Optimizing Three-Dimensional Constrained Ordered Weighted Averaging Aggregation Problem with Bounded Variables

Abstract

A single constrained ordered weighted averaging aggregation (COWA) problem is of considerable importance in many disciplines. Two models are considered: the maximization COWA problem with lower bounded variables and the minimization COWA problem with upper bounded variables. For a three-dimensional case of these models, we present the explicitly optimal solutions theoretically and empirically. The bounds and weights can affect the optimal solution of the three-dimensional COWA problem with bounded variables.

1. Introduction

An ordered weighted averaging (OWA) operator [1] is a general class of parametric aggregation operators that appears in many research fields such as decision making [2,3,4,5,6], fuzzy system [7,8], statistics [9,10,11], risk analysis [12] and others [13,14]. For more details, see Carlsson and Fullér [15], Emrouznejad and Marra [16] and Yager et al. [17]. A constrained OWA aggregation problem (COWA) attempts to optimize the OWA aggregation problem with multiple constraints. Yager [18] developed a mixed integer linear programming problem to solve a single COWA problem. Later, Carlsson et al. [19] proposed an algorithm to solve the single constrained OWA optimization problem for any dimensions. Furthermore, Coroianu and Fullér [20] presented an explicitly optimal solution for the single COWA problem with any constraint coefficients. In addition, there are other important references [21,22,23,24,25,26,27] dedicated to constrained OWA optimization problem with multiple constraints. However, the decision variables are usually bounded for the most practical problems. Recently, Chen and Tang [28] proposed a three-dimensional COWA problem with lower bounded variables. This paper presents the explicitly optimal solutions for the three-dimensional COWA problem with bounded variables. Two models are considered. One is a maximizing three-dimensional constrained OWA aggregation problem with lower bounded variables (3COWAL). The other is a minimizing three-dimensional constrained OWA aggregation problem with upper bounded variables (3COWAU).

The organization of this paper is as follows. Section 2 briefly reviews the COWA problem. For maximizing 3COWAL, there are two parameters $(w_1,w_2,w_3)$ and $(l_1,l_2,l_3)$ that affect the optimal solution types. We discuss the optimal solution behaviors in Section 3 for $w_1\ge w_2\ge w_3$ and Section 4 for $l_1\ge l_2\ge l_3$. Section 5 analyzes the optimal solution behaviors of minimizing 3COWAU. Finally, some concluding remarks are presented.

2. Constrained OWA Aggregation Problem

An OWA operator of dimension n is a mapping $F:{\mathcal{R}}^n\to \mathcal{R}$ that has an associated weighting vector $W=(w_1,w_2,\dots ,w_n)$ satisfying

$$w_1+w_2+\dots +w_n=1,0\le w_i\le 1,i=1,2,\dots ,n$$

and such that

$$G(x_1,x_2,\dots ,x_n)={\displaystyle \sum }_{i=1}^n{w}_i{y}_i$$

(1)

with $y_i$ being the ith largest of $\left\{x_1,x_2,\dots ,x_n\right\}$. For simplicity, we will denote this expression as $F(y_1,y_2,\dots ,y_n)$.

Consider the following single COWA problem:

$$\mathrm{Max}{W}^{T}Y,\mathrm{s}.\mathrm{t}.{\mathcal{I}}^{T}X\le 1,X\ge \mathbf{0},$$

(2)

where the column vectors X, Y, W and $\mathcal{I}$ are

$$X=\left[\begin{array}{c}{x}_1\\ x_2\\ \begin{array}{c}⋮\\ x_n\end{array}\end{array}\right],Y=\left[\begin{array}{c}{y}_1\\ y_2\\ \begin{array}{c}⋮\\ y_n\end{array}\end{array}\right],W=\left[\begin{array}{c}{w}_1\\ w_2\\ \begin{array}{c}⋮\\ w_n\end{array}\end{array}\right],\mathcal{I}=\left[\begin{array}{c}1\\ 1\\ \begin{array}{c}⋮\\ 1\end{array}\end{array}\right].$$

By introducing the $(n-1)\times n$ matrix

$$G=\left[\begin{array}{ccc}\begin{array}{ccc}-1& 1& \begin{array}{cc}0& 0\end{array}\\ 0& -1& \begin{array}{cc}1& 0\end{array}\end{array}& \cdots & \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ ⋮& \ddots & ⋮\\ \begin{array}{ccc}0& 0& \begin{array}{cc}0& 0\end{array}\end{array}& \cdots & \begin{array}{cc}-1& 1\end{array}\end{array}\right]$$

and the column binary vectors $Z_i\in {\left\{0,1\right\}}^n$, $i=1,2,\dots ,n$, Yager [18] transformed the nonlinear programming single COWA problem (2) to the following mixed integer linear programming problem (MILP):

$$\begin{gathered} \mathrm{Max}{W}^{T}Y,\mathrm{s}.\mathrm{t}.{\mathcal{I}}^{T}X\le 1,GY\le 0,y_i\mathcal{I}-X-M{Z}_i\le 0,i=1,2,\dots ,n-1, \\ {y}_n\mathcal{I}-X\le 0,{\mathcal{I}}^T{Z}_i\le n-i,i=1,2,\dots ,n-1, \\ {Z}_i\in {\left\{0,1\right\}}^n,i=1,2,\dots ,n-1,X\ge \mathbf{0}, \end{gathered}$$

(3)

where M is a very large positive number. To reduce the multiple solutions of the MILP (3), Chen and Tang [28] introduced the following constraints:

$$Z_{i+1}\le Z_i,i=1,2,\dots ,n-2.$$

Then, the more efficient MILP of a single COWA problem is as follows:

$$\begin{gathered} \mathrm{Max}{W}^{T}Y,\mathrm{s}.\mathrm{t}.{\mathcal{I}}^{T}X\le 1,GY\le 0,y_i\mathcal{I}-X-M{Z}_i\le 0,i=1,2,\dots ,n-1, \\ {y}_n\mathcal{I}-X\le 0,{\mathcal{I}}^T{Z}_i\le n-i,i=1,2,\dots ,n-1,Z_{i+1}\le Z_i,i=1,2,\dots ,n-2, \\ {Z}_i\in {\left\{0,1\right\}}^n,i=1,2,\dots ,n-1,X\ge \mathbf{0}. \end{gathered}$$

(4)

3. Maximizing a Three-Dimensional Constrained OWA Aggregation Problem with Lower Bounded Variables for w₁ ≥ w₂ ≥ w₃

It is fairly common in practical optimization problems that the decision variables are usually bounded. A typical decision variable $x_i$ is bounded from below by $l_i$ and from above by $u_i$, where $l_i\le u_i$. Section 3 and Section 4 analyze the maximization COWA problem with the lower bound constraints. Section 5 analyzes the minimization COWA problem with the upper bound constraints. Chen and Tang [28] proposed the following COWAL:

$$\begin{gathered} \mathrm{Max}{W}^{T}Y,\mathrm{s}.\mathrm{t}.{\mathcal{I}}^{T}X\le 1,GY\le 0,,y_i\mathcal{I}-X-M{Z}_i\le 0,i=1,2,\dots ,n-1, \\ {y}_n\mathcal{I}-X\le 0,{\mathcal{I}}^T{Z}_i\le n-i,i=1,2,\dots ,n-1,Z_{i+1}\le Z_i,i=1,2,\dots ,n-2, \\ {Z}_i\in {\left\{0,1\right\}}^n,i=1,2,\dots ,n-1,X\ge L, \end{gathered}$$

(5)

where the lower bounded vector

$$L=\left[\begin{array}{c}{l}_1\\ l_2\\ \begin{array}{c}⋮\\ l_n\end{array}\end{array}\right].$$

The lower bounded vector can be transformed into the zero vector by using the standard transformations $X^{\prime }=X-L$. The COWAL is as follows:

$$\begin{gathered} \mathrm{Max}{W}^{T}Y,\mathrm{s}.\mathrm{t}.{\mathcal{I}}^T{X}^{\prime }\le 1-{\mathcal{I}}^{T}L,GY\le 0,y_i\mathcal{I}-X^{\prime }-M{Z}_i\le L,i=1,2,\dots ,n-1, \\ {y}_n\mathcal{I}-X^{\prime }\le L,{\mathcal{I}}^T{Z}_i\le n-i,i=1,2,\dots ,n-1,Z_{i+1}\le Z_i,i=1,2,\dots ,n-2, \\ {Z}_i\in {\left\{0,1\right\}}^n,i=1,2,\dots ,n-1,X^{\prime }\ge \mathbf{0}. \end{gathered}$$

(6)

If $1-{\mathcal{I}}^{T}L<0$, then the COWAL has no feasible solution. If $1-{\mathcal{I}}^{T}L=0$, then $X^{\prime }=\mathbf{0}$ is the unique optimal solution, so $X=L$. The following will discuss the case that $1-{\mathcal{I}}^{T}L>0$.

Consider the 3COWAL for the case of

$$l_1+l_2+l_3\le 1.$$

(7)

Two parameters $(w_1,w_2,w_3)$ and $(l_1,l_2,l_3)$ are considered in 3COWAL. This section discusses 3COWAL with

$$w_1\ge w_2\ge w_3.$$

(8)

There are six permutations of $(l_1,l_2,l_3)$. First, consider the case of

$$l_1\ge l_2\ge l_3.$$

(9)

At optimality, the first constraint of model (6) becomes

$$x_1^{\prime }+x_2^{\prime }+x_3^{\prime }=1-l_1-l_2-l_3.$$

(10)

There are three types (A, B, C) of $(x_1^{\prime },x_2^{\prime },x_3^{\prime })$ according to the number of zero components.

The number of zero components of first type A is two. The possible values of $(x_1^{\prime },x_2^{\prime },x_3^{\prime })$ are

$$(1-l_1-l_2-l_3,0,0),(0,1-l_1-l_2-l_3,0)\mathrm{and}(0,0,1-l_1-l_2-l_3).$$

For the case of $(x_1^{\prime },x_2^{\prime },x_3^{\prime })=(1-l_1-l_2-l_3,0,0)$, we have

$$(x_1,x_2,x_3)=(1-l_2-l_3,l_2,l_3)$$

and

$$\begin{gathered} (y_1,y_2,y_3)=(1-l_2-l_3,l_2,l_3),(l_2,1-l_2-l_3,l_3),(l_2,l_3,1-l_2-l_3), \\ (1-l_2-l_3,l_3,l_2),(l_3,1-l_2-l_3,l_2)\mathrm{or}(l_3,l_2,1-l_2-l_3). \end{gathered}$$

Consider $(y_1,y_2,y_3)=(l_2,1-l_2-l_3,l_3)$, so $l_2\ge 1-l_2-l_3\ge l_3$, implying $2l_2+l_3\ge 1$. From Label (7), it follows that $l_2\ge l_1$, in contradiction to assumption (9). The same contradiction is also for $(y_1,y_2,y_3)=(l_2,l_3,1-l_2-l_3)$, $(1-l_2-l_3,l_3,l_2)$, $(l_3,1-l_2-l_3,l_2)$ and $(l_3,l_2,1-l_2-l_3)$. Therefore, for $(x_1^{\prime },x_2^{\prime },x_3^{\prime })=(1-l_1-l_2-l_3,0,0)$, the reasonable candidate for the optimal solution is $(y_1,y_2,y_3)=(1-l_2-l_3,l_2,l_3)$. For cases of $(0,1-l_1-l_2-l_3,0)$ and $(0,0,1-l_1-l_2-l_3)$, the reasonable candidates of $(y_1,y_2,y_3)$ are shown in

Table 1. The possible values of $(x_1^{\prime },x_2^{\prime },x_3^{\prime })$, $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ for 3COWAL with $l_1\ge l_2\ge l_3$.

	$({\mathit{x}}_{\mathbf{1}}^{\prime },{\mathit{x}}_{\mathbf{2}}^{\prime },{\mathit{x}}_{\mathbf{3}}^{\prime })$	$({\mathit{x}}_{\mathbf{1}},{\mathit{x}}_{\mathbf{2}},{\mathit{x}}_{\mathbf{3}})$	$({\mathit{y}}_{\mathbf{1}},{\mathit{y}}_{\mathbf{2}},{\mathit{y}}_{\mathbf{3}})$
A-1	$(1-l_1-l_2-l_3,0, 0)$	$(1-l_2-l_3, l_2, l_3)$	$(1-l_2-l_3, l_2, l_3)$
A-2	$(0,1-l_1-l_2-l_3,0)$	$(l_1,1-l_1-l_3, l_3)$	$(1-l_1-l_3, l_1, l_3)$, $(l_1,1-l_1-l_3, l_3)$
A-3	$(0,0,1-l_1-l_2-l_3)$	$(l_1, l_2, 1-l_1-l_2)$	$(1-l_1-l_2, l_1, l_2)$, $(l_1, 1-l_1-l_2, l_2)$,$(l_1, l_2, 1-l_1-l_2)$
B1-1	$(\frac{1-2l_1-l_3}{2}, \frac{1-2l_2-l_3}{2}, 0)$	$(\frac{1-l_3}{2}, \frac{1-l_3}{2}, l_3)$	$(\frac{1-l_3}{2},\frac{1-l_3}{2},l_3)$
B1-2	$(\frac{1-2l_1-l_2}{2},0,\frac{1-l_2-2l_3}{2})$	$(\frac{1-l_2}{2},l_2,\frac{1-l_2}{2})$	$(\frac{1-l_2}{2},\frac{1-l_2}{2},l_2)$
B1-3	$(0,\frac{1-l_1-2l_2}{2},\frac{1-l_1-2l_3}{2})$	$(l_1,\frac{1-l_1}{2},\frac{1-l_1}{2})$	$(l_1,\frac{1-l_1}{2},\frac{1-l_1}{2})$
B2-1	$(l_3-l_1,1-l_2-2l_3,0)$	$(l_3,1-2l_3,l_3)$
B2-2	$(1-l_1-2l_3,l_3-l_2,0)$	$(1-2l_3,l_3, l_3)$
B2-3	$(l_2-l_1,0, 1-2l_2-l_3)$	$(l_2,l_2,1-2l_2)$
B2-4	$(1-l_1-2l_2,0, l_2-l_3)$	$(1-2l_2,l_2, l_2)$	$(1-2l_2, l_2, l_2)$
B2-5	$(0, l_1-l_2,1-2l_1-l_3)$	$(l_1,l_1,1-2l_1)$	$(l_1, l_1,1-2l_1)$
B2-6	$(0,1-2l_1-l_2, l_1-l_3)$	$(l_1,1-2l_1,l_1)$	$(l_1, l_1,1-2l_1)$
C	$(1/3-l_1,1/3-l_2,1/3-l_3)$	$(1/3, 1/3, 1/3)$	$(1/3, 1/3, 1/3)$

.

In Table 1, there are six candidates for optimal solution $(y_1,y_2,y_3)$ for type A. Among these six candidates, we will show that the largest objective function $F(Y)=w_1{y}_1+w_2{y}_2+w_3{y}_3$ is that of $(y_1,y_2,y_3)=(1-l_2-l_3,l_2,l_3)$. Before we prove this result in detail, we present a well-known fact.

Theorem 1.

For $(x_1,x_2\dots ,x_n)$, $(x_1^{\prime },x_2^{\prime }\dots ,x_n^{\prime })$, $s_k={\displaystyle \sum }_{i=1}^k{x}_i$ and $s_k^{\prime }={\displaystyle \sum }_{i=1}^k{x}_i^{\prime }$, $k=1,2,\dots ,n$, if $s_k\ge s_k^{\prime }$, $k=1,2,\dots ,n,$ then for all $(w_1,w_2,\dots ,w_n)$ with $w_k\ge w_{k+1}$, $k=1,2,\dots ,n-1,$ we have

$${{\displaystyle \sum }}_{i=1}^n{w}_i{x}_i\ge {{\displaystyle \sum }}_{i=1}^n{w}_i{x}_i^{\prime }.$$

Comparing the objective function value of $(y_1,y_2,y_3)=(1-l_2-l_3,l_2,l_3)$ with that of $(1-l_1-l_3,l_1,l_3)$, we get that

$$s_1=1-l_2-l_3\ge s_1^{\prime }=1-l_1-l_3,$$

$$s_2=1-l_3\ge s_2^{\prime }=1-l_{3,}$$

$$s_3=1\ge s_3^{\prime }=1.$$

It implies that the most favorable value of the objective function is that with $(1-l_2-l_3,l_2,l_3)$. A similar argument shows that $F(1-l_2-l_3,l_2,l_3)$ is larger than those of $(l_1,1-l_1-l_3,l_3)$, $(1-l_1-l_2,l_1,l_2)$, $(l_1,1-l_1-l_2,l_2)$ and $(l_1,l_2,1-l_1-l_2)$. Therefore, the optimal solution for type A is $(1-l_2-l_3,l_2,l_3)$.

For the one zero component of $(x_1^{\prime },x_2^{\prime },x_3^{\prime })$, the possible values are

$$(x_1^{\prime },x_2^{\prime },0),(x_1^{\prime },0,x_3^{\prime })\mathrm{and}(0,x_2^{\prime },x_3^{\prime }).$$

At optimal, the possible values of $x_1^{\prime }$, $x_2^{\prime }$ and $x_3^{\prime }$ with at least one $x_i^{\prime }=0$, $i=1,2,3$ satisfy

$$x_1^{\prime }+l_1=x_2^{\prime }+l_2,x_1^{\prime }+l_1=x_3^{\prime }+l_3\mathrm{or}{x}_2^{\prime }+l_2=x_3^{\prime }+l_3.$$

We choose $x_i^{\prime }+l_i=x_j^{\prime }+l_j$ for type B1, and $x_i^{\prime }+l_i=l_k$ or $x_j^{\prime }+l_j=l_k$, $i\ne j\ne k$, $i, j, k=1,2,3$, for type B2. A similar argument shows that all possible candidates for optimal solutions $(y_1,y_2,y_3)$ are

$$(\frac{1-l_3}{2},\frac{1-l_3}{2},l_3),(\frac{1-l_2}{2},\frac{1-l_2}{2},l_2),(l_1,\frac{1-l_1}{2},\frac{1-l_1}{2})$$

for type B1, and

$$(1-2l_2,l_2,l_2),(l_1,l_1,1-2l_1)$$

for type B2. The largest objective function value is that with $(\frac{1-l_3}{2},\frac{1-l_3}{2},l_3)$ for type B1 and $(l_1,l_1,1-2l_1)$ for type B2. Furthermore, $F(\frac{1-l_3}{2},\frac{1-l_3}{2},l_3)\ge F(l_1,l_1,1-2l_1)$. Therefore, the optimal solution for type B is $(\frac{1-l_3}{2},\frac{1-l_3}{2},l_3)$.

Type C is the nonzero components. From Label (10), it follows that

$$(x_1^{\prime },x_2^{\prime },x_3^{\prime })=(1/3-l_1,1/3-l_2,1/3-l_3),(x_1,x_2,x_3)=(1/3,1/3,1/3)\mathrm{and}(y_1,y_2,y_3)=(1/3,1/3,1/3).$$

Therefore, there are six candidate optimal solutions for type A, five candidate optimal solutions for type B and one candidate optimal solution for type C. Detailed results of $(x_1^{\prime },x_2^{\prime },x_3^{\prime })$, $(x_1,x_2,x_3)$, $(y_1,y_2,y_3)$, $F(y_1,y_2,y_3)$ and condition for 3COWAL with $l_1\ge l_2\ge l_3$ are presented in

Table 2. The candidate optimal solutions of $(x_1^{\prime },x_2^{\prime },x_3^{\prime })$, $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$, $F(y_1,y_2,y_3)$ and condition for 3COWAL with $l_1\ge l_2\ge l_3$.

	$({\mathit{x}}_{\mathbf{1}}^{\prime },{\mathit{x}}_{\mathbf{2}}^{\prime },{\mathit{x}}_{\mathbf{3}}^{\prime })$	$({\mathit{x}}_{\mathbf{1}},{\mathit{x}}_{\mathbf{2}},{\mathit{x}}_{\mathbf{3}})$	$({\mathit{y}}_{\mathbf{1}},{\mathit{y}}_{\mathbf{2}},{\mathit{y}}_{\mathbf{3}})$	$\mathit{F}({\mathit{y}}_{\mathbf{1}},{\mathit{y}}_{\mathbf{2}},{\mathit{y}}_{\mathbf{3}})$	Condition
A1	$(1-l_1-l_2-l_3,0, 0)$	$(1-l_2-l_3, l_2, l_3)$	$(1-l_2-l_3, l_2, l_3)$	$w_1+l_2(- w_1+w_2)+l_3(-w_1+w_3)$
A2	$(0,1-l_1-l_2-l_3,0)$	$(l_1,1-l_1-l_3, l_3)$	$(1-l_1-l_3, l_1, l_3)$	$w_1+l_1(-w_1+w_2)+l_3(-w_1+w_3)$	$2l_1+l_3\le 1$
A3	$(0,1-l_1-l_2-l_3,0)$	$(l_1,1-l_1-l_3, l_3)$	$(l_1,1-l_1-l_3, l_3)$	$w_2+l_1(w_1-w_2)+l_3(-w_2+w_3)$	$2l_1+l_3\ge 1$
A4	$(0,0,1-l_1-l_2-l_3)$	$(l_1, l_2,1-l_1-l_2)$	$(1-l_1-l_2, l_1, l_2)$	$w_1+l_1(-w_1+w_2)+l_2(-w_1+w_3)$	$2l_1+l_2\le 1$
A5	$(0,0,1-l_1-l_2-l_3)$	$(l_1, l_2,1-l_1-l_2)$	$(l_1,1-l_1-l_2, l_2)$	$w_2+l_1(w_1-w_2)+l_2(-w_2+w_3)$	$2l_1+l_2\ge 1,$$l_1+2l_2\le 1$
A6	$(0,0,1-l_1-l_2-l_3)$	$(l_1, l_2,1-l_1-l_2)$	$(l_1, l_2,1-l_1-l_2)$	$w_3+l_1(w_1-w_3)+l_2(w_2-w_3)$	$l_1+2l_2\ge 1$
B1	$(\frac{1-2l_1-l_3}{2},\frac{1-2l_2-l_3}{2},0)$	$(\frac{1-l_3}{2},\frac{1-l_3}{2}, l_3)$	$(\frac{1-l_3}{2},\frac{1-l_3}{2}, l_3)$	$\frac{1-w_3-l_3+3l_3{w}_3}{2}$	$l_3\le 1/3$, $2l_1+l_3\le 1$
B2	$(\frac{1-2l_1-l_2}{2},0,\frac{1-l_2-2l_3}{2})$	$(\frac{1-l_2}{2}, l_2,\frac{1-l_2}{2})$	$(\frac{1-l_2}{2},\frac{1-l_2}{2}, l_2)$	$\frac{1-w_3-l_2+3l_2{w}_3}{2}$	$l_2\le 1/3$, $2l_1+l_2\le 1$
B3	$(0,\frac{1-l_1-2l_2}{2},\frac{1-l_1-2l_3}{2})$	$(l_1,\frac{1-l_1}{2},\frac{1-l_1}{2})$	$(l_1,\frac{1-l_1}{2},\frac{1-l_1}{2})$	$\frac{1-w_1-l_1+3l_1{w}_1}{2}$	$l_1\ge 1/3$, $l_1+2l_2\le 1$
B4	$(1-l_1-2l_2,0,l_2-l_3)$	$(1-2l_2, l_2, l_2)$	$(1-2l_2, l_2, l_2)$	$w_1+l_2-3l_2{w}_1$	$l_2\le 1/3$, $l_1+2l_2\le 1$
B5	$(0, l_1-l_2, 1-2l_1-l_3)$	$(l_1, l_1,1-2l_1)$	$(l_1, l_1,1-2l_1)$	$w_3+l_1-3l_1{w}_3$	$l_1\ge 1/3$, $2l_1+l_3\le 1$
C	$(1/3-l_1, 1/3-l_2,1/3-l_3)$	$(1/3, 1/3, 1/3)$	$(1/3, 1/3, 1/3)$	1/3	$l_1\le 1/3$, $l_2\le 1/3$, $l_3\le 1/3$

.

The largest objective function value is that of $\mathrm{A}1(1-l_2-l_3,l_2,l_3)$ for type A, $\mathrm{B}1(\frac{1-l_3}{2},\frac{1-l_3}{2},l_3)$ for type B and $\mathrm{C}(1/3,1/3,1/3)$ for type C. A similar argument shows that

$$F(\mathrm{A}1)\ge F(\mathrm{B}1)\mathrm{and}F(\mathrm{A}1)\ge F(\mathrm{C}).$$

Therefore, for the case of $w_1\ge w_2\ge w_3$, the optimal solution for $l_1\ge l_2\ge l_3$ is $\mathrm{A}1$. Similarly, the optimal solutions of the remaining five permutations $l_1\ge l_3\ge l_2$, $l_2\ge l_1\ge l_3$, $l_2\ge l_3\ge l_1$, $l_3\ge l_1\ge l_2$ and $l_3\ge l_2\ge l_1$ can be derived. Detailed optimal solutions are described as follows.

Theorem 2.

For $w_1\ge w_2\ge w_3$ and $l_1+l_2+l_3\le 1$, the optimal solution of 3COWAL is as follows:

$$(y_1^{*},y_2^{*},y_3^{*})=\{\begin{array}{c}(1-l_2-l_3,l_2,l_3), \mathrm{if} l_1\ge l_2\ge l_3\\ (1-l_2-l_3,l_3,l_2), \mathrm{if} l_1\ge l_3\ge l_2\\ (1-l_1-l_3,l_1,l_3), \mathrm{if} l_2\ge l_1\ge l_3\\ (1-l_1-l_3,l_3,l_1), \mathrm{if} l_2\ge l_3\ge l_1\\ (1-l_1-l_2,l_1,l_2), \mathrm{if} l_3\ge l_1\ge l_2\\ (1-l_1-l_2,l_2,l_1), \mathrm{if} l_3\ge l_2\ge l_1.\end{array}$$

(11)

4. Maximizing Three-Dimensional Constrained OWA Aggregation Problem with Lower Bounded Variables for l₁ ≥ l₂ ≥ l₃

This section considers the optimal solution behaviors for 3COWAL with $l_1\ge l_2\ge l_3$. The main result is described as follows.

Theorem 3.

For $l_1\ge l_2\ge l_3$ and $l_1+l_2+l_3\le 1$, the optimal solution $Y^{*}$ of 3COWAL is as follows.

(1) 

For $w_1\ge w_2\ge w_3$ or $w_1\ge w_3\ge w_2$, the optimal solution is $A1(1-l_2-l_3,l_2,l_3)$.

(2) 

For $w_2\ge w_1\ge w_3$, the optimal solution $Y^{*}$ is

$$if2l_1+l_3\ge 1,then{Y}^{*}=A3(l_1,1-l_1-l_3,l_3)else{Y}^{*}=B1 (\frac{1-l_3}{2},\frac{1-l_3}{2},l_3).$$

(3) 

For $w_2\ge w_3\ge w_1$, the optimal solution $Y^{*}$ is

$$if2l_1+l_3\ge 1,then{Y}^{*}=A3(l_1,1-l_1-l_3,l_3).$$

$$elseif{w}_3\le 1/3then{Y}^{*}=B1 (\frac{1-l_3}{2},\frac{1-l_3}{2},l_3);$$

$$elseif{l}_1\ge 1/3,then{Y}^{*}=B5(l_1,l_1,1-2l_1)else{Y}^{*}=C(1/3,1/3,1/3).$$

(4) 

For $w_3\ge w_1\ge w_2$, the optimal solution $Y^{*}$ is

$$if{l}_1+2l_2\ge 1,then{Y}^{*}=A6(l_1,l_2,1-l_1-l_2).$$

$$elseif{w}_1\ge 1/3then{Y}^{*}=B4 (1-2l_2,l_2,l_2);$$

$$elseif{l}_1\ge 1/3then{Y}^{*}=B3(l_1,\frac{1-l_1}{2},\frac{1-l_1}{2})else{Y}^{*}=C(1/3,1/3,1/3).$$

(5) 

For $w_3\ge w_2\ge w_1$, the optimal solution $Y^{*}$ is

$$if{l}_1+2l_2\ge 1,then{Y}^{*}=A6(l_1,l_2,1-l_1-l_2).$$

$$elseif{l}_1\ge 1/3,then{Y}^{*}=B3(l_1,\frac{1-l_1}{2},\frac{1-l_1}{2})else{Y}^{*}=C(1/3,1/3,1/3).$$

Proof. 

There are six permutations of $(w_1,w_2,w_3)$. From Theorem 2, for $w_1\ge w_2\ge w_3$, the optimal solution is $\mathrm{A}1(1-l_2-l_3,l_2,l_3)$. Similarly, the optimal solution also is A1 for $w_1\ge w_3\ge w_2$.

We now consider the case that $w_2\ge w_1\ge w_3$. From Table 2, all of the twelve candidates are divided into two categories to obtain optimal solution: (I) A1, A3, A4, A5, A6 and C; (II) A2, B1, B2, B3, B4 and B5. We will show that the largest objective function value is that of $\mathrm{A}3(l_1,1-l_1-l_3,l_3)$ for category I.

Comparing the objective function value of A3 with that of $\mathrm{A}1$, from Theorem 1, since $w_2\ge w_1\ge w_3$, we have to compare $s_1=y_2$, $s_2=y_1+y_2$ and $s_3=y_1+y_2+y_3$ with those of $s_1^{\prime }=y_2^{\prime }$, $s_2^{\prime }=y_1^{\prime }+y_2^{\prime }$ and $s_3^{\prime }=y_1^{\prime }+y_2^{\prime }+y_3^{\prime }$. From

$$s_1=1-l_1-l_3\ge s_1^{\prime }=l_2,$$

$$s_2=1-l_3\ge s_2^{\prime }=1-l_3,$$

$$s_3=1\ge s_3^{\prime }=1,$$

the comparison results imply that A3 is superior to A1. Similarly, $F(\mathrm{A}3)$ is larger than those of A2, A5, A6 and C. Therefore, the optimal solution for category I is A3. A similar argument shows that the optimal solution for category II is $\mathrm{B}1(\frac{1-l_3}{2},\frac{1-l_3}{2},l_3)$. From the conditions of A3 and B1 displayed in Table 2, the optimal solution for $w_2\ge w_1\ge w_3$ is A3 for $2l_1+l_3\ge 1$, and B1 for $2l_1+l_3\le 1$.

Similarly, for $w_2\ge w_3\ge w_1$, $w_3\ge w_1\ge w_2$ and $w_3\ge w_2\ge w_1$, the optimal solutions can be derived. □

We next present two numerical experiments to evaluate the optimal solutions of 3COWAL with $l_1\ge l_2\ge l_3$.

Table 3. The number of instances satisfying solution type and weight for 3COWAL with $-1\le l_3<l_2<l_1\le 1$.

	$({\mathit{y}}_{\mathbf{1}},{\mathit{y}}_{\mathbf{2}},{\mathit{y}}_{\mathbf{3}})$	${\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{3}}$	${\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{2}}$	${\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{3}}$	${\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{1}}$	${\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{2}}$	${\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{1}}$	Total
A1	$(1-l_2-l_3, l_2, l_3)$	8744	8744	0	0	0	0	17,488
A2	$(1-l_1-l_3, l_1, l_3)$	0	0	0	0	0	0	0
A3	$(l_1,1-l_1-l_3, l_3)$	0	0	1632	1632	0	0	3264
A4	$(1-l_1-l_2, l_1, l_2)$	0	0	0	0	0	0	0
A5	$(l_1,1-l_1-l_2, l_2)$	0	0	0	0	0	0	0
A6	$(l_1, l_2,1-l_1- l_2)$	0	0	0	0	1920	1920	3840
B1	$(\frac{1-l_3}{2},\frac{1-l_3}{2}, l_3)$	0	0	7112	5474	0	0	12,586
B2	$(\frac{1-l_2}{2},\frac{1-l_2}{2}, l_2)$	0	0	0	0	0	0	0
B3	$(l_1,\frac{1-l_1}{2},\frac{1-l_1}{2})$	0	0	0	0	3026	3912	6938
B4	$(1-2l_2, l_2, l_2)$	0	0	0	0	1614	0	1614
B5	$(l_1, l_1,1-2l_1)$	0	0	0	910	0	0	910
C	$(1/3, 1/3, 1/3)$	0	0	0	728	2184	2912	5824

and

Table 4. The number of instances satisfying solution type and weight for 3COWAL with $-2\le l_3<l_2<l_1\le 2$.

	$({\mathit{y}}_{\mathbf{1}},{\mathit{y}}_{\mathbf{2}},{\mathit{y}}_{\mathbf{3}})$	${\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{3}}$	${\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{2}}$	${\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{3}}$	${\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{1}}$	${\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{2}}$	${\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{1}}$	Total
A1	$(1-l_2-l_3, l_2, l_3)$	57,464	57,464	0	0	0	0	114,928
A2	$(1-l_1-l_3, l_1, l_3)$	0	0	0	0	0	0	0
A3	$(l_1,1-l_1-l_3, l_3)$	0	0	19,352	19,352	0	0	38,704
A4	$(1-l_1-l_2, l_1, l_2)$	0	0	0	0	0	0	0
A5	$(l_1,1-l_1-l_2, l_2)$	0	0	0	0	0	0	0
A6	$(l_1, l_2,1-l_1-l_2)$	0	0	0	0	15,640	17,008	32,648
B1	$(\frac{1-l_3}{2},\frac{1-l_3}{2}, l_3)$	0	0	38,112	29,004	0	0	67,116
B2	$(\frac{1-l_2}{2},\frac{1-l_2}{2}, l_2)$	0	0	0	0	0	0	0
B3	$(l_1, \frac{1-l_1}{2},\frac{1-l_1}{2})$	0	0	0	0	19,566	28,312	47,878
B4	$(1-2l_2, l_2, l_2)$	0	0	0	0	10,114	0	10,114
B5	$(l_1, l_1,1-2l_1)$	0	0	0	5060	0	0	5060
C	$(1/3, 1/3, 1/3)$	0	0	0	4048	12,144	12,144	28,336

entries correspond to a pair (S, W) and give the number of different instances of $(l_1,l_2, l_3,w_1,w_2, w_3)$ satisfying type of candidate solution (S) and weight (W). The adopted measure is the number of instances. From Table 2, we adopt twelve types of candidate solutions and six permutations of weight. More precisely, $S\in \left\{\mathrm{A}1,\mathrm{A}2,\mathrm{A}3,\mathrm{A}4,\mathrm{A}5,\mathrm{A}6,\mathrm{B}1,\mathrm{B}2,\mathrm{B}3,\mathrm{B}4,\mathrm{B}5,\mathrm{C}\right\}$, $W\in \left\{w_1>w_2>w_3,w_1>w_3>w_2,w_2>w_1>w_3, w_2>w_3>w_1, w_3>w_1>w_2, w_3>w_2>w_1\right\}$, $l_i\in \left\{-lb,-0.9,-0.8,\dots ,lb\right\}$ and $w_i\in \left\{0,0.1,0.2,\dots ,1\right\}$, $i =1,2,3$.

The value of bound $lb$ is $lb=$1 for Table 3 and $lb=$2 Table 4. For each weight, the instances $(l_1,l_2, l_3,w_1,w_2, w_3)$ of 3COWAL are 8744 for $lb=1$ and 57,464 for $lb=2$. The total instances of 3COWAL are 397,248. An examination of Table 3 and Table 4 reveals that the largest number of instances is $\mathrm{A}1$ for $w_1>w_2>w_3$ and $w_1>w_3>w_2$, $\mathrm{B}1$ for $w_2>w_1>w_3$ and $w_2>w_3>w_1$, and $\mathrm{B}3$ for $w_3>w_1>w_2$ and $w_3>w_2>w_1$. Among all the instances of 3COWAL, the zero number is A2, A4, A5 and B2. Therefore, A1, B1 and B3 are superior in the number of the instances, while A2, A4, A5 and B2 are inferior ones for 3COWAL with $l_1\ge l_2\ge l_3$.

5. Minimizing Three-Dimensional Constrained OWA Aggregation Problem with Upper Bounded Variables

Consider the minimizing COWAU problem described as follows:

$$\begin{gathered} \mathrm{Min}{W}^{T}Y,\mathrm{s}.\mathrm{t}.{\mathcal{I}}^{T}X\ge 1,GY\le 0,y_i\mathcal{I}-X-M{Z}_i\le 0,i=1,2,\dots ,n-1, \\ {y}_n\mathcal{I}-X\le 0,{\mathcal{I}}^T{Z}_i\le n-i,i=1,2,\dots ,n-1,Z_{i+1}\le Z_i,i=1,2,\dots ,n-2, \\ {Z}_i\in {\left\{0,1\right\}}^n,i=1,2,\dots ,n-1,X\le U \end{gathered}$$

(12)

where the column vector

$$U=\left[\begin{array}{c}{u}_1\\ u_2\\ \begin{array}{c}⋮\\ u_n\end{array}\end{array}\right].$$

Using the standard transformations $X^{\prime }=U-X$, these lead to the following model

$$\begin{gathered} \mathrm{Min}{W}^{T}Y,\mathrm{s}.\mathrm{t}.{\mathcal{I}}^T{X}^{\prime }\le {\mathcal{I}}^{T}U-1,GY\le 0,y_i\mathcal{I}+X^{\prime }-M{Z}_i\le U,i=1,2,\dots ,n-1, \\ {y}_n\mathcal{I}+X^{\prime }\le U,{\mathcal{I}}^T{Z}_i\le n-i,i=1,2,\dots ,n-1,Z_{i+1}\le Z_i,i=1,2,\dots ,n-2, \\ {Z}_i\in {\left\{0,1\right\}}^n,i=1,2,\dots ,n-1,X^{\prime }\ge \mathbf{0}. \end{gathered}$$

(13)

If ${\mathcal{I}}^{T}U-1<0$, we conclude that the COWAU has no feasible solutions. If ${\mathcal{I}}^{T}U-1=0$, then $X^{\prime }=\mathbf{0}$ is the unique optimal solution, so $X=U$.

This section considers 3COWAU for

$${\mathcal{I}}^{T}U-1=u_1+u_2+u_3\ge 1.$$

(14)

Similar analyses to Section 3 and Section 4 can be derived. The results are described as follows.

At optimality, the first constraint of the model (13) becomes

$$x_1^{\prime }+x_2^{\prime }+x_3^{\prime }=u_1+u_2+u_3-1.$$

(15)

There are three types (A′, B′, C′) of $(x_1^{\prime },x_2^{\prime },x_3^{\prime })$ according to the number of zero components.

For 3COWAU with

$$u_1\ge u_2\ge u_3,$$

(16)

there are six candidate optimal solutions for type A′, seven candidate optimal solutions for type B′ and one candidate optimal solution for type C′. Detailed results of $(x_1^{\prime },x_2^{\prime },x_3^{\prime })$, $(x_1,x_2,x_3)$, $(y_1,y_2,y_3)$, $F(y_1,y_2,y_3)$ and condition are presented in

Table 5. The candidate optimal solutions of $(x_1^{\prime },x_2^{\prime },x_3^{\prime })$, $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$, $F(y_1,y_2,y_3)$ and condition for 3COWAU with $u_1\ge u_2\ge u_3$.

	$({\mathit{x}}_{\mathbf{1}}^{\prime },{\mathit{x}}_{\mathbf{2}}^{\prime },{\mathit{x}}_{\mathbf{3}}^{\prime })$	$({\mathit{x}}_{\mathbf{1}},{\mathit{x}}_{\mathbf{2}},{\mathit{x}}_{\mathbf{3}})$	$({\mathit{y}}_{\mathbf{1}},{\mathit{y}}_{\mathbf{2}},{\mathit{y}}_{\mathbf{3}})$	$\mathit{F}({\mathit{y}}_{\mathbf{1}},{\mathit{y}}_{\mathbf{2}},{\mathit{y}}_{\mathbf{3}})$	Condition
A′1	$(u_1+u_2+u_3-1, 0, 0)$	$(1-u_2-u_3, u_2, u_3)$	$(1-u_2-u_3, u_2, u_3)$	$w_1+u_2(-w_1+w_2)+u_3(-w_1+w_3)$	$2u_2+u_3\le 1$
A′2	$(u_1+u_2+u_3-1, 0, 0)$	$(1-u_2-u_3, u_2, u_3)$	$(u_2,1-u_2-u_3,u_3)$	$w_2+u_2(w_1-w_2)+u_3(-w_2+w_3)$	$2u_2+u_3\ge 1$, $u_2+2u_3\le 1$
A′3	$(u_1+u_2+u_3-1, 0, 0)$	$(1-u_2-u_3, u_2, u_3)$	$(u_2,u_3,1-u_2-u_3)$	$w_3+u_2(w_1-w_3)+u_3(w_2-w_3)$	$u_2+2u_3\ge 1$$u_2\ge 1/3$
A′4	$(0, u_1+u_2+u_3-1, 0)$	$(u_1,1-u_1-u_3, u_3)$	$(u_1,1-u_1-u_3,u_3)$	$w_2+u_1(w_1-w_2)+u_3(-w_2+w_3)$	$u_1+2u_3\le 1,$$u_3\le 1/3$
A′5	$(0, u_1+u_2+u_3-1, 0)$	$(u_1,1-u_1-u_3, u_3)$	$(u_1, u_3,1-u_1-u_3)$	$w_3+u_1(w_1-w_3)+u_3(w_2-w_3)$	$u_1+2u_3\ge 1$
A′6	$(0,0, u_1+u_2+u_3-1)$	$(u_1,u_2,1-u_1-u_2)$	$(u_1, u_2,1-u_1-u_2)$	$w_3+u_1(w_1-w_3)+u_2(w_2-w_3)$
B′1	$(\frac{2u_1+u_3-1}{2}, \frac{2u_2+u_3-1}{2},0)$	$(\frac{1-u_3}{2},\frac{1-u_3}{2}, u_3)$	$(\frac{1-u_3}{2},\frac{1-u_3}{2}, u_3)$	$\frac{1-w_3-u_3+3u_3{w}_3}{2}$	$u_3\le 1/3,$$2u_2+u_3\ge 1$
B′2	$(\frac{2u_1+u_3-1}{2},\frac{2u_2+u_3-1}{2},0)$	$(\frac{1-u_3}{2},\frac{1-u_3}{2}, u_3)$	$(u_3,\frac{1-u_3}{2},\frac{1-u_3}{2})$	$\frac{1-w_1-u_3+3u_3{w}_1}{2}$	$u_3\ge 1/3$
B′3	$(\frac{2u_1+u_2-1}{2},0,\frac{2u_3+u_2-1}{2})$	$(\frac{1-u_2}{2}, u_2,\frac{1-u_2}{2})$	$(u_2,\frac{1-u_2}{2},\frac{1-u_2}{2})$	$\frac{1-w_1-u_2+3u_2{w}_1}{2}$	$u_2\ge 1/3,$$u_2+2u_3\ge 1$
B′4	$(0,\frac{2u_2+u_1-1}{2},\frac{2u_3+u_1-1}{2})$	$(u_1,\frac{1-u_1}{2},\frac{1-u_1}{2})$	$(u_1,\frac{1-u_1}{2},\frac{1-u_1}{2})$	$\frac{1-w_1-u_1+3u_1{w}_1}{2}$	$u_1+2u_3\ge 1$
B′5	$(u_1-u_3,u_2+2u_3-1, 0)$	$(u_3,1-2u_3, u_3)$	$(u_3, u_3,1-2u_3)$	$w_3+u_3-3u_3{w}_3$	$u_3\ge 1/3$
B′6	$(u_1-u_3,u_2+2u_3-1, 0)$	$(u_3,1-2u_3, u_3)$	$(1-2u_3, u_3, u_3)$	$w_1+u_3-3u_3{w}_1$	$u_3\le 1/3$, $u_2+2u_3\ge 1$
B′7	$(u_1-u_2,0,2u_2+u_3-1)$	$(u_2,u_2,1-2u_2)$	$(u_2, u_2,1-2u_2)$	$w_3+u_2-3u_2{w}_3$	$u_2\ge 1/3,$$2u_2+u_3\ge 1$
C′	$(u_1-1/3,u_2-1/3,u_3-1/3)$	$(1/3, 1/3, 1/3)$	$(1/3, 1/3, 1/3)$	1/3	$u_1\ge 1/3,$$u_2\ge 1/3,$$u_3\ge 1/3$

.

Theorem 4.

For 3COWAU with $u_1\ge u_2\ge u_3$ and $w_1\ge w_2\ge w_3$, the smallest objective function value is that of $(y_1,y_2,y_3)=A^{\prime }1(1-u_2-u_3,u_2,u_3)$ for type A′, $B^{\prime }1(\frac{1-u_3}{2},\frac{1-u_3}{2},u_3)$ for type B′ and $C^{\prime }(1/3,1/3,1/3)$ for type C′.

For 3COWAU with $u_1\ge u_2\ge u_3$, the optimal solutions of different permutations of weight are described as follows.

Theorem 5.

For 3COWAU with $u_1\ge u_2\ge u_3$ and $u_1+u_2+u_3\ge 1$, the optimal solution $Y^{*}$ is described as follows.

(1) 

For $w_1\ge w_2\ge w_3$, the optimal solution $Y^{*}$ is

$$\begin{gathered} if{u}_3\ge 1/3,then{Y}^{*}=C^{\prime }(\frac{1}{3},\frac{1}{3},\frac{1}{3})elseif2u_2+u_3\le 1 \\ then{Y}^{*}=A^{\prime }1(1-u_2-u_3,u_2,u_3)else{Y}^{*}=B^{\prime }1(\frac{1-u_3}{2},\frac{1-u_3}{2},u_3). \end{gathered}$$

(2) 

For $w_1\ge w_3\ge w_2$, the optimal solution $Y^{*}$ is

$$if2u_2+u_3\le 1,then{Y}^{*}=A^{\prime }1 (1-u_2-u_3,u_2,u_3).$$

$$elseif{w}_3\ge 1/3,then{Y}^{*}=B^{\prime }7 (u_2,u_2,1-2u_2);$$

$$elseif{u}_3\le 1/3,then{Y}^{*}=B^{\prime }1(\frac{1-u_3}{2},\frac{1-u_3}{2},u_3)else{Y}^{*}=C^{\prime }(\frac{1}{3},\frac{1}{3},\frac{1}{3}).$$

(3) 

For $w_2\ge w_1\ge w_3$, the optimal solution $Y^{*}$ is

$$if{u}_1+2u_3\le 1,then{Y}^{*}=A^{\prime }4 (u_1,1-u_1-u_3,u_3).$$

$$elseif{w}_1\le 1/3,then{Y}^{*}=B^{\prime }4 (u_1,\frac{1-u_1}{2},\frac{1-u_1}{2});$$

$$elseif{u}_3\le 1/3,then{Y}^{*}=B^{\prime }6(1-2u_3,u_3,u_3)else{Y}^{*}=C^{\prime }(\frac{1}{3},\frac{1}{3},\frac{1}{3}).$$

(4) 

For $w_2\ge w_3\ge w_1$, the optimal solution $Y^{*}$ is

$$if{u}_1+2u_3\le 1,then{Y}^{*}=A^{\prime }4(u_1,1-u_1-u_3,u_3)else{Y}^{*}=B^{\prime }4(u_1,\frac{1-u_1}{2},\frac{1-u_1}{2}).$$

(5) 

For $w_3\ge w_1\ge w_2$ and $w_3\ge w_2\ge w_1$, the optimal solution is $A^{\prime }6(u_1,u_2,1-u_1-u_2)$.

To compare the optimal solution behaviors of maximizing 3COWAL with those of minimizing 3COWAU, the performance is shown in Figure 1. The first bar corresponds to the number of optimal solutions of maximizing 3COWAL and second to the minimizing 3COWAU, where the weight type $\mathrm{W}\left[i\right]$ denotes the ith component of $W=\left\{w_1>w_2>w_3,w_1>w_3>w_2,w_2>w_1>w_3, w_2>w_3>w_1, w_3>w_1>w_2, w_3>w_2>w_1\right\}$. The number of optimal solutions of $\mathrm{W}\left[i\right]$, $i=1,2,\dots ,6$, for maximizing 3COWAL is the same as that of $\mathrm{W}\left[6-i\right]$ for minimizing 3COWAU. Therefore, the numbers of optimal solutions for maximizing 3COWAL are same as those of minimizing 3COWAU but in reverse order. The correspondence between the optimal solution of these two mathematical models is worthy of future research.

We perform two numerical experiments to evaluate the optimal solution behaviors of 3COWAU with $u_1\ge u_2\ge u_3$.

Table 6. The number of instances satisfying solution type and weight for 3COWAU with $-1\le u_3\le u_2\le u_1\le 1$.

	$({\mathit{y}}_{\mathbf{1}},{\mathit{y}}_{\mathbf{2}},{\mathit{y}}_{\mathbf{3}})$	${\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{3}}$	${\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{2}}$	${\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{3}}$	${\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{1}}$	${\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{2}}$	${\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{1}}$	Total
A′1	$(1-u_2-u_3,u_2,u_3)$	624	624	0	0	0	0	1248
A′2	$(u_2,1-u_2-u_3,u_3)$	0	0	0	0	0	0	0
A′3	$(u_2,u_3,1-u_2-u_3)$	0	0	0	0	0	0	0
A′4	$(u_1,1-u_1-u_3,u_3)$	0	0	912	888	0	0	1800
A′5	$(u_1, u_3,1-u_1-u_3)$	0	0	0	0	0	0	0
A′6	$(u_1,u_2,1-u_1-u_2)$	0	0	0	0	1632	1632	3264
B′1	$(\frac{1-u_3}{2}, \frac{1-u_3}{2}, u_3)$	728	546	0	0	0	0	1274
B′2	$(u_3, \frac{1-u_3}{2}, \frac{1-u_3}{2})$	0	0	0	0	0	0	0
B′3	$(u_2, \frac{1-u_2}{2}, \frac{1-u_2}{2})$	0	0	0	0	0	0	0
B′4	$(u_1, \frac{1-u_1}{2}, \frac{1-u_1}{2})$	0	0	558	744	0	0	1302
B′5	$(u_3,u_3,1-2u_3)$	0	0	0	0	0	0	0
B′6	$(1-2u_3, u_3, u_3)$	0	0	92	0	0	0	92
B′7	$(u_2, u_2,1-2u_2)$	0	252	0	0	0	0	252
C′	$(1/3, 1/3, 1/3)$	280	210	70	0	0	0	560

and

Table 7. The number of instances satisfying solution type and weight for 3COWAU with $-2\le u_3\le u_2\le u_1\le 2$.

	$({\mathit{y}}_{\mathbf{1}},{\mathit{y}}_{\mathbf{2}},{\mathit{y}}_{\mathbf{3}})$	${\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{3}}$	${\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{2}}$	${\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{3}}$	${\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{1}}$	${\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{1}}>{\mathit{w}}_{\mathbf{2}}$	${\mathit{w}}_{\mathbf{3}}>{\mathit{w}}_{\mathbf{2}}>{\mathit{w}}_{\mathbf{1}}$	Total
A′1	$(1-u_2-u_3, u_2, u_3)$	9184	9184	0	0	0	0	18,368
A′2	$(u_2,1-u_2-u_3, u_3)$	0	0	0	0	0	0	0
A′3	$(u_2, u_3,1-u_2-u_3)$	0	0	0	0	0	0	0
A′4	$(u_1,1-u_1-u_3, u_3)$	0	0	13,472	13,328	0	0	26,800
A′5	$(u_1, u_3,1-u_1-u_3)$	0	0	0	0	0	0	0
A′6	$(u_1, u_2,1-u_1-u_2)$	0	0	0	0	26,352	26,352	52,704
B′1	$(\frac{1-u_3}{2}, \frac{1-u_3}{2}, u_3)$	11,728	8796	0	0	0	0	20,524
B′2	$(u_3,\frac{1-u_3}{2},\frac{1-u_3}{2})$	0	0	0	0	0	0	0
B′3	$(u_2, \frac{1-u_2}{2}, \frac{1-u_2}{2})$	0	0	0	0	0	0	0
B′4	$(u_1, \frac{1-u_1}{2}, \frac{1-u_1}{2})$	0	0	9768	13,024	0	0	22,792
B′5	$(u_3, u_3,1-2u_3)$	0	0	0	0	0	0	0
B′6	$(1-2u_3, u_3,u_3)$	0	0	1752	0	0	0	1752
B′7	$(u_2, u_2,1-2u_2)$	0	4292	0	0	0	0	4292
C′	$(1/3, 1/3, 1/3)$	5440	4080	1360	0	0	0	10,880

give the number of different instances $(u_1,u_2, u_3,w_1,w_2, w_3)$ satisfying type of candidate solution (S) and weight (W). More precisely,

$$\begin{gathered} S\in \left\{{\mathrm{A}}^{\prime }1,{\mathrm{A}}^{\prime }2,\dots ,{\mathrm{A}}^{\prime }6,{\mathrm{B}}^{\prime }1,{\mathrm{B}}^{\prime }2,\dots ,{\mathrm{B}}^{\prime }7,{\mathrm{C}}^{\prime }\right\}, \\ W\in \left\{w_1>w_2>w_3,w_1>w_3>w_2, w_2>w_1>w_3, \\ {w}_2>w_3>w_1, w_3>w_1>w_2, w_3>w_2>w_1\right\}, \\ {u}_i\in \left\{-ub,-0.9,-0.8,\dots ,ub\right\}\mathrm{and}{w}_i\in \left\{0, 0.1, 0.2,\dots , 1\right\}i=1,2,3. \end{gathered}$$

The value of bound $ub$ is $ub=1$ for Table 6 and $ub=2$ Table 7. For each weight, the number of the instances of 3COWAU is 8744 for $ub=1$ and 57,464 for $ub=2$. The total instances of 3COWAU are 397,248. From Table 6 and Table 7, the largest number of instances is ${\mathrm{B}}^{\prime }1(\frac{1-u_3}{2},\frac{1-u_3}{2},u_3)$ for $w_1>w_2>w_3$, ${\mathrm{A}}^{\prime }1(1-u_2-u_3,u_2,u_3)$ for $w_1>w_3>w_2$, ${\mathrm{A}}^{\prime }4(u_1,1-u_1-u_3,u_3)$ for $w_2>w_1>w_3$ and $w_2>w_3>w_1$, and ${\mathrm{A}}^{\prime }6(u_1,u_2,1-u_1-u_2)$ for $w_3>w_1>w_2$ and $w_3>w_2>w_1$. We also notice that the number of instances is zero for A′2, A′3, A′5, B′2, B′3 and B′5. Therefore, ${\mathrm{A}}^{\prime }1$, ${\mathrm{A}}^{\prime }4$, ${\mathrm{A}}^{\prime }6$ and ${\mathrm{B}}^{\prime }1$ are the best candidates, while A′2, A′3, A′5, B′2, B′3 and B′5 are inferior ones for 3COWAU with $u_1\ge u_2\ge u_3$.

6. Conclusions

This paper presents the optimal solutions for both maximizing 3COWAL and minimizing 3COWAU. For maximizing 3COWAL with $l_1\ge l_2\ge l_3$, there are six candidate optimal solutions for type A, five candidate optimal solutions for type B and one candidate optimal solution for type C. Theoretically and empirically, the largest number of instances is $\mathrm{A}1 (1-l_2-l_3,l_2,l_3)$ for $w_1>w_2>w_3$ and $w_1>w_3>w_2$, $\mathrm{B}1(\frac{1-l_3}{2},\frac{1-l_3}{2},l_3)$ for $w_2>w_1>w_3$ and $w_2>w_3>w_1$, and $\mathrm{B}3(l_1,\frac{1-l_1}{2},\frac{1-l_1}{2})$ for $w_3>w_1>w_2$ and $w_3>w_2>w_1$. For minimizing 3COWAU with $u_1\ge u_2\ge u_3$, there are six candidate optimal solutions for type A′, seven candidate optimal solutions for type B′ and one candidate optimal solution for type C′. The largest number of instances is ${\mathrm{B}}^{\prime }1(\frac{1-u_3}{2},\frac{1-u_3}{2},u_3)$ for $w_1>w_2>w_3$, ${\mathrm{A}}^{\prime }1 (1-u_2-u_3,u_2,u_3)$ for $w_1>w_3>w_2$, ${\mathrm{A}}^{\prime }4(u_1,1-u_1-u_3,u_3)$ for $w_2>w_1>w_3$ and $w_2>w_3>w_1$, and ${\mathrm{A}}^{\prime }6(u_1,u_2,1-u_1-u_2)$ for $w_3>w_1>w_2$ and $w_3>w_2>w_1$. Therefore, the best candidate optimal solutions are A1, B1 and B3 for maximizing 3COWAL with $l_1\ge l_2\ge l_3$, and ${\mathrm{A}}^{\prime }1$, ${\mathrm{A}}^{\prime }4$, ${\mathrm{A}}^{\prime }6$ and ${\mathrm{B}}^{\prime }1$ for minimizing 3COWAU with $u_1\ge u_2\ge u_3$.

Extending the analysis to high dimensions is worthy of future research in addition to analysis of the correspondence between the optimal solution of maximizing 3COWAL and minimizing 3COWAU. Thus, the analysis of maximizing COWAL and minimizing COWAU is a subject of considerable ongoing research.