Computation of a multi-choice goal programming problem

Applied Mathematics and Computation 271 (2015) 489–501

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Computation of a multi-choice goal programming problem Kanan K. Patro a,b,∗, M.M. Acharya a, M.P. Biswal c, Srikumar Acharya a a b c

Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar, India Department of Mathematics, Kendriya Vidyalaya, Bargarh, India Department of Mathematics, Indian Institute of Technology, Kharagpur, India

a r t i c l e

i n f o

Keywords: Multi-criteria decision making Multi-choice goal programming Multiple aspiration levels

a b s t r a c t The standard goal programming problem allows decision maker to assign an aspiration level to an objective function. In real life decision making problems, the decision maker always seeks for suitable aspiration level i.e. “the more suitable the better”. Therefore, a decision maker is allowed to assign multiple number of aspiration levels to an objective function. The aim of the decision maker is to select an appropriate aspiration level for an objective function that minimizes the deviations between the achievement of goal and the aspiration levels. The traditional goal programming techniques cannot be used for solving such type of multichoice goal programming problem. This paper presents an equivalent model of the multichoice goal programming problem by using Vandermonde’s interpolating polynomial, binary variables and least square approximation method. The equivalent model is solved by existing method/software. Two illustrative examples are presented in support of the proposed methodology. © 2015 Elsevier Inc. All rights reserved.

1. Introduction The term ‘decision’ carries different meanings, depending upon the nature of decision maker. A decision maker may be a lawyer, a businessman, a psychologist or a statistical or a general person. It might be behavioral action, mathematical model, or a specific kind of information processing. It is very difficult to represent decision making problems in mathematical models due to conflicts of resources and incompleteness of available information. In realistic situations decision making problems require to consider multiple objectives on one hand and various types of uncertainties on the other hand. There exists different methods to handle different uncertainties. Several techniques, namely utility function approach, goal programming approach, reference point method, interactive approach etc. exist for the solution of multi-criteria decision making problems. Among them the most popular is goal programming (GP) approach. GP is an analytic approach devised to address decision making problems where targets have been assigned to all the attributes. The decision maker (DM) is interested in minimizing the non-achievement of the goals. GP was first addressed by Charnes and Cooper [1]. Since the mid 70s, due to the seminal works by Lee [2] and Ignizio [3] an impressive revolution of GP applications and theoretical developments took place. Now-a-days, GP is the key-technique to work with multi-criteria decision making problems. Lee [2] and Ignizio [4] wrote impressive books on GP. Tamiz et al. [5] provided an up to date review on GP. The

∗

Corresponding author at: Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar, India. Tel.: +919861414866. E-mail addresses: [email protected] (K.K. Patro), [email protected] (M.M. Acharya), [email protected] (M.P. Biswal), [email protected], [email protected] (S. Acharya). http://dx.doi.org/10.1016/j.amc.2015.09.030 0096-3003/© 2015 Elsevier Inc. All rights reserved.

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contributions of Tamiz et al. [6], Romero et al. [7], Ijiri [8], Schniederjans [9], Zeleny and Cochrane [10] were remarkable for the development of GP. The distinction between various types of generalized goal programming, is made on the basis as how one actually measures the “goodness” of any solution (the value of X) to the set of goals. This is a typical method facilitated by means of the concept of “goal deviations” and the “achievement function”. By the philosophy of goal programming problem DM chooses a target value and decides whether to penalize positive or negative deviations from the target. However, DMs are not interested in the specific fixed deterministic targets associated with certain attributes in real-life. DMs prefer flexibility and suitability. It is observed that DMs are interested in a set of deterministic targets associated with an attribute. Keeping this in mind, Chang [11] proposed a new idea for programming the multi-choice aspiration level problem and named it as Multi-Choice Goal Programming (MCGP) Problem. He introduced the multiplicative terms of binary variables in order to tackle with multi-choice aspiration levels associated with each goal. The way he introduced the multiplicative terms of binary variables is too difficult to implement and is not easily understood by industrial participants. In Chang [12], although multiple aspiration levels are assigned to a goal, it is difficult to describe role of all aspiration levels in that goal. It replaces multiplicative terms of the binary variables by taking the help of a continuous variable, with a range of interval values as the lower and upper bound of each objective function. A new concept of constrained multi-choice goal programming is introduced for constructing the relationships between goals. This paper presents a new approach to search the appropriate set of aspiration levels from multiple sets of aspiration levels using multiplicative terms of binary variables. Biswal and Acharya [13,14], Acharya and Biswal [15] used binary variables in order to transform a multi-choice linear programming problem to an equivalent mathematical model. Using the concept of Chang [11], Liao [16] formulated multi-segment goal programming. Acharya and Acharya [17] generalized the transformation technique proposed by Biswal and Acharya [14]. Biswal and Acharya [18] used interpolating polynomial approach to solve multi-choice linear programming problem. After using the interpolation, the formulated mathematical model was a mixed integer nonlinear programming problem. Chang et al. [19] used his own technique to select a suitable house for homebuyer. Ustun [20] used conic scalarization function for formulating the MCGP. Fuzzy multi-choice goal programming problem was first addressed by Bankian-Tabrizi et al. [21]. Chang et al. [22] used multi-coefficients goal programming for group pricing problem. Multi choice mixed integer goal programming problem was carried out by Da Silva et al. [23]. They mainly focused on decisions on the choice of production process, including storage stages and distribution. The paper is organized as follows: Section 2 contains basic preliminaries followed by “mathematical model” in Section 3. In Section 4 equivalent models for the proposed MCGP are presented. Two numerical examples are provided in Section 5 to justify the methodology. In Section 6, results and discussions are presented. Finally concluding remarks are made following supporting references. 2. Basic preliminaries The aim of the goal programming is to minimize the deviations between the achievement of goals and their aspiration levels. According to the philosophy of satisficing we are interested in measuring the non-achievement of each goal. This is the unwanted deviations from the aspiration levels (i. e. the value of each goal ‘g’). We let

di = the deviation between ‘the aspiration level’ and ‘the acheivement o f goal’. ‘or’di = gi − fi (X ). Where gi is the aspiration level and fi (X) is the objective, which is to be achieved. Hence, we can express the general goal programming as:

min :

| fi (X ) − gi | for i = 1, 2, . . . , m

subject to X ∈R where R is a feasible set. The three oldest and still most widely used forms of GP are used to minimize the unwanted deviations. The methods are as follows: 1. Lexicographic Goal Programming (LGP), also known as non-Archimedean or preemptive GP. 2. Weighted GP (WGP), also known as Archimedean GP. 3. Min–Max GP (MGP) also known as Chebyshev or Fuzzy Programming. The mathematical formulations for LGP is expressed as follows:

lexicographically min : a¯ = (a1 , a2 , a3 , . . . , ak , . . . , aK ) subject to fi (X ) + ηi − βi = bi X, η¯ , ρ¯ ≥ 0

∀i

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where ak = gk (η¯ , ρ) ¯ and a¯ is the achievement vector for which we seek the lexicographic minimum; k is the ranking or priority ¯ is the usual linear function of goal in deviational variables that are to be minimized at priority level for k=1, 2, . . . , K; gk (η¯ , ρ) level k. The mathematical formulation of WGP might be expressed as:

min : a =



(αi ηi + βi ρi )

i∈G

subject to fi (X ) + ηi − ρi = gi for i ∈ G X ∈R X, η¯ , ρ¯ ≥ 0 Where ‘R’ is the feasible set and a is the measure of achievement (actually, of the non-achievement) of given flexible goals in G; α i is the weight associated with the negative deviational variable (ηi ) of ith goal; β i is the weight associated with the positive deviational variable (ρ i ) of ith goal. The weighted goal programming problem approach leads to a single linear goal programming model which may be solved via conventional (single objective) mathematical programming methods. Finally, let us consider the case of min–max philosophy, which leads to MGP. For convenience we first convert all goals (i. e. f (X ) ≤ g, f (X ) ≥ g, or f (X ) = g) into the Type-1 form (i. e. f(X) ≤ g). The model and achievement function may then be written as:

min :

λ

subject to

( fi (X ) − gi ) − λ ≤ 0 ∀ i X ∈R X, λ ≥ 0 Here we have a feasible set ‘R’. Where λ is the maximum deviation from any goal. This problem is again in the form of conventional, single objective model and thus any appropriate single objective solution methodology can be used. 3. Mathematical model We present a general multi-choice goal programming model as:



(Rk )

min : |Zk (X ) − gk |, gk ∈ g(k1) , g(k2) , g(k3) , . . . , gk



, k = 1, 2, 3, . . . , K

(3.1)

subject to n 

ai j x j ≤ bi , i = 1, 2, . . . , m

(3.2)

j=1

x j ≥ 0, j = 1, 2, . . . , n.

(3.3)

The first, second, third etc. elements of the set represent the aspiration levels for first, second, third etc. objective functions respectively. The goal programming model given in (3.1)–(3.3) can be expressed as:

min :

K 

ωk (ρk + ηk )

(3.4)

k=1

subject to



(Rk )

Zk (X ) + ηk − ρk = gk , gk ∈ g(k1) , g(k2) , g(k3) , . . . , gk n 



, k = 1, 2, 3, . . . , K

(3.5)

ai j x j ≤ bi , i = 1, 2, . . . , m

(3.6)

wk = 1

(3.7)

j=1 K  k=1

ρk , ηk , x j ≥ 0, j = 1, 2, . . . , n; k = 1, 2, 3, . . . , K ωk , ρ k , ηk represent the weights, positive and negative deviations associated with kth goal respectively.

(3.8)

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4. Equivalent models In this section, three different approaches are discussed. 4.1. Interpolating polynomial approach A polynomial that passes through a given set of points is called an interpolating polynomial. There are ‘n’ points in a plane

(i, yi ), i = 0, 1, 2, 3 . . . , n − 1 with distinct i then there exists a unique polynomial in x whose degree is n − 1. Let the polynomial be

PRk −1 (zi ) = a0 + a1 zi + a2 (zi )2 + a3 (zi )3 + · · · + an−2 (zi )n−2 + an−1 (zi )n−1

(4.9)

after substituting the n points in the polynomial we have,

P (0) = a0 + a1 (0) + a2 (0)2 + a3 (0)3 + . . . + an−2 (0)n−2 + an−1 (0)n−1 = y0

(4.10)

P (1) = a0 + a1 (1) + a2 (1)2 + a3 (1)3 + . . . + an−2 (1)n−2 + an−1 (1)n−1 = y1

(4.11)

P (2) = a0 + a1 (2) + a2 (2)2 + a3 (2)3 + . . . + an−2 (2)n−2 + an−1 (2)n−1 = y2

(4.12)

.. .

(4.13)

P (n − 1) = a0 + a1 (n − 1) + a2 (n − 1)2 + a3 (n − 1)3 + . . . + an−1 (n − 1)n−1 = yn−1

(4.14)

The system of equations can be written as AX = B where

⎛

1

⎜1 ⎜ ⎜1 A=⎜ ⎜ ⎜.. ⎝. 1

⎞

···

0

12

···

1n−2

1n−1

2

···

2

n−2

n−1

..

.. .

.. .

(n − 1)n−2

(n − 1)n−1

0

0

1 2

2

.. .

.. .

(n − 1)

(n − 1)

2

.

···

0 2

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎛a

0

⎜a1 ⎜ ⎜ X = ⎜a2 ⎜. ⎝.. an−1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎛y

0

⎜ y1 ⎜ ⎜ B = ⎜ y2 ⎜. ⎝..

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(4.15)

yn−1

The value of a1 , a2 , . . . , an can be found out by solving AX = B. If A is non singular then the system can be solved. Now, we formulate a multi-choice goal programming model for the multi-objective linear programming problem by using Vandermonde’s interpolating polynomial as:

min :

K 

ωk (ρk + ηk )

(4.16)

k=1

subject to

Zk (X ) + ηk − ρk = PRk −1 (z), k = 1, 2, 3, . . . , K n 

ai j x j ≤ bi , i = 1, 2, . . . , m

(4.17)

(4.18)

j=1

z is an integer and 0 ≤ z ≤ Rk − 1

(4.19)

ρk , ηk , x j , z ≥ 0, j = 1, 2, . . . , n; k = 1, 2, 3, . . . , K

(4.20)

4.2. Binary variable approach Theorem 1. Every natural number can be expressed as sum of 2k number of terms and each term is a power of 2, where k ∈ N ∪ {0}. The proof of the above theorem is obvious. Now, we formulate a multi-choice goal programming model for the multi-objective linear programming problem by using binary variables as:

min :

K  k=1

ωk (ρk + ηk )

(4.21)

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493

Table 1 Nodes.

subject to

0

1

2

…

n−1

f(x)

f0

f1

f2

…

fn−1



θk(1) , θk(2) , k = 1, 2, 3, . . . , K

Zk (X ) + ηk − ρk = Pk n 

x

ai j x j ≤ bi , i = 1, 2, . . . , m

(4.22)

(4.23)

j=1

θk(1) , θk(2) ∈ {0, 1}

(4.24)

ρk , ηk , x j , z ≥ 0, j = 1, 2, . . . , n; k = 1, 2, 3, . . . , K

(4.25)

4.3. Linear least square approximation approach Let a function f(x) be given by the following Table 1 at a discrete set of points i; i = 0(1)n − 1. By using the set of points we want to find out a least square line such that the sum of the square of the vertical distance of the points (i, fi ), i = 0(1)n − 1 from the line is the minimum. In this case the degree of the least square polynomial is 1 and

li (z) = a0 + a1 z

(4.26)

the least square error E is given by

E=

n−1 

( fi − a0 − a1 i)2

(4.27)

i=0

The necessary conditions for E to be minimum are

a0

n−1 



+ a1

i

i=0

n−1 



2

=

i

i=0

a0 n + a1

n−1  i=0

 i

=

n−1 

n−1 

ifi

(4.28)

i=0

fi

(4.29)

i=0

which is a system of two linear equations in a0 , a1 and are the normal equations for the least square linear polynomial. By solving the normal equation given by (4.28) and (4.29), we get the two coefficients a0 and a1 and the linear least square polynomial li (z) = a0 + a1 z. Now, we formulate a multi-choice goal programming model for the multi-objective linear programming problem by using linear least square approximation approach as:

min :

K 

ωk (ρk + ηk )

(4.30)

k=1

subject to

Zk (X ) + ηk − ρk = lk (z), k = 1, 2, 3, . . . , K n 

(4.31)

ai j x j ≤ bi , i = 1, 2, . . . , m

(4.32)

z is an integer and 0 ≤ z ≤ Rk − 1

(4.33)

ρk , ηk , x j , z ≥ 0, j = 1, 2, . . . , n; k = 1, 2, 3, . . . , K

(4.34)

j=1

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K.K. Patro et al. / Applied Mathematics and Computation 271 (2015) 489–501 Table 2 Investment opportunities (Eatman and Sealey Jr [24]). Investment category

Rate of return (%)

Liquid part (%)

Required capital (%)

Risk asset

1. Cash 2. Short term 3. Government: over 1 to 5 years 4. Government: over 1 to 5 years 5. Government: over 10 years 6. Installment loans 7. Mortgage loans 8. Commercial loans

0.0 4.0 4.5 5.5 7.0 10.5 8.5 9.2

100.0 99.5 96 90 85.0 0.0 0.0 0.0

0.0 0.5 4.0 5.0 7.5 10.0 10.0 10.0

No No No No No Yes Yes Yes

5. Numerical example Example 1. Consider a weighted goal programming problem which was proposed by Eatman and Sealey Jr [24]. Bank Three has a modest $20 million capital, with $150 million in demand deposits(checking accounts) and $80 million in time deposits (saving accounts and certificates of deposit). Table 2 displays the categories among which the Bank must divide its capital and deposited funds. Rates of return are also provided for each category together with other information related to risk. We model Bank Three’s investment decisions with a decision variable A, B, C, D, E, F, G and H for each category of investment in the given table. The first goal of any private business is to maximize profit. Using rates of return from the Table 2, we have the objective function

max: Z1 = 0.04B + 0.045C + 0.055D + 0.07E + 0.105F + 0.085G + 0.092H It is less clear how to quantify investment risk. We employ two common ratio measures. One is capital-adequacy ratio, expressed as the ratio of required capital for bank solvency to actual capital. A low value indicates minimum risk. Bank Three’s present capital is $20 million. Thus, we express a second objective as

min: Z2 = 0.05(0.005B + 0.04C + 0.05D + 0.075E + 0.1F + 0.1G + 0.1H ) Another measure of risk focuses on liquid risk assets. A low risk asset/capital ratio indicates a financially secure institution. For our example, this third measure of success is expressed as

min Z3 = 0.05(F + G + H ) To complete the model of Bank Three’s investment plans, we must describe the relevant constraints. Our example will assume five types: 1. 2. 3. 4. 5.

Investments must sum to the available capital and deposit funds. Cash reserves must be at least 14% of demand deposit plus 4% of time deposits. The portion of investments considered liquid should be at least 47% of demand deposit plus 36% of time deposits. At least 5% of funds should be invested in each of the eight categories, for diversity. At least 30% of funds should be invested in commercial loans, to maintain the bank’s community status.

Combining the three objective functions above with these five systems of constraints gives rise to multi-objective linear programming model

max : Z1 = 0.04B + 0.045C + 0.055D + 0.07E + 0.105F + 0.085G + 0.092H

(5.35)

min : Z2 = 0.05(0.005B + 0.04C + 0.05D + 0.075E + 0.1F + 0.1G + 0.1H )

(5.36)

min : Z3 = 0.05(F + G + H )

(5.37)

subject to

A + B + C + D + E + F + G + H = 250

(5.38)

A ≥ 24.2

(5.39)

1.0A + 0.995B + 0.96C + 0.9D + 0.85E ≥ 99.3

(5.40)

A, B, C, D, E, F, G ≥ 12.5

(5.41)

H ≥ 75

(5.42)

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The above multi-objective programming model can be written in goal format as follows:

min : |0.04B + 0.045C + 0.055D + 0.07E + 0.105F + 0.085G + 0.092H − g1 |, g1 ∈ {17, 20, 22, 24, 27}

(5.43)

min : |0.05(0.005B + 0.04C + 0.05D + 0.075E + 0.1F + 0.1G + 0.1H ) − g2 |, g2 ∈ {0.6, 0.7, 0.8, 0.9} min : |0.05(F + G + H ) − g3 |, g3 ∈ {5, 6, 7}

(5.44) (5.45)

subject to

A + B + C + D + E + F + G + H = 250

(5.46)

A ≥ 24.2

(5.47)

1.0A + 0.995B + 0.96C + 0.9D + 0.85E ≥ 99.3

(5.48)

A, B, C, D, E, F, G ≥ 12.5

(5.49)

H ≥ 75

(5.50)

5.1. Solution by binary variable approach By using the concept of multi-choice goal programming, it can be represented as:

min : η1 + ρ2 + ρ3

(5.51)

subject to

0.04B + 0.045C + 0.055D + 0.07E + 0.105F + 0.085G + 0.092H + η1 − ρ1 = 17θ11 (1 − θ12 )θ13 + 20(1 − θ11 )θ12 θ13 + 22θ11 (1 − θ12 )(1 − θ13 ) + 24(1 − θ11 )θ12 (1 − θ13 ) + 27(1 − θ11 )(1 − θ12 )θ13

(5.52)

0.05(0.005B + 0.04C + 0.05D + 0.075E + 0.1F + 0.1G + 0.1H ) + η2 − ρ2 = 0.6θ21 θ22 + 0.7θ21 (1 − θ22 ) + 0.8(1 − θ21 )θ22 + 0.9(1 − θ21 )(1 − θ22 )

(5.53)

0.05(F + G + H ) + η3 − ρ3 = 5θ31 θ32 + 6θ31 (1 − θ32 ) + 7(1 − θ31 )θ32

(5.54)

A + B + C + D + E + F + G + H = 250

(5.55)

A ≥ 24.2

(5.56)

1.0A + 0.995B + 0.96C + 0.9D + 0.85E ≥ 99.3

(5.57)

A, B, C, D, E, F, G ≥ 12.5

(5.58)

H ≥ 75

(5.59)

θ11 + θ12 + θ13 ≥ 1

(5.60)

θ11 + θ12 + θ13 ≤ 2

(5.61)

θ11 + θ12 ≤ 1

(5.62)

θ31 + θ32 ≥ 1

(5.63)

A, B, C, D, E, F, G, H ≥ 0

(5.64)

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ρi , ηi ≥ 0; i = 1, 2, 3

(5.65)

θ11 , θ12 , θ13 ∈ {0, 1}

(5.66)

θ21 , θ22 ∈ {0, 1}

(5.67)

θ31 , θ32 ∈ {0, 1}

(5.68)

The above mathematical model (5.51)–(5.68) is solved with the help of LINGO 11.0 [25] software to obtain the optimal solution as (A, B, C, D, E, F, G, H) = (39.7, 12.5, 12.5, 12.5, 32.8, 12.5, 12.5, 115), (θ 11 , θ 12 , θ 13 , g1 ) = (1.0, 0.0, 1.0, 17), (θ 21 , θ 22 , g2 ) = (0.0, 0.0, 0.9), (θ 31 , θ 32 , g3 ; Z1 , Z2 , Z3 ) = (0.0, 1.0; 17.0, 0.88, 7.0). 5.2. Solution by Vandermonde’s interpolating polynomial approach By using the concept of multi-choice goal programming, it can be represented as:

min : η1 + ρ2 + ρ3

(5.69)

subject to

0.04B + 0.045C + 0.055D + 0.07E + 0.105F + 0.085G + 0.092H + η1 − ρ1 = 17 + 3.83z1 − z12 + 0.17z13

(5.70)

0.05(0.005B + 0.04C + 0.05D + 0.075E + 0.1F + 0.1G + 0.1H ) + η2 − ρ2 = 0.6 + 0.1z2

(5.71)

0.05(F + G + H ) + η3 − ρ3 = 5 + z3

(5.72)

A, B, C, D, E, F, G, H ≥ 0

(5.73)

A + B + C + D + E + F + G + H = 250

(5.74)

A ≥ 24.2

(5.75)

1.0A + 0.995B + 0.96C + 0.9D + 0.85E ≥ 99.3

(5.76)

A, B, C, D, E, F, G ≥ 12.5

(5.77)

H ≥ 75

(5.78)

ρi , ηi ≥ 0, i = 1, 2, 3

(5.79)

0 ≤ z1 ≤ 4

(5.80)

0 ≤ z2 ≤ 3

(5.81)

0 ≤ z3 ≤ 2.

(5.82)

zi ∈ Z, i = 1, 2, 3

(5.83)

The above mathematical model (5.69)–(5.83) is solved with the help of LINGO 11.0 [25] software to obtain the solution as (A, B, C, D, E, F, G, H) = (39.89, 12.5, 12.5, 12.5, 32.6, 13.46, 12.5, 114.03); (z1 , g1 ) = (0.0, 17); (z2 , g2 ) = (3, 0.9); (z3 , g3 ; Z1 , Z2 , Z3 ) = (2.0, 7.0; 17.0, 0.88, 7.0).

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5.3. Solution by linear least square approximation approach By using the concept of multi-choice goal programming, it can be represented as:

min : η1 + ρ2 + ρ3

(5.84)

subject to

0.04B + 0.045C + 0.055D + 0.07E + 0.105F + 0.085G + 0.092H + η1 − ρ1 = 17.2 + 2.4z1

(5.85)

0.05(0.005B + 0.04C + 0.05D + 0.075E + 0.1F + 0.1G + 0.1H ) + η2 − ρ2 = −1.5 + 1.5z2

(5.86)

0.05(F + G + H ) + η3 − ρ3 = 5 + z3

(5.87)

5A + 7B + 8C + η3 − ρ3 = 56.8 + 5.4z2

(5.88)

+ − 56.8 + 5.4z2 + d21 − d22 =0

(5.89)

A, B, C ≥ 0

(5.90)

A + B + C + D + E + F + G + H = 250

(5.91)

A ≥ 24.2

(5.92)

1.0A + 0.995B + 0.96C + 0.9D + 0.85E ≥ 99.3

(5.93)

A, B, C, D, E, F, G ≥ 12.5

(5.94)

H ≥ 75

(5.95)

ρi , ηi ≥ 0, i = 1, 2, 3

(5.96)

0 ≤ z1 ≤ 2

(5.97)

0 ≤ z2 ≤ 4; z1 , z2 ∈ Z

(5.98)

The above mathematical model (5.84)–(5.98) is solved with the help of LINGO 11.0 [25] software to obtain the solution as (A, B, C, D, E, F, G, H) = (39.89, 12.5, 12.5, 12.5, 32.6, 13.46, 12.5, 114.03); (z1 , g1 ) = (0.0, 17);(z2 , g2 ) = (3, 0.9); (z3 , g3 ; Z1 , Z2 , Z3 ) = (2.0, 7.0; 17.0, 0.88, 7.0). Example 2. Let us consider a multi-choice goal programming problem [11] with the following goals and constraints, which cannot be solved by the known GP techniques.

Goals : (G1)3A + 2B + C ≤ g1 , g1 ∈ {100, 120}

(G2)4A + 3B + 2C ≥ g2 , g2 ∈ {80, 100} (G3)3.5A + 5B + 3C ≥ g3 , g3 ∈ {70, 90, 110} subject to B + C ≥ 10; B ≥ 4; A + B + C ≥ 15. 5.4. Solution by binary variable approach Based on the proposed binary variable approach, this problem can be formulated as:

min : ρ1 + η2 + η3

(5.99)

subject to

3A + 2B + C + ρ1 − η1 = 100θ41 + 120(1 − θ41 )

(5.100)

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K.K. Patro et al. / Applied Mathematics and Computation 271 (2015) 489–501

4A + 3B + 2C + ρ2 − η2 = 80θ51 + 100(1 − θ51 )

(5.101)

3.5A + 5B + 3C + ρ3 − η3 = 70θ61 θ62 + 90θ61 (1 − θ62 ) + 110(1 − θ61 )θ62

(5.102)

A, B, C ≥ 0

(5.103)

B + C ≥ 10

(5.104)

B≥4

(5.105)

A + B + C ≥ 15

(5.106)

θ41 ≤ 1

(5.107)

θ51 ≤ 1

(5.108)

ρi , ηi ≥ 0; i = 1, 2, 3

(5.109)

θ61 + θ62 ≥ 1

(5.110)

θ61 + θ62 ≤ 2

(5.111)

¯ 7.77, ¯ This problem is solved by using LINGO 11.0 [25] to obtain the optimal solution as (A, B, C, θ 41 , θ 51 , θ 61 , θ 62 ) = (17.55, ¯ 1, 0, 0, 1). From the results we get that goal G1 has achieved 71.44 and reached the aspiration level 100, whereas goals G2 3.22, and G3 reached the aspiration level 100 and 110 respectively.

5.5. Solution by Vandermonde’s interpolating polynomial approach Based on the proposed Vandermonde’s interpolating polynomial approach, this problem can be formulated as:

min : ρ1 + η2 + η3

(5.112)

subject to

3A + 2B + C + ρ1 − η1 = 100 + 20z1

(5.113)

4A + 3B + 2C + ρ2 − η2 = 80 + 20z2

(5.114)

3.5A + 5B + 3C + ρ3 − η3 = 70 + 20z3

(5.115)

A, B, C ≥ 0

(5.116)

B + C ≥ 10

(5.117)

B≥4

(5.118)

A + B + C ≥ 15

(5.119)

zi ∈ Z, i = 1, 2, 3

(5.120)

0 ≤ z1 ≤ 1

(5.121)

0 ≤ z2 ≤ 1

(5.122)

0 ≤ z3 ≤ 2

(5.123)

ρi , ηi ≥ 0; i = 1, 2, 3

(5.124)

¯ 8.88, ¯ 1.11, ¯ 0, 1, 2). This problem is solved by using LINGO 11.0 [25] to obtain the optimal solution as (A, B, C, z1 , z2 , z3 ) = (17.77, From the results we realized that goal G1 has reached 72.22 and achieved the aspiration level 100, goals G2 and G3 reached the aspiration level respectively 100 and 110.

K.K. Patro et al. / Applied Mathematics and Computation 271 (2015) 489–501

499

5.6. Solution by linear least square approximation approach Based on the proposed linear least square approximation approach, this problem can be formulated as:

min : ρ1 + η2 + η3

(5.125)

subject to

3A + 2B + C + ρ1 − η1 = 100 + 20z1

(5.126)

4A + 3B + 2C + ρ2 − η2 = 80 + 20z2

(5.127)

3.5A + 5B + 3C + ρ3 − η3 = 70 + 20z3

(5.128)

A, B, C ≥ 0

(5.129)

B + C ≥ 10

(5.130)

B≥4

(5.131)

A + B + C ≥ 15

(5.132)

zi ∈ Z, i = 1, 2, 3

(5.133)

0 ≤ z1 ≤ 1

(5.134)

0 ≤ z2 ≤ 1

(5.135)

0 ≤ z3 ≤ 2

(5.136)

ρi , ηi ≥ 0; i = 1, 2, 3

(5.137)

¯ 8.88, ¯ 1.11, ¯ 0, 1, 2). This problem is solved by using LINGO 11.0 [25] to obtain the optimal solution as (A, B, C, z1 , z2 , z3 ) = (17.77, From the results we realized that goal G1 has reached 72.22 and achieved the aspiration level 100, goals G2 and G3 reached the aspiration level exactly 100 and 110 respectively. 6. Results, discussions and conclusions This paper uses three approaches, namely: Binary variable approach, Vandermonde’s interpolating polynomial approach and linear least square approximation approach. From the computational results, it is concluded that in all the three methods, goals are the same. The objective values in all the three cases are approximately equal. In Chang [11] model, multiplicative terms of binary variables were also used but repetitions occur during solution procedure, whereas, in our model the binary variable do not allow any repetition. The use of binary variables in multi-choice goal programming model increases the complexity. The complexity of solution also increases if the number of objective functions and the number of choice increases. Here we use two sets of binary variable. The first set of binary variables searches a group which the appropriate choice belongs. Second set of binary variables searches the appropriate parameter from the searched group. Comparison of results of three approaches for Example 1 presented in Table 3. Chang [11] model represents the time complexity in non-polynomial time(exponential/logarithmic) whereas Chang [12] model maintain the time complexity in polynomial time. Biswal and Acharya [14] used binary variable and some auxiliary constraints to deal with multi-choice parameters. But the proposed binary variable approach does not need any extra constraints. It is observed that our proposed Vandermonde’s interpolating polynomial approach has equal computational complexity as that of Acharya et al. [26]. Although the proposed linear least square approximation approach gives approximate choices of the multi-choice parameter, but computational complexity is equal to simple linear integer programming problem. It can also be solved by linear integer programming method. The method suggested by Chang [12] also do the same thing but unable to take care of all the multi-choice parameters. Here a comparative study of time complexity for 8, 9 and 10 number of goals having 2, 3, 4, 5, and 8 choices of multiple aspiration levels is presented in Tables 4–6. It shows the proposed methods perform better. An attempt is to be made to linearize binary variable approach and Vandermonde’s interpolating polynomial approach.

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K.K. Patro et al. / Applied Mathematics and Computation 271 (2015) 489–501 Table 3 Comparative analysis of three approaches for Example 1. Investment category

PM1

PM2

PM3

1. Cash(A) 2. Short term(B) 3. Government: over 1 to 5 years(C) 4. Government: over 5 to 10 years(D) 5. Government: over 10 years(E) 6. Installment loans(F) 7. Mortgage loans(G) 8. Commercial loans(H) 9. Profit(Z1 ) 10.Capital - adequacy ratio(Z2 ) 11.Risk - asset ratio(Z3 )

39.7 12.5 12.5 12.5 32.8 12.5 12.5 115 17 0.88 7.0

39.89 12.5 12.5 12.5 32.6 13.46 12.5 114.03 17 0.88 7.0

17.7 12.5 54.7 12.5 12.5 12.5 52.5 75.0 17.2 0.89 7.0

PM1 : Binary variable approach PM2 : Vandermonde’s interpolating polynomial approach PM3 : Linear least square approach Table 4 Relative performance of 8 number of Goals(CPU Time)(hh:mm:ss). No. of A.L.

Chang [11]

Chang [12]

Biswal and Acharya [14]

Biswal and Acharya [18]

PM1

PM2

PM3

2 3 4 5 8

00:00:01 00:00:06 00:00:05 00:03:45 00:03:31

00:00:01 00:00:01 00:00:01 00:00:01 00:00:01

00:00:01 00:00:04 00:00:05 00:00:15 00:00:12

00:00:01 00:00:04 00:00:05 00:00:15 00:00:12

00:00:01 00:00:02 00:00:03 00:00:10 00:00:07

00:00:01 00:00:04 00:00:05 00:00:15 00:00:12

00:00:01 00:00:01 00:00:01 00:00:01 00:00:01

Table 5 Relative performance of 9 number of Goals(CPU Time)(hh:mm:ss). No. of A.L.

Chang [11]

Chang [12]

Biswal and Acharya [14]

Biswal and Acharya [18]

PM1

PM2

PM3

2 3 4 5 8

00:00:01 00:00:07 00:00:06 01:55:37 01:43:34

00:00:01 00:00:01 00:00:01 00:00:01 00:00:01

00:00:01 00:00:05 00:00:06 00:07:38 00:06:31

00:00:01 00:00:05 00:00:06 00:07:38 00:06:31

00:00:01 00:00:03 00:00:01 00:06:43 00:00:01

00:00:01 00:00:05 00:00:06 00:07:38 00:06:31

00:00:01 00:00:01 00:00:01 00:00:01 00:00:01

Table 6 Relative performance of 10 number of Goals(CPU Time)(hh:mm:ss). No. of A.L.

Chang [11]

Chang [12]

Biswal and Acharya [14]

Biswal and Acharya [18]

PM1

PM2

PM3

2 3 4 5 8

00:00:01 00:00:07 00:00:06 15:53:25 15:13:05

00:00:01 00:00:01 00:00:01 00:00:01 00:00:01

00:00:01 00:00:05 00:00:05 01:02:14 00:58:45

00:00:01 00:00:05 00:00:05 01:02:14 00:58:45

00:00:01 00:00:03 00:00:03 57:03:12 55:01:35

00:00:01 00:00:05 00:00:05 01:02:14 00:58:45

00:00:01 00:00:01 00:00:01 00:00:01 00:00:01

Acknowledgments The authors would like to thank the Editor-in-Chief, Associate Editor, and the reviewers for their helpful suggestions and comments which have led to an improvement in both quality and clarity of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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