for i = 1, 2, 3, ..., m, given by ⎛ S − S+ ⎞ ⎛ R i − R+ ⎞ Q i = γ ⎜ −i + ⎟ + (1 − γ )⎜ − +⎟ ⎝S − S ⎠ ⎝R − R ⎠

Fij = (10) C

max i ∈ m fij − fij max i ∈ m fij − mini ∈ m fij

(13) DOI: 10.1021/acs.iecr.6b03453 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article


of ELECTRE II is to rank the extent of one objective outranking other objectives. Then, select the strongest one based on the rank values (i.e., the solution with the largest ranking score).17 This method requires the user to input weightage for each objective as well as two pairs of threshold values (C*, D* and C−, D− in Step 4). In the Excel program developed, C* = 0.5, D* = 0.1, C− = 0.4, and D− = 0.2 were used as default values; these can be changed, if required. Step 1. Modify all objective values to maximization as follows. For the maximization objectives,

Step 2. Find the reference network points:

F j+ = max i ∈ m Fij

(14)

Step 3. Find the difference between F+j and Fij:

ΔIij = |F j+ − Fij|

(15)

Step 4. Find the value of GRC of each optimal solution: GRCi =

1 m

n

∑ j=1

Δmin +Δmax ΔIij + Δmax

(16)

Fij = fij

where Δmax = maxi∈m,j∈n(ΔIij) and Δmin = mini∈m,j∈n(ΔIij). Step 5. Find the largest GRCi; the corresponding solution is the recommended optimal solution. 2.6. Simple Additive Weighting (SAW). The score of each optimal solution, i.e., row i in the objective matrix is obtained by summing the product of normalized objective j and its weight, wj. The selected solution is that having the largest score.15 Step 1. Construct the normalized objective matrix of m rows and n columns by applying Fij =

fij f j+

For the minimization objectives,

Fij = −fij

f j− fij

n

C(a , b) =

for a minimization criterion, where f j− = Mini ∈ mfij

To avoid division by zero in the above definitions, the maximum value of a maximization objective and values of a minimum objective cannot be zero. Step 2. Construct the weighted normalized objective matrix by applying

⎧ 0 if Fj(a) ≥ Fj(b) ∀ j ⎪ D(a , b) = ⎨ Max [F (b) − F (a)] j j j ⎪ otherwise ⎩ δ

(19)

(26)

Step 3. Find the score of each optimal solution, i.e., sum of weighted values by

Here, δ is the maximum difference of any criterion, maxa,b,j[Fj(b) − Fj(a)], which is the largest difference in the pair comparison of all solutions for each of the objectives. The discordance (matrix) element measures the strength of the argument that, when solution a is compared to solution b for a given objective j, the value of Fj(a) is significantly worse than Fj(b). Step 3. Make the selection. After the concordance and discordance indices (i.e., elements in the matrices) for each pair of Pareto-optimal solutions are computed, two types of outranking relations are built by comparing the indices with two pairs of threshold values: (C*, D*) and (C−, D−). The pair (C*, D*) is the concordance and discordance thresholds for the strong outranking relation, and the pair (C−, D−) is the thresholds for the weak outranking relation; note that C* > C− and D* < D−. Next, outranking relations are built according to the following two rules: (1) if C(a, b) ≥ C*, D(a, b) ≤ D*, and C(a, b) ≥ C(b, a), then alternative a is regarded as strongly outranking alternative b; and (2) if C(a, b) ≥ C−, D(a, b) ≤ D−, and C(a, b) ≥ C(b, a), then alternative a is regarded as weakly outranking alternative b. Once the outranking relationship is decided, mark SF at the corresponding position; here, SF stands for strong outranking relation. For instance, if C(1,2) satisfies the above rule 1, then mark C(1,2) as SF. In the end, count the number of SF in each row, which represents the extent of one

n

∑ vij j=1

(20)

Step 4. Find the largest Ai, and the solution i is the recommended optimal solution. 2.7. Multiplicative Exponent Weighting (MEW). The score of each optimal solution, i.e., row i in the objective matrix is obtained by the product of weighted exponent of normalized objective i. The chosen solution is that having the largest score.15 Algorithm of MEW is the same as that of SAW except that eqs 19 and 20 are replaced by the following equations. vij = Fij wj

(21)

n

Ai =

∏ vij j=1

(25)

If solution a is equal or preferred to (i.e., at least as good as) solution b for objective j (i.e., Fj(b) − Fj(a) ≤ 0), then wj* = wj, else w*j = 0. The individual concordance index, C(a, b) measures the strength of the argument that, when solution a is compared to solution b for a given objective j, the value of Fj(a) is at least as good as Fj(b). Construct discordance matrix with m rows and n columns, whose elements are given by

(18)

Ai =

∑ w*j j=1

for a maximization criterion, where f j+ = Max i ∈ mfij

vij = Fij × wj

(24)

Step 2. Construct concordance matrix with m rows and n columns. In the following, solution a and solution b represent any two Pareto-optimal solutions selected for comparison. An element in the concordance matrix is given by

(17)

Fij =

(23)

(22)

In eq 21, Fij cannot be negative because wj is between 0 and 1. Therefore, this method is not suitable for situations in which negative values are present in the objective matrix. 2.8. Elimination and Choice Translating Priority II (ELCTRE II). This method is the first of ELECTRE methods especially designed to deal with ranking problems. The principle D


Article


Step 4. The credibility matrix of m rows and n columns is given by

objective outranking other objectives. Then, by the rule of intersection,17 the Pareto-optimal solution with the most SF in the horizontal row (i.e., it outranks other solutions) and least SF in the corresponding vertical column (i.e., it is outranked by other solutions) is the recommended optimal solution. 2.9. Elimination and Choice Translating Priority III (ELECTRE III). The basic principle ELECTRE III is to rank the extent of one solution outranking other solutions.18 Then, select the strongest one based on the rank values (i.e., the solution with the largest ranking score). As compared to ELECTRE II, ELECTRE III employs a different set of equations to calculate the concordance and discordance matrix elements. This method requires the user to input weightage for each objective, indifference threshold (Qj), preference threshold (Pj), veto threshold (Vj), and a cutoff value (0−1). By default, the Excel program developed uses the indifference, preference, and veto thresholds as 10%, 20%, and 80% of the range of optimal values of each objective, respectively,19 and zero cutoff value (i.e., cutoff was not applied); these can be changed, if required. Step 1. Modify the objective matrix as in ELECTRE II. Step 2. As before, solutions a and b represent any two selected solutions to be compared. Indifference threshold (Qj) is the range of variation of each objective, for which it is not possible for the decision maker to favor the objective of one solution over that of another solution. It therefore represents the range of values over which the particular objective of two solutions is indiscernible. If the difference between two values for a given objective exceeds preference threshold (Pj), then preference is given to the solution with the better value. Veto threshold (Vj) serves to reject a solution relative to another solution if the difference between the values of a particular objective is too large to be acceptable. An element of the concordance matrix of m rows and columns is computed by

S(a , b) ⎧ ⎪ ⎪ =⎨ ⎪C(a , b) ⎪ ⎩

j ∈ J(a , b)

1 − Dj(a , b)

otherwise

1 − C(a , b)

(30)

n

∑ wC j j(a , b)

D(a , b) =

(27)

j=1

∏ [1 − (Dj(a , b))3 ] (31)

j=1

where

where ⎧ ⎪ ⎪ ⎪ Dj(a , b) = ⎨ ⎪ Fj(b) − ⎪ ⎪ Vj ⎩

⎧ 1 if Fj(b) − Fj(a) ≤ Q j ⎫ ⎪ ⎪ ⎪ ⎪ − > 0 if F ( b ) F ( a ) P ⎪ ⎪ j j ⎬ Cj(a , b) = ⎨ ⎪ Pj − [Fj(b) − Fj(a)] ⎪ ⎪ if Q j < Fj(b) − Fj(a) ≤ Pj ⎪ ⎪ ⎪ Pj − Q j ⎩ ⎭ (28)

if Fj(b) − Fj(a) > Vj

(32)

⎫ ⎪ 0 if Fj(b) − Fj(a) ≤ Pj ⎪ ⎪ ⎬ ⎪ Fj(a) − Pj if Pj < Fj(b) − Fj(a) ≤ Vj ⎪ ⎪ − Pj ⎭ (29) 1

⎫ ⎪ 0 if Fj(b) − Fj(a) ≤ Pj ⎪ ⎪ ⎬ ⎪ Fj(a) − Pj if Pj < Fj(b) − Fj(a) ≤ Vj ⎪ ⎪ − Pj ⎭ 1

The discordance matrix element, D(a, b), measures the strength of the argument that, when solution a is compared to solution b for the jth objective, the value of Fj(a) is significantly worse than Fj(b). Step 4. Construct the credibility matrix by multiplying the concordance element and discordance element at the corresponding location.

The individual concordance index, C(a, b), measures the strength of the argument that, when solution a is compared to solution b for a given objective j, the value of Fj(a) is at least as good as Fj (b). Step 3. Construct elements of discordance matrix by ⎧ ⎪ ⎪ ⎪ Dj(a , b) = ⎨ ⎪ Fj(b) − ⎪ ⎪ Vj ⎩

∏

if Dj(a , b) ≤ C(a , b) ∀ j

where J(a, b) is a set of objectives for which Dj(a, b) > C(a, b). Step 5. Finally, make the selection based on the credibility matrix by calculating the difference between the strength (sum of row) and weakness (sum of column) of each solution, with or without cutoff values. If applying the cutoff value, every element of the credibility matrix with a value greater than the cutoff value is rounded up to 1.0, and those with a value smaller than the cutoff value are rounded down to 0.0. 2.10. Net Flow Method (NFM). The principle of NFM, developed from ELECTRE methods, is to rank the difference of the extent of one objective outranking other objectives and the extent of other objectives outranking this objective.20 Then, the solution having the largest rank value is recommended. NFM is different from ELECTRE III in the equations to construct the credibility matrix. It too requires the user to input weightage for each objective, indifference threshold (Qj), preference threshold (Pj), and veto threshold (Vj). By default, the Excel program developed uses the indifference, preference, and veto thresholds as 10%, 20% and 80% of the range of optimal values of each objective, respectively.19 Step 1. Modify the decision matrix as in ELECTRE II. Step 2. Construct the concordance matrix as in ELECTRE III. Step 3. Construct discordance matrix of m rows and n columns by

n

C(a , b) =

C(a , b)

if Fj(b) − Fj(a) > Vj

σ(a , b) = C(a , b) ·D(a , b)

(33)

Step 5. Make the selection to calculate the ranking score of each solution, i, as follows: m

Si =

k=1

The discordance matrix element measures the strength of the argument that, when solution a is compared to solution b for a given objective j, the value of Fj(a) is significantly worse than Fj(b).

m

∑ σ ( i , k ) − ∑ σ (k , i ) k=1

(34)

The first term evaluates the extent to which solution i performs relative to all other solutions, while the second term E


Article


Figure 1. Main interface of the selection methods program in Excel-VBA.

the last column provides the ranking score of each optimal solution; the selected solution (row) is shaded for ready reference, and the selected solution is shown as a red dot on plots of one objective against another objective. Plots with selected solutions by all methods are available in the worksheet “All Charts”, for visual comparison of optimal solutions recommended by different methods. Implemented code of each method was validated by application to at least one example available in the literature, as summarized in Table 1. Each method in the developed program recommended the same optimal solution as in the literature.

evaluates the performance of all other solutions relative to solution i. The solution with the highest ranking score (Si) is the recommended optimal solution.

3. IMPLEMENTATION OF METHODS AND VALIDATION All 10 methods presented in the previous section are implemented in a program, developed in MS Excel using Visual Basic Application (VBA). The reason for choosing MS Excel is that it is probably the most readily available and most commonly used computational tool in both academia and industry. Hence, researchers and engineers familiar with MS Office can utilize the program developed in this study. Furthermore, the selection methods program described here can be used in conjunction with our programs for MOO, namely, Excel-based MOO21 and integrated multiobjective differential evolution.3 All these programs are available at no cost to interested readers by contacting the corresponding author of this paper. The user interface of selection methods program is organized in a number of worksheets in Excel. The overview worksheet provides brief guidelines on using the program. Pareto-optimal solutions (i.e., values of all objectives with or without corresponding values of decision variables and constraints) obtained from a MOO program are provided by the user in the worksheet entitled “Results from MOO”. On the “Main Interface” worksheet (Figure 1), the user needs to give the number of objectives, number of decision variables (zero if their values are not given), number of constraints (zero if their values are not given), population size (i.e., number of optimal solutions), and type (i.e., maximization or minimization) and weight for each objective. Then, the user can execute selection methods, one by one or all by clicking on the corresponding icon (Figure 1). The results of executing the selection methods are presented in both numbers and plots, with one worksheet for each method. In the worksheet for the results of a selection method,

Table 1. Summary of Examples Used for Validating the Selection Methods Program method TOPSIS

LINMAP

VIKOR FUCA

GRA SAW MEW ELECTRE II ELECTRE III NFM

F

example from the literature

reference

solving the traffic problem in the busiest highway by proposing 4 different solutions and considering 15 objectives design of powered Stirling heat engine with 3 objectives (namely, maximize power and thermal efficiency and minimize pressure loss) and 40 solutions a mountain climber must choose 1 of 3 destinations, considering 2 objectives of risk and altitude a design problem with 3 objectives (namely, maximize net present worth and minimize payback period and flow rate) and 15 solutions sport score evaluation problem with 5 five contestants attending a decathlon competition and 10 objectives personnel selection problem with 5 candidates and 7 objectives decision making among 6 candidate networks with 4 objectives: delay, bandwidth, cost, and jitter finding the best location for a wastewater treatment plant in Ireland, with 5 alternatives and 7 objectives choosing one of 6 hotels for MCDM conference considering 6 objectives volatile organic compound recovery process design for 5 economic and environmental objectives from 100 nondominated solutions

22

14

23 24

25 26 27 17 18 19


Article

Industrial & Engineering Chemistry Research Table 2. Summary of 12 Mathematical Problems Used for Evaluating the Selection Methods S. No.

mathematical Problems

effect of weightage: cases where recommended solution is similar (see section 4.1 for details)

1

ZDT1 function with 2 objectives and 500 nondominated solutions

similar solution by FUCA in cases a and c, and by ELECRE II and NFM in cases a and b

2


similar solution by TOPSIS, ELECTRE III, and NFM in cases a and b

3


similar solution by FUCA in cases a and c

4


similar solution by FUCA in cases a and c, and by ELECTRE II in cases a and b

5


similar solution by FUCA, SAW, MEW, and NFM in cases a and c

6


similar solution by TOPSIS and MEW in all cases; by ELECTRE II in cases a and b; and by FUCA, SAW, ELECTRE III, and NFM in cases a and c

7

Tanaka function with 2 objectives and 90 nondominated solutions

similar solution by TOPSIS and FUCA in cases a and c, by SAW and MEW in cases a and b, and by NFM in cases a and c

8

Osyczka function with 2 objectives and 100 nondominated solutions

similar solution by TOPSIS, LINMAP, and VIKOR in cases a and b; by ELECTRE II in cases a and c; and by ELECTRE III and NFM in cases a and b

9

Belegundu function with 2 objectives and 100 nondominated solutions

similar solution by SAW in cases a, b, and c

10

Binh function with 2 objectives and 669 nondominated solutions

similar solution by FUCA in cases a and b and by SAW, MEW, and ELECTRE II in cases a and b

11

Tamaki function with 3 objectives and 337 nondominated solutions

similar solution by TOPSIS in cases a and d and by SAW, MEW, and ELECTRE III in cases a and d

12

DTLZ3 function with 3 objectives and 411 nondominated solutions

similar solution by TOPSIS in cases a and c; by FUCA in cases a, b, and d; and by MEW in cases a and c

Table 3. Summary of 13 Chemical Engineering Problems Used for Evaluating Selection Methods S. No.

effect of weightage: cases where recommended solution is similar (see sections 4.2 and 4.3 for details)

brief details of the application

reference

1

cumene production process optimization to minimize material loss and total capital cost; 70 nondominated solutions

similar solution by VIKOR, SAW, MEW, ELECTRE III, and NFM in cases a and c

28

2

dividing-wall column design optimization to minimize total capital cost and utility cost; 55 nondominated solutions

no similar solution for different weightages

29

3

biodiesel production process optimization to maximize profit and minimize heat duty; 100 nondominated solutions

similar solution by SAW, MEW, ELECTRE III and NFM in cases a and b

30

4

a chemical engineering design problem to maximize net present worth and profit before tax and minimize flow rate of the heavy oil; 200 nondominated solutions

similar solution by FUCA in cases a and c

31

5

a chemical engineering design problem to minimize annual cost, eutrophication potential, and atmospheric acidification potential; 200 nondominated solutions

similar solution by SAW and MEW in cases a and c and by NFM in cases a and d

31

6

a chemical engineering design problem to maximize net present worth and minimize payback period and flow rate of the heavy oil; 200 nondominated solutions

similar solution by ELECTRE II in cases a and d

31

7

agrofood supply chain design problem to maximize net present value and minimize global warming potential and investment; 200 nondominated solutions

no similar solution

32

8

agrofood supply chain design problem to maximize net present value and minimize global warming potential and variable unit cost; 200 nondominated solutions

similar solution by ELECTRE III and NFM in cases a and d

32

9

agrofood supply chain design problem to maximize net present value and minimize global warming potential and variable sales price strategy; and 200 nondominated solutions

similar solution by ELECTRE III in cases a and d

32

10

regression of phase equilibrium data at 40 °C with 4 objectives to minimize; 95 nondominated solutions

similar solution by ELECTRE III and NFM in cases a and c

5

11

regression of phase equilibrium data at 50 °C with 4 objectives to minimize; 100 nondominated solutions

similar solution by NFM in cases a and b and in cases c and d

5

12

volatile organic compound recovery process design based on 5 economic and environmental objectives; 100 nondominated solutions

similar solution by SAW in cases a, b, and d and by ELECTRE III and NFM in cases c, d, and e

19

13

design of an industrial water network for 2 objectives: fresh water flow rate and total flow rate through regeneration units

similar solutions by TOPSIS, LINMAP, VIKOR in cases a and c; by ELECTRE II in cases a and c; by ELECTR III and NFM in cases a, b and c; by FUCA in cases a and c; and by SAW in cases a and b

33

G


Article


Figure 2. DTLZ3 function. Filled triangle is the recommended solution by selection methods using equal weightage for F1, F2, and F3.

objectives to minimize.34 The number of nondominated solutions available at http://www.cs.cinvestav.mx/~emoobook is 4106. To decrease the computational effort, one set of objective values is selected from every 10 sets, and finally the objective matrix with 411 nondominated solutions is generated. Four sets of weightage values are used to assess the effect of weightage on the recommended optimal solution by selection methods. The four cases are (a) equal weightage for each objective; (b) 0.8 for F1, 0.1 for F2, and 0.1 for F3; (c) 0.1 for F1, 0.8 for F2, and 0.1 for F3; and (d) 0.1 for F1, 0.1 for F2, and 0.8 for F3. The last three sets have very large weight on one objective in turn and smaller weight on other objectives. Similarly, for all the problems in Tables 2 and 3, case a is always for equal weightage for each objective, and the following cases have very large weight (0.8) on one objective according to the sequence in the text and the remaining weight (0.2) is distributed equally for the other objectives. In DTLZ3, there will be many plots because there are three objectives. To present them for all 10 methods in a single figure, only one of plot of F3 versus F1 for each method is chosen (Figure 2). The bar chart in Figure 2 shows fractional objective values from the respective minimum (eq 35) of the solution rec-

4. APPLICATION TO MATHEMATICAL AND CHEMICAL ENGINEERING PROBLEMS The selection methods program has been applied to choosing one optimal solution from the nondominated solutions of 12 mathematical and 13 chemical engineering problems, as summarized in Tables 2 and 3. The mathematical problems have 2 or 3 objectives and 31−669 nondominated solutions, and the chemical engineering problems have 2−5 objectives and 9−200 nondominated solutions. Data files of all 12 mathematical problems are available from http://www.cs. cinvestav.mx/~emoobook. Nondominated solutions (only objective function values for brevity) of all 13 chemical engineering problems are compiled as part of this study and are available from https://drive.google.com/open?id= 0BzO5dE3XATtyYmdxcEIzNkoxZk0. Results from the application of selection methods to one mathematical problem and for two chemical engineering applications are presented and discussed in this section. These include the effect of weightage to be provided by the user on the optimal solution chosen by each method; see Tables 2 and 3. 4.1. Example of a Mathematical Problem: DTLZ3 Function. DTLZ3 is a mathematical problem with three H


Article


Figure 3. DTLZ3 function. Fractional objective values of the recommended solution by selection methods using different weightages: (a) equal; (b) F1 = 0.8, F2 = F3 = 0.1; (c) F2 = 0.8, F1 = F3 = 0.1; and (d) F3 = 0.8, F1 = F2 = 0.1.

in weightage has negligible effect in these scenarios. The same thing was observed for other applications but in different methods and/or cases, as summarized in Table 2. These results indicate the nonlinear effect of weightage in these methods on the recommended solution. 4.2. Example of Chemical Engineering Problem: Dividing-Wall Column Design. This is a dividing-wall column design problem with two objectives to minimize utility cost and total capital cost and 55 nondominated solutions. Three sets of weightages, namely, (a) equal weightage; (b) 0.8 for utility cost, 0.2 for total capital cost; and (c) 0.2 for utility cost, 0.8 for total capital cost, are tried, and the results are shown in Figures 4 and 5. As shown in these figures, the bar height of an objective decreases with increasing weightage of that objective. However, in other applications studied, weightage has negligible effect in some methods and/or cases, as summarized in Table 3. 4.3. Example of Chemical Engineering Problem: Industrial Water Network Design. This is an industrial water network design problem with two objectives to minimize fresh water flow rate (FW) and total flow rate through regeneration units (TRU). For this design problem, nine nondominated solutions were obtained earlier using the ε-constraint method and BARON in GAMS for solving the resulting single-objective optimization problem.33 For choosing one optimal solution by the selection methods, three sets of weightages, namely, (a) equal weightage; (b) 0.8 for FW and 0.2 for TRU, and (c) 0.2 for FW and 0.8 for TRU are tried. For equal weightage (Figure 6), TOPSIS, LINMAP, VIKOR, GRA, ELECTRE III, and NFM recommend the same optimal solution; FUCA recommends an extreme solution with high FW; SAW and MEW

ommended by each of the 10 methods. Fractional objective values are given by Fractional Fj =

Fj − Fj ,min Fj ,max − Fj ,min

where j = 1, 2, ..., n (35)

The normalization factor (i.e., denominator in the above) for each objective is chosen as the difference between the maximum and minimum of that column in the objective matrix. This allows presenting objective values of significantly different magnitude concisely in a single plot. Equation 35 can be used for both minimization and maximization objective. In the case of minimization objective (which is common in many examples tested), the value of that objective of the recommended solution by a selection method is closer to the minimum if the bar is negligible; otherwise, it is far from the minimum of that objective. On the other hand, in the case of maximization objective, the value of that objective of the recommended solution by a selection method is closer to the maximum if the bar is long (i.e., closer to 1.0). In the DTLZ3 function, the objectives are to be minimized; hence, in Figures 2 and 3, the particular objective is optimized greatly if its bar is very short or absent. As shown in these two figures, the bar height of an objective generally decreases with increasing weightage of that objective. Recall that GRA does not require any weight input from the user; hence, the recommended solution by it is the same in all cases and plots in Figure 3. Among other methods, the recommended solution by TOPSIS is similar in cases a and c; the recommended solution by FUCA is similar in cases a, b, and c; and the recommended solution by MEW is similar in cases a and c. In effect, change I


Article


Figure 4. Dividing-wall column design optimization for utility cost (UC) and total capital cost (TCC). Filled triangle is the recommended solution by selection methods using equal weightage for objectives.

from the three groups and are placed in a fourth group. All four groups are discussed in the following sections. 5.1. TOPSIS, LINMAP, and VIKOR. Both TOPSIS and LINMAP usually recommend similar solutions, sometimes even exactly the same solution. Sometimes, VIKOR also recommends a solution similar to that of the other two, but in the majority of cases, it chooses a different optimal solution. The major difference between TOPSIS, LINAMP, and VIKOR is that TOPSIS requires that the selected optimal solution should be closest to the positive ideal point and simultaneously farthest from the negative ideal point (section 2). On the other hand, both VIKOR and LINMAP require the selected optimal solution to be the closest to the positive ideal point. Hence, TOPSIS is more suitable for those decision makers who would like to avoid risk and simultaneously maximize benefit, whereas LINMAP and VIKOR are suitable for users who like to achieve the maximum benefit only. TOPSIS has one disadvantage, as described by Yuan et al.35 This can be explained considering Figure 8, where A* is the positive ideal point and A− is the negative ideal point. Recall that the relative closeness of TOPSIS is given by eq 7, where Si− is the distance to the negative ideal point and Si+ is the

suggest a similar solution with high TRU; and the solution recommended by ELECTRE II is different from all others. The general trend in Figure 7 is that bar height of an objective decreases with increasing weight of that objective. Application of selection methods to the industrial water network design is the same as that for other examples even though the former is solved using the ε-constraint method and a deterministic optimization method for solving the resulting single-objective optimization problem. The only difference is that the number of nondominated solutions is small in this example. This is similar to many examples in Table 1 (used for validating the selection methods program), which have just 3−6 nondominated solutions (or alternatives or candidates) for selection.

5. COMPARISON AND DISCUSSION OF SIMILAR METHODS Based on the principle and algorithm (described in section 2) as well as recommended solutions in the applications studied (section 4), 8 of the 10 methods studied can be put into 3 groups. The remaining two (GRA and FUCA) are different J


Article


Figure 5. Dividing-wall column design optimization for utility cost (UC) and total capital cost (TCC). Fractional objective values of the recommended solution by selection methods using different weightages for objectives: (a) equal weightage; (b) TCC = 0.8, UC = 0.2; and (c) TCC = 0.2, UC = 0.8.

distance to the positive ideal point. When solution Am, which is the midpoint between positive and negative ideal points, and solution Ai in Figure 8 are compared, Ai is always preferred to Am as long as the distance from Ai to A− (i.e., Si−) is greater than that from Ai to A* (i.e., Si+). This will be ridiculous when Ai is very far from both positive and negative ideal points. In addition to objective weights, VIKOR involves weight of strategy. Greater than 0.5 for this represents the decisionmaking process that uses the strategy of maximum group utility. It is found that the objective with higher weightage is the one to be optimized with the increase of weight of strategy. In addition, the objective with higher absolute difference between adjacent solutions (e.g., Fi+1,1 − Fi,1 and Fi+1,2 − Fi,2, gain or drop because the absolute value is employed) is the chosen objective j to be optimized when the weight of the strategy in VIKOR increases. After comparison and analysis, TOPSIS is recommended over LINMAP and VIKOR because it requires fewer user inputs and considers both positive and negative impacts. 5.2. SAW and MEW. These two methods share many similarities and usually recommend very similar solutions, as Figures 3 and 5 reflect. Among all the problems and weightages considered, only for the Osyczka function are the solutions recommended by SAW and MEW very different (see Figure 9). For using either SAW or MEW, the minimization objective value of any optimal solution and the maximum value of the maximization objective cannot be 0 because they are in the denominator in the normalization equations (eqs 17 and 18). In addition, MEW cannot be employed for problems with a negative value in the objective matrix. There are some situations (e.g., annual profit of a process) where negative values

may be involved. Hence, SAW is recommended over MEW because the former can handle more problems. 5.3. ELECTRE II, ELECTRE III, and NFM. These three methods have a similar fundamental concept of pairwise comparison between every objective of each pair of solutions. This makes them computationally intensive, particularly for problems with numerous nondominated solutions. As shown in Figures 3, 5, and 10, the recommended solutions by ELECTRE III and NFM are similar in all applications and cases studied. However, the recommended solutions by ELECTRE II with the default threshold values used are generally different from those of ELECTRE III and NFM. As mentioned in section 2, the user needs to give C*, D*, C−, and D− threshold values for ELECTRE II and indifference threshold (Qj), preference threshold (Pj), and veto threshold (Vj) for both ELECTRE III and NFM. ELECTRE III needs an additional cutoff value. Values of C*, D*, C−, and D− do not have clear operational and physical meaning in practical problems. Hence, it is not easy for the user to decide these threshold values. Therefore, NFM, which requires fewer and physically meaningful inputs, is recommended from this group of similar methods. 5.4. FUCA and GRA. The advantage of FUCA as compared with other methods is that its principle is relatively simple. However, it still requires weightage values from the user, which makes it less attractive than GRA. Among the 10 methods tested, GRA is the only method that does not require any inputs from decision makers. It is found that the result generated by GRA is usually quite close to (sometimes exactly the same as) those obtained by other methods under the case of equal weightage for objectives involved. Therefore, to reduce K


Article


Figure 6. Industrial water network design optimization for fresh water flow rate (FW) and total flow rate through regeneration units (TRU). Filled triangle is the recommended solution by selection methods using equal weightage for objectives.

the subjectivity of selecting an optimal solution, whenever the user inputs are required, GRA is recommended from this group.

mean that it is better to choose a method whose principle and the required inputs are easy to understand. The selected optimal solution will depend not only on required inputs but also on the selection method. Hence, choosing a selection method and its inputs is itself an optimization problem involving some subjectivity. For this and for minimizing the subjectivity, we propose the following approach. It can be modified suitably according to the complexity and level of MOO application under consideration. 1. Assemble several decision makers, some having relevant technical expertise (such as process technology, operation and control, safety, and environmental impact) and others from management to bring in knowledge on company policy and government regulations. 2. Brief the decision makers on the short-listed selection methods, their principle, and required inputs. 3. The decision makers can discuss and decide together the weights for objectives because they are easy to understand and their values depend on company policy, government regulations, and technical benefits.

6. DISCUSSION MOO and selection of an optimal solution have been studied in the field of economics, management, and strategy planning for several decades. In the chemical engineering field, MOO is employed increasingly in the past 15 years, but selection of an optimal solution is not yet commonly studied. Application of 10 selection methods presented above, with the exception of GRA, for selecting one optimal solution from the set of nondominated solutions, requires inputs from the user(s), often referred to as decision makers. The most common inputs are weight for each of the objectives involved; other inputs may be required depending on the selection method. Although significance of weight for each objective is obvious, there will be some subjectivity in giving a suitable value to it. Moreover, the effect of weight on the selected optimal solution depends on the selection method. On the other hand, significance of other inputs may not be as simple to understand as weights. All these L


Article


Figure 7. Industrial water network design optimization for fresh water flow rate (FW) and total flow rate through regeneration units (TRU). Fractional objective values of the recommended solution by selection methods using different weightages for objectives: (a) equal weightage; (b) FW = 0.8, TRU = 0.2; and (c) FW = 0.2, TRU = 0.8.

optimal solutions and other factors, if any, not captured by the objective functions in the optimization problem.

7. CONCLUSIONS In this study on methods for selecting one of the nondominated solutions obtained from MOO, principles and algorithms of 10 selection methods are elaborated using consistent terminology. All these methods are implemented in an MS Excel-based software package, which is verified against at least one solved example from the literature. The 10 selection methods with different objective weightages are applied to 12 benchmark mathematical problems and 13 chemical engineering problems. Results show that objective weightages have nonlinear effect on the optimal solution recommended by

Figure 8. Selection of an optimal solution by TOPSIS.

4. Ask each decision maker to choose independently a selection method. Apply the method to find the optimal solution recommended. 5. Review the recommended optimal solutions obtained in the previous step, and choose one of them for implementation. This review should also consider relative feasibility of decision variable values corresponding to the

Figure 9. Selected solution of Osyczka function by SAW (left plot) and MEW (right plot) under three different sets of weightages: square for F1 = 0.2, F2 = 0.8; triangle for F1 = F2 = 0.5; and circle for F1 = 0.8, F2 = 0.2. M


Article


Figure 10. Recommended solutions by ELECTRE II, ELECTRE III, and NFM for cumene process to minimize total capital cost and material loss with equal weightage. processes and power generation. Curr. Opin. Chem. Eng. 2015, 10, 49− 62. (7) Shirazi, A.; Najafi, B.; Aminyavari, M.; Rinaldi, F.; Taylor, R. A. Thermal-economic-environmental analysis and multi-objective optimization of an ice thermal energy storage system for gas turbine cycle inlet air cooling. Energy 2014, 69, 212−226. (8) Sanaye, S.; Modarrespoor, D. Thermal-economic multi-objective optimization of heat pipe heat exchanger for energy recovery in HVAC application using genetic algorithm. Thermal Science 2014, 18, 375− 391. (9) Ouattara, A.; Pibouleau, L.; Azzaro-Pantel, C.; Domenech, S. Economic and environmental impacts of the energy source for the utility production system in the HDA process. Energy Convers. Manage. 2013, 74, 129−139. (10) Alsayed, M.; Cacciato, M.; Scarcella, G.; Scelba, G. Design of hybrid power generation systems based on multi criteria decision analysis. Sol. Energy 2014, 105, 548−560. (11) Sharafi, M.; Elmekkawy, T. Y. Multi-objective optimal design of hybrid renewable energy systems using PSO-simulation based approach. Renewable Energy 2014, 68, 67−79. (12) Niknam, T.; Bornapour, M.; Gheisari, A. Combined heat, power and hydrogen production optimal planning of fuel cell power plants in distribution networks. Energy Convers. Manage. 2013, 66, 11−25. (13) Hwang, C. L.; Yoon, K. Multiple Attribute Decision Making: Methods and Applications; Springer-Verlag: Berlin, 1981. (14) Ahmadi, M. H.; Hosseinzade, H.; Sayyaadi, H.; Mohammadi, A. H.; Kimiaghalam, F. Application of the multi-objective optimization method for designing a powered stirling heat engine: design with maximized power, thermal efficiency and minimized pressure loss. Renewable Energy 2013, 60, 313−322. (15) Martınez-Morales, J. D.; Pineda-Rico, U.; Stevens-Navarro, E. Performance comparison between madm algorithms for vertical handoff in 4g networks. https://pdfs.semanticscholar.org/cd67/ f9b9fee3845e4a9a16113fbdfade0fec8e29.pdf (accessed September 5, 2016), 2010. (16) Fernado, M. M. L.; Escobedo, J. L. P.; Azzaro-Pantel, C.; Pibouleau, L.; Domenech, S.; Aguillar-Lasserre, A. Selecting the best portfolio alternative from a hybrid multiobjective GA-MCDM approach for new product development in the pharmaceutical industry. http://ieeexplore.ieee.org/document/5949271/ (accessed September 5, 2016), 2011. (17) Wang, X.; Triantaphyllou, E. Ranking irregularities when evaluating alternatives by using some electre methods. Omega 2008, 36, 45−63. (18) Ros, J. C. Introduction to decision deck-diviz: Examples and user guide. http://www.decision-deck.org/diviz/_static/ ReportDecisionDeck-DEIM-URV.pdf (accessed September 5, 2016), 2011. (19) Lee, E. S. Q.; Rangaiah, G. P. Optimization of recovery processes for multiple economic and environmental objectives. Ind. Eng. Chem. Res. 2009, 48, 7662−7681. (20) Thibault, J. Net flow and rough sets: Two methods for ranking the pareto domain. In Multi-objective Optimization: Techniques and

a method. Based on the similarities and results, the 10 methods tested are put into four groups, and one method from each group is recommended as the most suitable method. Considering user inputs, simplicity of principle, and applicability, TOPSIS, GRA, and SAW are better for choosing one of the nondominated solutions. Choosing a selection method and giving required user inputs are somewhat subjective. Hence, domain knowledge and optimal values of decision variables should be combined with one or more of these methods for selecting an optimal solution. For this, an approach involving several decision makers is outlined.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Gade Pandu Rangaiah: 0000-0001-8108-6608 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank Dr. Shivom Sharma of Ecole Polytechnique Fédérale de Lausanne, Sion, Switzerland, for his suggestions on the manuscript and Prof. Catherine Azzaro-Pantel of École Nationale Supérieure des Ingénieurs en Arts Chimiques et Technologiques, Toulouse, France, for providing numerical data and her student’s Ph.D. thesis.

■

REFERENCES

(1) Bhaskar, V.; Gupta, S. K.; Ray, A. K. Applications of multiobjective optimization in chemical engineering. Rev. Chem. Eng. 2000, 16, 1−54. (2) Masuduzzaman; Rangaiah, G. P. Multi-objective optimization applications in chemical engineering. In Multi-objective Optimization: Techniques and Applications in Chemical Engineering; Rangaiah, G. P., Ed.; World Scientific: Singapore, 2009. (3) Sharma, S.; Rangaiah, G. P. An improved multi-objective differential evolution with a termination criterion for optimizing chemical processes. Comput. Chem. Eng. 2013, 56, 155−173. (4) Rangaiah, G. P. Introduction. In Multi-Objective Optimization: Techniques and Applications in Chemical Engineering; Rangaiah, G. P., Ed.; World Scientific: Singapore, 2009. (5) Bonilla-Petricioloet, A.; Sharma, S.; Rangaiah, G. P. Phase equilibrium data reconciliation using multi-objective differential evolution with tabu list. In Multi-Objective Optimization in Chemical Engineering: Developments and Applications; Rangaiah, G. P., BonillaPetriciolet, A., Eds.; John Wiley: Chichester, U.K., 2013. (6) Rangaiah, G. P.; Sharma, S.; Sreepathi, B. K. Multi-objective optimization for the design and operation of energy efficient chemical N


Article

Industrial & Engineering Chemistry Research Applications in Chemical Engineering; Rangaiah, G. P., Ed.; World Scientific: Singapore, 2009. (21) Sharma, S.; Rangaiah, G. P.; Cheah, K. S. Multi-objective optimization using ms excel with an application to design of a fallingfilm evaporator system. Food Bioprod. Process. 2012, 90, 123−134. (22) Munier, N. State of the art in decision-making. In A Strategy for Using Multicriteria Analysis in Decision-Making; Springer: New York, 2011. (23) Opricovic, S.; Tzeng, G. H. Compromise solution by mcdm methods: a comparative analysis of vikor and topsis. European Journal of Operational Research 2004, 156, 445−455. (24) Ouattara, A. Methods of eco-design methodology for multiobjective optimization and multiple-criteria decision analysis. Translated from Méthodologie d’éco-conception de procédés par optimization multiobjectif et aide à la décision multicritére. Ph.D. Thesis, Univerity de Toulouse, Toulouse, France, 2011. (25) Chang, C. L.; Tsai, C. H.; Chen, L. Applying grey relational analysis to the decathlon evaluation model. International Journal of The Computer, The Internet and Management 2003, 11, 54−62. (26) Afshari, A.; Mojahed, M.; Yusuff, R. M. Simple additive weighting approach to personnel selection problem. International Journal of Innovation, Management and Technology 2010, 1, 511−515. (27) Savitha, K.; Chandrasekar, C. Vertical handover decision schemes using saw and wpm for network selection in heterogeneous wireless networks. Global Journal of Computer Science and Technology 2011, 11, 19−24. (28) Flegiel, F.; Sharma, S.; Rangaiah, G. P. Development and multiobjective optimization of improved cumene production processes. Mater. Manuf. Processes 2015, 30, 444−457. (29) Lin, H. W. Evaluation of Excel-based multi-objective optimizers for solving chemical process problems. B.Eng. (Chemical Engineering) Dissertation, Department of Chemical and Biomolecular Engineering, National University of Singapore, Singapore, 2016. (30) Patle, D. S.; Sharma, S.; Ahmad, Z.; Rangaiah, G. P. Multiobjective optimization of two alkali catalyzed processes for biodiesel from waste cooking oil. Energy Convers. Manage. 2014, 85, 361−372. (31) Azzaro-Pantel, C.; Ouattara, A.; Pibouleau, L. Ecodesign of chemical processes with multi-objective genetic algorithms. In MultiObjective Optimization in Chemical Engineering: Developments and Applications; Rangaiah, G. P., Bonilla-Petriciolet, A., Ed.; John Wiley: Chichester, U.K., 2013. (32) Ackerman, M. A. Multi-objective optimization for the management and design of a green supply chain: application to the case of food filiere orange juice. Translated from Optimisation multiobjectif pour la gestion et la conception d’une chaine logistique verte: application au cas de la filiere agroalimentaire du jus d’orange. Ph.D. Thesis, Univerity de Toulouse, 2015. (33) Sharma, S.; Rangaiah, G. P. Designing, retrofitting, and revamping water networks in petroleum refineries using multiobjective optimization. Ind. Eng. Chem. Res. 2016, 55, 226−236. (34) Deb, K.; Thiele, L.; Laumanns, M.; Zitzler, E. Scalable multiobjective optimization test problems; http://e-collection.library.ethz. ch/eserv/eth:24696/eth-24696-01.pdf, 2001 (accessed 5th September 2016). (35) Yuan, Y.; Guan, T.; Yan, X. B.; Li, Y. J. Selection of supplier decision model based on mixed-vikor method. Translated. Control and Decision 2014, 29, 551−560.

O


Application and Analysis of Methods for Selecting an Optimal Solution

Recommend Documents