Article pubs.acs.org/IECR
Intuitionistic Fuzzy Chance Constrained Programming for Handling Parametric Uncertainty: An Industrial Grinding Case Study Nagajyothi Virivinti and Kishalay Mitra* Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Ordnance Factory Estate, Yeddumailaram 502205, India ABSTRACT: Uncertainty in the parameters of an optimization problem has a large impact on the outcome of the optimization results. Intuitionistic fuzzy chance constrained programming (IFCCP) is one technique used to handle optimization under uncertainty (OUU) problems. This technique assumes uncertain parameters as intuitionistic fuzzy numbers, which is a super set of conventional fuzzy numbers. With the assumption of intuitionistic fuzzy numbers, different degrees of risk can be modeled considering different viewpoints, e.g., optimistic, pessimistic, and mixed approaches. This generic concept of IFCCP has been applied on two different multiobjective optimization problems: the Binh korn test function, where uncertain parameters are linearly related, and a real life case study of an industrial grinding process, in which uncertain parameters are nonlinearly related. The proposed approach is generic and can be applied to any OUU problem. In addition to the trade-off between solution optimality and quality, sensitivity analysis has also been carried out. the system might have experienced because of the first-stage decision. The optimal policy from such a model is a single firststage decision and expected cost of the optimal recourse action based on several scenarios associated with the second stage. This approach has been profoundly explored in the literature.5,6 One of the major drawbacks of two-stage stochastic programming is that the problem size increases exponentially with an increase in number of uncertain parameters because of the greater number of scenarios to be considered during the recourse decision. Unlike SP, where the decisions have to be feasible for all the outcomes of uncertain parameters, chance constrained programming (CCP), pioneered by Charnes and Cooper,3 requires feasibility of solutions with at least some probability of constraint satisfaction. Chance constraint allows the decision maker to consider the objectives in terms of their attainment probability, δ, which can be a predetermined confidence level decided by the decision maker. This indicates that the constraint should have a probability of satisfaction δ. Thus, the uncertain problem can be converted into an equivalent deterministic optimization problem with the introduction of a probability of constraint satisfaction (δ), and thereby the problem can be solved in a deterministic fashion. The probabilistic nature of this approach leads to multiobjective analysis, where maximization of the reliability of the solution becomes the other objective, in addition to the objective functions that are already present. The deterministic equivalent optimization problem is relatively small even in the presence of a large number of uncertain parameters, which is the main advantage of this method. However, the consequent disadvantage of distribution-based methods is the assumption that the
1. INTRODUCTION Over the past five decades, optimization theory has found numerous applications in a wide variety of fields, e.g., the study of physical, chemical, and biological systems; the gamut of operations research problems; and engineering design, to name a few. Most of these problems are deterministic in nature, i.e., they solve the optimization problem ignoring the variance involved in the parameters. However, many of these models have parameters that are changing with time, e.g., fuel price appearing in the cost function of the model etc., and the model is optimized using a fixed value of that parameter. The results of these problems, therefore, when implemented in real time, become unrealistic because at the time of implementation, the changing parameters are no longer at the values for which the optimization problem was solved and the results were obtained. Conventionally, these uncertainties can be handled by overdesigning the equipment or overestimating the tuning parameters or replacing the uncertain parameters with their nominal values, thereby solving the deterministic optimization problem. Both of these methods lead to solutions that are suboptimal or sometimes infeasible. The area of optimization under uncertainty, therefore, emerged aiming at developing approaches or methodologies that create reliable solutions even in the presence of parametric uncertainty.1 Beginning with the inspiring work of several researchers,2−4 there have been three broad philosophies on which several methods for optimization under uncertainties can be categorized: stochastic programming, chance constrained programming, and fuzzy mathematical programming. Stochastic programming (SP) assumes that the probability distributions governing the uncertain parameters can be estimated. The most widely applied stochastic programming model is two-stage stochastic programming (TSSP). Here, the decision variables are deployed in two stages, where the decision maker takes some action in the first stage. In the second stage, upon the realization of uncertainty, a recourse decision can be made, which compensates for the effects that © 2015 American Chemical Society
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October May 15, May 28, May 28,
17, 2014 2015 2015 2015 DOI: 10.1021/ie504109v Ind. Eng. Chem. Res. 2015, 54, 6291−6304
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Industrial & Engineering Chemistry Research
uncertain parameters that are appearing nonlinearly in the model. The uncertain multiobjective problem under consideration needs treatment of uncertainty propagation in constraints and as well as in objectives, and the equivalent deterministic problem can be solved using fuzzy simulation based approaches, where the uncertain parameters are treated as the intuitionistic fuzzy numbers. Although several works on optimization under uncertainty are available in the literature, the IFCCP technique is relatively new, and extension of IFCCP to a multiobjective optimization problem with nonlinearly related parameters is not very straightforward. To the best of our knowledge, such an analysis has not been considered in any of the earlier work reported in the literature. A modified version of the nondominated sorting genetic algorithm, i.e., intuitionistic fuzzy chance constrained nondominated sorting genetic algorithm II (IFCC-NSGAII), is used to solve the equivalent deterministic multiobjective optimization problem. The rest of this paper is organized as follows: In section 2, preliminary knowledge on fuzzy sets and intuitionistic fuzzy sets has been introduced. This is followed by an overview of intuitionistic fuzzy chance constrained programming technique and fuzzy simulation based approaches in section 3. Section 4 comprises the results of IFCCP on the Binh−Korn test function,14 whereas in section 5, a brief overview of the grinding process optimization formulation is presented in both deterministic as well as uncertain approaches, and the results of IFCPP have been analyzed and discussed thoroughly.
uncertain parameters have to follow some well-behaved probability distribution, which may not always be true in reallife situations. On the contrary, fuzzy mathematical programming (FMP),7 introduced by Zimmermaan,7 neither assumes any statistical distribution for uncertain parameters to follow nor leads to an exponential increase in problem size even in the presence of a large number of uncertain parameters. The uncertain problem can be converted into an equivalent deterministic optimization problem by expressing uncertain parameters as fuzzy numbers. The fuzzy set theory initiated by Zadeh8 uses a membership function, as it assists as a useful mathematical implement to describe the uncertain behavior of intricate systems. The extent of constraint violation can be expressed by a membership function, which varies between 0 and 1, where 0 signifies the maximum extent of constraint violation and 1 signifies the minimum extent of constraint violation, i.e., the location at the constraint boundary. Intermediate points can be interpolated linearly or nonlinearly between 0 and 1 based on the adopted fuzzification approach. Several applications of fuzzy set theory and fuzzy mathematical programming (FMP) appear in the literature from various fields.11,12 As a generalization of Zadeh’s fuzzy sets,8 Atanassov13 proposed intuitionistic fuzzy sets (IFS), a perfect implement to represent the imperfect or uncertain information. The Zadeh fuzzy set uses membership degree to represent the belongingness of an element to the set, whereas intuitionistic fuzzy sets use membership as well as nonmembership degrees, which are more or less independent, cognate by the constraint that the sum of both membership and nonmembership degrees must not exceed one. The membership and nonmembership values define an indeterminacy index, which quantifies the hesitation expected when a decision maker must decide the degree to which an element belongs to a set. In a variety of voting events, support and abnegation are not sufficient to declare the result. Along with them, there is usually abstention, which indicates hesitation or nondeterminacy. However, a fuzzy set is used to represent the satisfied aspects, while IFS is used to represent the satisfied, unsatisfied, and uncertain information. Hence, the intuitionistic fuzzy set is the ideal tool to represent the uncertain or imprecise information that the fuzzy set alone is not able to represent. In this paper, the more pragmatic scenarios of handling uncertainty in parameters for a given optimization problem have been considered to explore the opportunity of using intuitionistic fuzzy chance constrained programming (IFCCP) as an uncertainty-handling tool while analyzing its impact on the overall optimization problem. The concept of IFS has been derived and applied for handling parametric uncertainty in a nonlinear multiobjective optimization problem under a different framework (IFCCP), and various aspects of it have been demonstrated using two motivating examples. First, the technique is applied to the Binh−Korn test function,14 the first motivating example, that has two conflicting objectives and a few linearly related uncertain parameters. Next, it has been applied to an industrial grinding model, the second motivating example, toward analyzing the impact of uncertain parameters, nonlinearly appearing in the model, on the overall optimization of the grinding system. The industrial grinding process, adopted from the previous work of Mitra and Gopinath,15 contains two conflicting objectives, e.g., throughput and midsize fraction, which are to be maximized simultaneously and subjected to certain practical constraints. This grinding model has several
2. PRELIMINARIES In this section, some basic notations used in the study have been introduced. Definition 1: Let xi be a fuzzy number which is associated with a membership degree within an interval [0 1] representing the degree of association of the fuzzy number with fuzzy set A. A = {⟨xi , μA (xi)⟩|xi ∈ X }
(1)
Definition 2: Let xi be a fuzzy number in a finite universal set X. IFS A is an ordered set of triplets, A = {⟨xi , μA (xi), νA(xi)⟩|xi ∈ X }
(2)
where μA(xi),νA(xi) are the membership and nonmembership degree functions mapping from X → [0,1], such that 0 ≤ μA(xi) + νA(xi) ≤ 1. Yager16 defined the intuitionistic fuzzy number (IFN), (μA,νA), which contains two elements, membership degree and nonmembership degree. If μA(xi) + νA(xi) = 1, for all elements x ∈ X, then the traditional fuzzy concept is recovered. Definition 3: Let A and B be two IFSs defining a finite universal set X. The intersection of A and B is defined as follows: A ∩ B = {⟨xi , min(μA (xi), μB (xi)), max(νA(xi), νB(xi))⟩ |xi ∈ X , }
(3)
Definition 4: For each intuitionistic fuzzy set A, the amount πA(x) = 1 − μA(x) − νA(x) is called the hesitancy degree or indeterminacy (nondeterminacy or undecidedness), which may cater to membership value, nonmembership value, or both.16 Yager16 called each triplet (μA,νA,πA) an intuitionistic fuzzy number (IFN), which contains a membership degree, nonmembership degree, and nondeterminacy, where μA + νA + πA =1 6292
DOI: 10.1021/ie504109v Ind. Eng. Chem. Res. 2015, 54, 6291−6304
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Industrial & Engineering Chemistry Research
Yager16 found some flaws in this approach and suggested an alternative approach to solve the above optimization problem. Let us consider two cases of membership and nonmembership degrees μk(x) = 0.4, νk(x) = 0.5 and μk(x) = 0,νk(x) = 0. The values of α − β for the two cases are −0.1, 0, respectively, and the second case appears to be optimal, which has a higher value of α − β. Although the membership degree of the first case is higher than that of the second case, the latter appears to be optimal. Yager suggested an alternative way to solve such an optimization problem, which takes nondeterminacy into account. Transformation of μk − νk follows the function Fkλ(x) = μk(x) + λπk(x), λ∈(0,1]. The value λ represents how much nondeterminacy can be resolved. The larger value of λ indicates that nondeterminacy gets resolved more in favor of the membership degree, and the smaller value indicates that the nondeterminacy gets resolved in favor of the nonmembership degree. The value of λ completely depends on the decision maker’s choice. In this paper, λ = 1/2 is used, which resolves half of the nondeterminacy in favor of the membership degree. The new membership function of IFS for a fuzzy variable x is as given below.
Each IFN has a physical interpretation (for example, 0.5,0.3,0.2) which can be interpreted as “the vote for the resolution is 5 in favor, 3 against, 2 abstentions”. A general optimization problem contains some uncertain information, which are the constant coefficients in the objective function or constraints. The uncertain optimization problem can be solved using intuitionistic fuzzy optimization theory, which maximizes the degree of acceptance that can be calculated using the membership function, and minimizes the nonmembership function that is used to represent the degree of rejection of the intuitionistic fuzzy objective and the constraints, as shown below. max μk (x) min νk(x) 0 ≤ μk (x), νk(x) ≤ 1 μk (x) + νk(x) ≤ 1
(4)
where μk(x) denotes the membership degree of the k IFS, while the νk(x) denotes the nonmembership degree of the kth IFS (objective or constraint). However, an optimization problem contains several objectives and constraints. Angelov17 proposed an optimization formulation, which can be described by a two-stage process including aggregation of objectives and constraints and defuzzification into a crisp optimization problem. Let us consider an optimization problem that contains m objectives and n constraints. By Bellman and Zadeh4 extension principle, the membership degree and nonmembership degree of the intuitionistic fuzzy decision set D can be written as follows: th
Fk(x) = μk (x) +
pos(gj(X, ξ) ≤ 0) = sup(min(μ(ξ1), μ(ξ2), ......, μ(ξr ))), where ξj ∈ ξ
i
j
Here, possibility is defined as the supremum of membership functions of the constraint satisfaction cases.
(5)
where f i,gj are objective functions and constraints of an optimization problem, respectively, where i = 1,2,...m and j = 1,2,...n. The formula can be transformed into the following crisp optimization problem
3. INTUITIONISTIC FUZZY CHANCE CONSTRAINED PROGRAMMING Let us consider a general multiobjective optimization problem, which has m objectives and n constraints with some uncertain parameters in constraints,
max α , min β α ≤ μk (x)
maximize f1 (X), f2 (X)...fm (X) where k = 1, 2, .... , m + n
subjected to gj(X, ξ) ≤ 0,
α≥β α+β≤1
X iL ≤ X i ≤ XUi ,
(6)
j = 1, ..., n
i = 1, ..., k
(10)
where X is a set of decision variables and ξ represents the set of uncertain parameters in the optimization problem. The optimization under uncertainty problem can be solved by Chance constrained programming (CCP).5 In CCP, it is assumed that the uncertain parameters are present in the constraints and the objectives, and due to this, there is no guarantee that a constraint should be satisfied every time for all realizations of the uncertain parameters. Hence, the constraints that are associated with uncertain parameters are required to have at least some specified level of probability of satisfaction, but not necessarily of probability 1. The uncertain optimization formulation as presented in eq 10 can be presented in CCP as follows:
where α denotes the minimal acceptable degree of objectives and constraints, and β denotes the maximal degree of rejection of objectives and constraints. The above nonfuzzy optimization formulation can also be written as shown below17 and can be solved by deterministic optimization methods. max α − β α ≤ μk (x) β ≥ νk(x) α≥β α+β≤1
(9)
j
νD(x) = max(ν f (x), νg (x))
β ≥ νk(x),
(8)
Definition 5: Possibility of a constraint gj(X,ξ) ≤ 0, which has a set of uncertain variables, ξ = {ξ1, ξ2,......, ξr} is given below,
μD (x) = min(μ f (x), μg (x)) and i
1 πk(x) 2
(7) 6293
DOI: 10.1021/ie504109v Ind. Eng. Chem. Res. 2015, 54, 6291−6304
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Industrial & Engineering Chemistry Research maximize f1 (X), f2 (X)...fm (X) subjected to Pr(gj(X, ξ) ≤ 0) ≥ δj , X iL ≤ X i ≤ XUi ,
Fk(x) = μk (x) +
j = 1, ...n
i = 1, ..., k
If the model is nonlinear in uncertain parameters, then it is proposed that the possibility calculation of the objective function and constraints can be carried out by fuzzy simulation based approaches18 with the new membership function, which takes the nondeterminacy into account. Fuzzy simulation based approaches for possibility calculation of both objective and constraints as proposed in this work are given below:
(11)
Pr represents the probability space of random parameters, and δj represents the confidence level of constraint satisfaction. For example, if the value of δj is 0.8, then at least 80% of the time the constraint should be satisfied. A more reliable system can be achieved with higher values of δj. This probabilistic approach leads to a multiobjective analysis, which has various objective functions f1(X), f 2(X)...f m(X) and minimization of risk or maximization of δj. However, with higher values of δj, the feasible set of x values decreases. The major advantage of this method is that the problem size does not blow up with the increase in number of uncertain parameters. However, the basic disadvantage of distribution-based methods is that the random numbers should follow some well-behaved statistical distribution, which may not be true in a real-life situation. To overcome this problem, uncertain parameters are assumed as fuzzy numbers and uncertainty based optimization with fuzzy numbers was introduced by Zimmermaan.7 Assumption of fuzzy numbers replaces the probability calculation of constraint satisfaction with the possibility of constraint satisfaction. The optimization formulation for fuzzy chance constrained programming (FCCP) is as shown below,
pos(fi (X, ξ) ≥ fi ̃ ) ≥ δf
X iL ≤ X i ≤ XUi ,
j = 1, ..., n
pos(gj(X, ξ) ≤ 0) ≥ δg
i = 1, ..., k
3.2. Calculation of Possibility of Constraint with Predetermined Level of Confidence. Step 1: Assume pos =0 Step 2: Randomly generate N points of μL and πL (L = 1, 2, ...., N) from ε-set of ξ ̂ = {ξ1̂ , ξ2̂ , ......, ξr̂ }, where ε is a sufficiently small number, μ is the membership function, and π is nondeterminacy; Step 3: Compute gj(X,ξ), j = 1, 2,....., n Step 4: Calculate intuitionistic fuzzy decision, FL = min(FL(ξ1̂), FL(ξ2̂ ), ........, FL(ξr̂ )) if gj(X,ξ) ≥ 0 Step 5: Updatepos = FL, if FL > pos Step 6: Repeat steps 3 to 5 for N (L = 1, 2, ....., N) number of times and return pos. The proposed IFCCP approach as described above is quite generic and can be applied for problems which have uncertainty anywhere in the system, e.g., in the feed streams, the output streams, or the system/model parameters. The concept of IFCCP can be applied to the optimization problems using three different viewpoints: optimistic, pessimistic, and mixed approaches.19 The membership function definition is same for these three approaches, whereas the nonmembership function definition is different for these three approaches. The membership and nonmembership functions for these three cases are given below. Let us consider uncertainty only on the lower side of deterministic value (x < b) of an uncertain variable b. If p > 0 and q > 0 are the tolerances for membership and nonmembership functions, respectively, then the membership function can be defined as a linear continuous function as follows:
maximize f1 (X, ξ), f2 (X, ξ).....fm (X, ξ)
X iL ≤ X i ≤ XUi ,
j = 1, ..., n
i = 1, ..., k
(13)
The FCCP formulation written by introducing auxiliary variables for the objective functions can be given as follows: maximize f1̃ , f2̃ , ....., fm̃ subjected to pos(fi (X, ξ) ≥ fi ̃ ) ≥ δ f , i
pos(gj(X, ξ) ≤ 0) ≥ δg , j
j
(12)
Pos(gj(X,ξ) ≤ 0) ≥ δj means that X is feasible if the possibility measure of constraint satisfaction should be greater than or equal to a predetermined confidence level δj. If the uncertain parameters are present in both the objective function and the constraints, then the uncertain optimization formulation can be shown as follows:
subjected to gj(X, ξ) ≤ 0,
i
3.1. Calculation of Possibility of Objective Function with Predetermined Level of Confidence. Step 1: Assume fi ̂ = −∞ or 0 Step 2: Randomly generate N points of μL and πL (L = 1, 2,...., N) from ε-set of ξ ̂ = {ξ1̂ , ξ2̂ , ......, ξr̂ }, where ε is a sufficiently small number, μ is the membership function, and π is nondeterminacy; Step 3: Compute f1(X,ξ), f 2(X,ξ)...f m(X,ξ) for the particular realization of uncertain parameters ξ ̂ = {ξ1̂ , ξ2̂ , ......, ξr̂ } Step 4: Calculate intuitionistic fuzzy decision FL = min(FL(ξ1̂), FL(ξ2̂ ), ........, FL(ξr̂ )) Step 5: Update fi ̃ = fi ̂, if fi ̃ ≤ fi ̂ Step 6: Repeat steps 3 and 4 for N number of times and return fi ̃
maximize f1 (X), f2 (X)...fm (X) subjected to pos(gj(X, ξ) ≤ 0) ≥ δj ,
1 πk(x) 2
i = 1, 2, ...., m
j = 1, 2, ....., n (14)
where δgj, δfi are the confidence levels of the j constraint and the ith objective function. As mentioned earlier, intuitionistic fuzzy numbers are the perfect tool to represent the uncertain information. So the concepts of fuzzy chance constrained programming have been extended to intuitionistic fuzzy chance constrained programming with a new membership function, as that given here. th
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DOI: 10.1021/ie504109v Ind. Eng. Chem. Res. 2015, 54, 6291−6304
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Industrial & Engineering Chemistry Research ⎧ 0, x≤b−p ⎪ ⎪ ( ) x b − μ(x) = ⎨1 + , b−p
