MODIFICATION OF SOME SCALARIZATION APPROACHES FOR MULTIOBJECTIVE OPTIMIZATION

. In this paper, we propose revisions of two existing scalarization approaches, namely the feasible-value constraint and the weighted constraint. These methods do not easily provide results on proper efficient solutions of a general multiobjective optimization problem. By proposing some novel modifications for these methods, we derive some interesting results concerning proper efficient solutions. These scalarization approaches need no convexity assumption of the objective functions. We also demonstrate the efficiency of the proposed method using numerical experiments. In particular, a rocket injector design problem involving four objective functions illustrates the performance of the proposed method.


Introduction
Multiobjective optimization refers to the mathematical problems in which we deal with minimization or maximization of competing objective functions over a feasible set of decisions.One challenging issue is that it is usually not possible to optimize multiple objective functions at the same time.This type of problem has been playing an important role in many applied fields, including economics, engineering, medicine, management, and etc. (see [6,20,23,[26][27][28][29]33]).One of the most important methods for solving multiobjective optimization problems is to use scalarization approaches.In these methods, a parametric single objective optimization problem corresponding to the multiobjective optimization problem is solved, and the relationship between optimal solutions of the single objective problem and efficient solutions of the multiobjective optimization problem is investigated.A broad range of scalarization approaches exists in the literature, including the weighted-sum scalarization approach [14,30], the epsilon-constraint approach [7], the Pascoletti Serafini approach [24], the modified Pascoletti Serafini methods [1,8,11,12,16], the weighted-constraint method [4] and the feasible-valueconstraint approach [5].It is important to note that references [4,5] provided results that were necessary and/or sufficient for (weakly) efficient solutions.However, these scalarization techniques have no result on proper efficiency.In this respect, our aim is to propose a modification of these methods by adding slack and surplus variables in the constraints and penalizing the violations in the objective function, the inflexibility of the con-straints is resolved.Here, we will investigate how adding slack and surplus variables in the constraints can help to identify conditions for (weak, proper) efficiency.Furthermore, we find several necessary and sufficient conditions for different types of efficient solutions of a multiobjective optimization problem.The proposed scalarization methods require no convexity assumption on objective functions.We demonstrate the efficiency of the proposed method using numerical experiments.Our first and second numerical examples are three-objective problems with nonconvex constraints.For these problems approximating the nondominated set (Pareto front) is a difficult task.As a practical problem, the rocket injector design problem will be investigated in the last numerical example.This problem was first presented in [18] with four objective functions.In [18], this problem is reduced to a three-objective problem.However, our obtained results show that such a reduction may lead to loosing some parts of the nondominated set.
The rest of the paper is organized as follows.We review preliminaries and basic definitions in Section 2. In Sections 3 and 4, we propose two modifications of the feasible-value constraint and weighted constraint approaches.Some necessary and sufficient conditions relating to weak efficient, efficient, and proper efficient solutions are proved by using these modifications.Section 5 provides some numerical examples that demonstrate the efficiency of our method in practice.

Basic notations and preliminaries
This section contains some notations and standard definitions from multiobjective optimization which are used throughout this paper.The general form of a multiobjective optimization problem is given by with vector-valued continuous functions  : R  −→ R ℓ ,  : R  −→ R  (ℓ, ,  ∈ N, ℓ ≥ 2) and a nonempty set of feasible points  ⊆ R  .We assume that the functions   are bounded from below on the set , with a known lower bound.A vector  ∈ R ℓ with entries   ∈ R,  = 1, . . ., ℓ, is written as ( 1 , . . .,  ℓ ).All relations in this paper should be read as component-wise, i.e., for ,  ′ ∈ R ℓ it is , for all  = 1, . . ., ℓ, and  ̸ =  ′ ,   ′ ⇐⇒   ≤  ′  , for all  = 1, . . ., ℓ.The following definitions are standard in multiobjective optimization [9,15].
(a) x is called an efficient solution for the MOP, if there exists no  ∈  such that  () ≤  (x).(b) x is a weakly efficient solution for the MOP, if there is no other  ∈  such that  () <  (x).(c) x is a strictly efficient solution for the MOP, if there exists no  ∈ ,  ̸ = x such that  ()  (x).(d) x is said to be a proper efficient solution in the Geoffrion's sense for the MOP, if it is efficient and there exists a real positive number  > 0 such that for each  ∈  and  ∈ {1, 2, . . ., ℓ} satisfying   () <   (x) there exists at least one  ∈ {1, 2, . . ., ℓ} such that   (x) <   () and There are different definitions for proper efficiency in the literature; see [2,3,13,17,21,25,32]. Throughout this paper, all proper efficient solutions of the MOP refers to Geoffrions proper efficient solutions.Hereafter, the set of weakly efficient, efficient, strictly efficient and proper efficient solutions of the MOP are denoted by  WE ,  E ,  SE and  PE , respectively.The image of the feasible set under the objective functions is called the image space, and it is denoted by  .The images of efficient solutions are called nondominated solutions.The set of nondominated solution is denoted by where   > 0, for every  = 1, . . ., ℓ, is called a utopia point of the MOP.
In the following, we give a brief review of the two exiting scalarization approaches, namely the feasible-value constraint and the weighted constraint.
The feasible-value constraint approach.For some ̂︀  ∈ , set the weight For each fixed  ∈ {1, 2, . . ., ℓ} and for a given  ∈  (̂︀ ), the mathematical model of the feasible-value constraint approach is stated as follows (see [5]): Define the feasible and solution set of ( The main characteristics of the feasible-value-constraint approach are given in the following two main theorems, which can be found in [5]. Theorem 2.3.Let ̂︀  ∈  and  ∈  (̂︀ ).-Suppose that for some , x ∈   ︀  .Then x ∈  WE .
-If x is the unique solution of (  ︀  ), for some , then x ∈   .
The weighted-constraint approach.For each fixed  ∈ {1, 2, . . ., ℓ}, the mathematical model of the weighted-constraint technique is stated as follows (see [4]): Define the set of positive weights and solution set of (   ) as: What follows, we state the principal results of the weighted-constraint approach obtained by Burachik et al.
Theorem 2.5.For every  ∈ {1, 2, . . ., ℓ} and some  ∈  , one has x ∈    if and only if x ∈  WE .We note that the scalarized problems (  ︀  ) and (   ) do not easily provide results on proper efficiency of optimal solutions.In the following, we present a modification of the scalarized problems (  ︀  ) and (   ) by adding slack and surplus variables, and derive necessary and sufficient conditions for weak efficient, efficient, and proper efficient solutions of the MOP.

The flexible and modified feasible-value constraint problems
By using Ehrgott and Ruzika's idea [10], we propose the following two scalarized problem by adding slack and surplus variables.In [10], the objective functions are bounded from above by a parameter .Therefore, improper selection of  may lead to poor performance of the approach.So, we need an informed choice of  using the structure of the problem.We allow the added constraints of (  ︀  ) to be violated and then penalize these violations in the objective function of (  ︀  ).For any fixed  ∈ {1, 2, . . ., ℓ}, let ̂︀  ∈  and  ∈  (̂︀ ).The modified feasible-value-constraint scalarization problem (MP  ︀  ) is formulated as follows: where  is a utopia vector respective to problem (MP  ︀  ), and   ,  ̸ = , are nonnegative weights.Recently, Hoseinpoor and Ghaznavi [19] proposed a modified objective-constraint scalarization technique and established sufficient conditions for (weakly, properly) efficient solutions of a general multiobjective optimization problem.This scalarizing can be obtained by choosing appropriate values for the scalarized problem (MP x) parameters ( = (0, 0, . . ., 0),     =   ).The reference point information of the decision maker is taken into consideration by the (MP  x) method, which is one of the characteristics of the (MP  x) scalarization method.The flexible feasible-value-constraint scalarization problem (FP  ︀  ) is formulated as follows: where  is a utopia vector respective to problem (FP  (1) If (x, š) ∈ SMP  x for some  and  0, Proof.(1) Suppose that x / ∈  WE .Therefore, there exists ̃︀  ∈  such that we break the proof into two cases.Case I. Let (̃︀ , š) be feasible for (MP  ︀  ).we have Since   > 0, from (3.1) we obtain Hence, equations (3.2) and (3.3) are contradictory to that (x, š) ∈ SMP  x.Therefore, case I cannot happen, and we are left only with Case II.Case II.Suppose that (̃︀ , š) is not feasible for (MP  ︀  ).Thus, there exists  ∈ {1, 2, . . ., ℓ}∖{} such that By (3.4) and (3.5), we conclude Therefor, for  ̸ = , we have   (̃︀ ) >   (x).This contradicts (3.1) for  = .Consequently, x ∈  WE .(2) Let x and  be as in the hypothesis, and assume  satisfies and there exists index  such that Now, we consider two cases.
In the following, by utilizing the proof of the part 3 of Theorem 3.1, we establish a sufficient condition for efficient and proper efficient solutions of the MOP.As the proof of the following theorem is similar to [10], the proof of parts 1 and 2 are omitted.(1) x and (, š) > 0, then x ∈  PE . Proof.
(1) The proof is similar to the Theorem 3.1 in [10]. ( The proof is similar to that of Lemma 3.2 in [10]. (3) From part 1 we have x ∈  E .By part 3 of Theorem 3.1, we can rewrite the objective function as with  > 0. Using Geoffrion's theorem [20], x is proper efficient for the MOP with feasible set   (̂︀ ).In view of part 2, x is a proper efficient solution of the MOP with feasible set .
The next theorem shows that, if  N is not externally stable, the result of part 3 of Theorem 3.2 is no longer true.
is not externally stable, then the MOP does not have any proper efficient solution.
Proof.The proof can be found in [31].
In Table 1, we summarize some of the results obtained for the scalarized problem (MP  ︀  ).

The revised feasible-value constraint problem
Using the idea of Ehrgott and Ruzika [10], we propose the following scalarized problem by a combination of (MP  x) and (FP  x), which is considered as follows.For any fixed  ∈ {1, 2, . . ., ℓ}, let x ∈  and  ∈  (x).The Revised feasible-value constraint problem (RP x) is stated as follows: in which u is a utopia vector respective to problem (RP  x), and   ,   ,  ̸ = , are nonnegative weights.Based on the work of Ehrgott and Ruzika [10], it should be noted that if there exist some  ∈  (x) for a given feasible solution (,  + ,  − ) of (RP  x), (,  + + ,  − + ) is also feasible for (RP  x), where  ∈ R  .Then, This shows that the objective function value depends on  ∈ R ℓ .Here,  can be chosen arbitrarily in R ℓ .If  < , then the objective function value of (RP  x) is unbounded.In this respect, in what follows, we assume  .
We recall that the set of solutions of problem (RP  ︀  ) is denoted by SRP  x.Below, we provide results characterizing (proper, weak) efficiency solutions of the MOP utilizing the scalarized problem (RP  x).
(1) The proof follows from the case 1 of Theorem 3.1.
(2) The proof is similar to case 2 of Theorem 3.1.
In the following, utilizing the proof of Theorem 3.5, we establish a sufficient condition for proper efficient solutions of the MOP.Theorem 3.6.Let x ∈  and  ∈  (̂︀ ). (1) ) is an optimal solution of (RP  x) and  > 0, then ̂︀  ∈  PE Proof.
(1) The proof follows from Theorem 3.5 and part 2 of Theorem 3.1.
(2) The proof is similar to that of Theorem 5.1 in [10], so we omit it here.
Similar to Theorem 3.4, we present a necessary condition for proper efficient solutions of the MOP.

The revised weighted-constraint problem
Burachik et al. [4] proposed the following scalarization method, named the weighted-constraint approach The scalarized problem (   ) does not easily provide results on (proper)efficiency of optimal solutions.In the following, motivated by the idea of Ehrgott and Ruzika [10], we allow the added constraints of (   ) to be violated, and then penalize these violations in the objective function of (   ).The modified weighted-constraint approach is formulated as follows: in which   ,  = 1, . . ., ℓ,  ̸ = , are nonnegative weights.The flexible weighted-constraint approach is formulated as follows: In the present research, we purpose a combination of the two modifications (MP   ) and (FP   ) to generate proper efficient solutions of the MOP.
In the following, we provide results characterizing (proper, weak) efficiency solutions of the MOP utilizing the scalarized problem (RP   ).
Proof.The proof is similar to the Theorem 5.2 in [10] and will be omitted here.
At the end of this section, similar to Theorems 3.4 and 3.7, we present a necessary condition for proper efficient solutions of the MOP.Theorem 4.3.Assume that there exist some  ∈  .

Numerical results
In this section, we demonstrate the efficiency of the proposed approaches through the results of some numerical experiments.To this end, we divide this section into three experiments.All experiments have been implemented in MATLAB (R2017b).MATLAB's optimization solver fmincon has been used, with default options, for solving the associated nonlinear problems.All numerical tests have been performed on PC Intel Core i7 2700 k CPU 3.4 GHz and 8 GB of RAM.
For comparison, Algorithms 1 and 2 in [5] are utilized, which implements the (  x ) and (   ), respectively.In addition, for the proposed approaches, the same steps as 1-5 of Algorithms 1 and 2 in [5] are done, except that in step 4 of those, our proposed approaches are solved instead of ( x ) and (   ).Example 5.1.We consider the following nonconvex three objective problem which has a Pareto front with a boundary that is difficult to construct [5].
For the algorithms that implement (MP  x)((MP   )) and (FP x) ((FP   )), the nonnegative weights   for  = 1, 2, 3 are selected as random.The point (−10, −10, −10) is taken as the utopia point.The algorithms are implemented with  = 21 ( = 35).Figure 2 shows the Pareto points generated by our proposed approaches applied to the given algorithms are extremely successful compared to Figures 5a and 5c of [5].All of the outer and inner end points of the Pareto front are generated among these produced points, as shown in the figure, and the provided approaches generate Pareto points that are distributed rather evenly in the approximation of the Pareto front.The distribution of points obtained by our approaches and the proposed method in [5] is depicted in Figure 2.
Example 5.2.The original problem in [22] has earlier been studied in [11].The Pareto front of the following test problem is non-convex and gaps appear in the boundary of the Pareto front.
The point (−100, −100, −100) is taken as the utopia point and we set  = 20 ( = 30).Uniformly distributed weights are provided and the resulting Pareto points are shown in Figure 3.Note that we plot again the negative objective function values.

Application to a liquid-rocket injector design
The liquid rocket single element injector design problem was previously studied as a multiobjective optimization problem in [5,18].This is a computationally expensive engineering design problem, involving design of a hybrid Boeing element injector.This problem has two primary goals: (i) the improvement of the performance (by minimizing combustion length), (ii) material sustainability (by minimizing face temperature).For the rocket injector design problem, four design variables are defined in [18], which are listed below:  : the hydrogen flow angle, ∆HA: the hydrogen area,   ∆OA: the oxygen area, OPTT: the oxidizer post tip thickness.
The variables range considered in the mathematical model of the rocket injector design problem are shown in Table 2.Note that the variables of the problem have been normalized.According to Goel et al. [18], the four objective functions to be considered for a rocket injector design problem consist of: TF max : the face temperature, which is the maximum temperature of the injector face, TT max : the tip temperature, which is the maximum temperature on the post tip of the injector,  CC : the combustion length, which is the distance from the inlet where 99% of the combustion is complete, TW 4 : the wall temperature, which is the wall temperature at three inches (at the fourth probe) from the injector face.
The rocket injector design problem was solved in [18] with three objectives, namely, with TF max , TT max , and  CC .This reduced problem (with TW 4 eliminated out) is justified by a claim in [18] that TF max and TW 4 are correlated after a correlation analysis.However, Figures 4c and 4d show that the projection of the nondominated set to  4  3  2 -space clearly differs from that in Figures 4a and 4b: the points facing  3  2 -plane, changes more rapidly with TT max and with  CC .This implies that incorporation of TW 4 in this multiobjective optimization problem is essential, or necessary, as opposed to its exclusion in [18].

Conclusions
In the present research, we proposed a modification of the scalarization approach introduced by Burachik et al. [4,5] for solving multiobjective programming problems.We proved necessary and sufficient conditions for various types of efficiency, in particular for proper efficiency.The proposed approach was applied to solve problems with convex, nonconvex, connected and disconnected feasible sets.We have demonstrated the efficiency of the proposed method using numerical experiments.Since the modifications resolve the inflexibility of the constraints of the feasible-value constraint and the weighted constraint methods leading to computational advantages, it will also be interesting to study applications of our approach to multiobjective mixed integer optimization problems.

︀
) and   ,  ̸ = , are nonnegative weights.It is recalled that we denote by SMP  ︀  the set of solutions of problem (MP  ︀  ).In the following, we provide results which characterize (proper, weak) efficiency solutions of the MOP utilizing the scalarized problems (MP  ︀  ).Theorem 3.1.Fix ̂︀  ∈ , and let  ∈  (̂︀ ).

Figure 4 .
Figure 4. Projected Nondominated set for the rocket injector design problem.(a) Projection of the nondominated set to the  1  3  2 -space.(b) Rotated view of the front in (a).(c) Projection of the nondominated set to the  4  3  2 -space.(d) Rotated view of the front in (c).

Table 2 .
Range of design variables ( is an acute angle in degrees and  is the thickness of OPTT in inches).