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Solving multichoice multiobjective transportation problem: a utility function approach
Journal of Uncertainty Analysis and Applications volume 2, Article number: 11 (2014)
Abstract
This paper explores the study of multichoice multiobjective transportation problem (MCMTP) under the environment of utility function approach. MCMTP is converted to multiobjective transportation problems (MOTP) by transforming the multichoice parameters like cost, demand, and supply to realvalued parameters. A general transformation procedure using binary variables is illustrated to reduce MCMTP into MOTP. Most of the MOTP are solved by goal programming (GP) approach. Using GP, the solution of MOTP may not be satisfied all the time by the decision maker (DM) when the proposed problem contains intervalvalued aspiration level. To overcome this difficulty, here we propose the approaches of revised multichoice goal programming (RMCGP) and utility function into the MOTP and then compared the solution between them. Finally, numerical examples are presented to show the feasibility and usefulness of our paper.
Introduction
The transportation problem is the central nerve system to keep the balance in economical world from ancient day until today. In earlier days, transportation problem developed with the assumption that the supply, demand, and cost parameters are exactly known. But in reallife applications, all the parameters of the transportation problem are not generally defined precisely. Keeping this point of view, in this paper, we have incorporated with multichoice multiobjective transportation problem (MCMTP) considering the parameters of transportation problem as multichoice type.
Instead of single choice, if there may be several choices involved associated with the transportation parameters like cost, supply, or demand, then the decision maker is confused to select the proper choice for these parameters. In this circumstances, the study of transportation problem creates a new direction which is called multichoice multiobjective transportation problem. Chang [1] proposed a multichoice goal programming approach to solve the mathematical programming. Again in the subsequent year, Chang [2] proposed another multichoice goal programming approach in revised form. Though the multichoice concept discussed in both the papers of Chang [1, 2] is totally related to the goals of objective functions, recently, Mahapatra et al. [3] and Roy et al. [4] discussed the multichoice stochastic transportation problem involving extreme value distribution and exponential distribution in which the multichoice concept involved only in the cost parameters.
In this paper, we have designed a general transformation technique to reduce the multichoice parameters like cost, time, and demand to singlevalued parameters. Using this technique, the MCMTP problem can be reduced to MOTP.
Goal programming (GP), an analytical approach, is devised to address the decisionmaking problem where targets have been assigned to all objective functions. The objective functions are conflicting and commensurable to each other, and the DM is interested to minimize the nonachievement of the corresponding goals. In other words, the DM derived an optimal solution with this strategy of GP which is satisfactory. However, using GP, the solution procedure for MOTP has some limitation. The main limitation behind GP is that the priority of goals for the DM is not easily considered. It seems far from reality. In the recent past, the notion of utility function is introduced by several researchers such as [5], Yu et al. [6], and Podinovski [7]. Recently, multichoice goal programming (MCGP) has been proposed by Chang [1, 2] to solve multiobjective decisionmaking problems with aspiration level. However, to the best of our knowledge, no works have been done on utility function to solve MOTP with the DM’s preferences. The main motivation of this study is to investigate the better solution of MOTP by using utility function approach and then compare the solution to other methods such as GP and RMCGP.
Charnes et al. [8] introduced the concept of GP further developed by several researchers such as Charnes and Cooper [9], Lee [10], Ignizio [11], Tamiz et al. [12], and Romero [13]. In long back, the main concept of GP was to minimize the deviation between the achievement goals and the achievement levels. The mathematical model of multiobjective decision making (MODM) can be considered in the following form:
GP
where w_{ i } are the weights attached to the deviation of the achievement function. Z^{i}(X) is the i th objective function of the i th goal, and g_{ i } is the aspiration level of the i th goal. Z^{i}(X)−g_{ i } represents the deviation of the i th goal. Later on, a modification on GP is provided and denoted as weighted goal programming (WGP) which can be displayed in the following form:
WGP
where ${d}_{i}^{+}$ and ${d}_{i}^{}$ are over and under achievements of the i th goal, respectively.
However, the conflicts of resources and the incompleteness of available information make it almost impossible for DMs to set the specific aspiration levels and choose the better decision. To overcome this situation, MCGP approach has been presented by Chang [1] with a new direction to solve MODM problem. In the next year, Chang [2] proposed the revised form of MCGP defined as RMCGP to solve MODM. The mathematical model of MODM using RMCGP is defined as follows:
RMCGP
where y_{ i } is the continuous variable associated with i th goal which restricted between the upper (g_{i,max}) and lower (g_{i,min}) bounds, ${e}_{i}^{+}$ and ${e}_{i}^{}$ are positive and negative deviations attached to the i th goal of y_{ i }−g_{i,max}, and α_{ i } is the weight attached to the sum of the deviations of y_{ i }−g_{i,max}; other variables are defined as in WGP.
The main motivation of this paper is to investigate the better solution of MCMTP by using utility function approach and then compare the solutions to other methods such as GP and RMCGP.
The remainder of the paper is organized as follows: In Section “Mathematical model”, mathematical models are formulated for MOTP and MCMTP and solution procedures have been discussed with utility function approach. In the next section “Numerical examples”, we demonstrate the usefulness of the proposed model with realistic examples. Finally, conclusion is presented regarding our consideration.
Mathematical model
The mathematical model of multiobjective transportation problem (MOTP) can be considered as follows:
Model 1
Here ${C}_{\mathit{\text{ij}}}^{t},{a}_{i},{b}_{j}$ are the cost, supply, and demand parameters of t th objective function in MOTP, respectively, and $\sum _{i=1}^{m}{a}_{i}\ge \sum _{j=1}^{n}{b}_{j}$ is the feasibility condition. According to the nature of the problem, the decision maker has the right to choose the goals of the objective functions. Assuming that these goals are g_{ t } (t=1,2,…, K) of K objective functions, the goals are defined as interval valued as g_{ t }= [ g_{t,min}, g_{t,max}], (t=1,2,…, K).
In many reallife situations, the multiple choices in the transportation parameters like cost, demand, and source create complexities for the DM to make a decision. Multichoice costs may occur due to several routes for transporting the goods. Due to weather condition or different seasons, the demands or the supply become multichoices in nature. In the atmosphere of multichoice transportation parameters, the mathematical model of MCMTP is defined as follows:
Model M1
Here $\left({\stackrel{~}{C}}_{\mathit{\text{ij}}}^{t1}\text{or}{\stackrel{~}{C}}_{\mathit{\text{ij}}}^{t2}\text{or}\cdots \text{or}{\stackrel{~}{C}}_{\mathit{\text{ij}}}^{\mathit{\text{tr}}}\right)$, $\left({\xe3}_{i}^{1}\text{or}{\xe3}_{i}^{2}\text{or}\cdots \text{or}{\xe3}_{i}^{p}\right)$, and $\left({\stackrel{~}{b}}_{j}^{1}\text{or}{\stackrel{~}{b}}_{j}^{2}\text{or}\cdots \text{or}{\stackrel{~}{b}}_{j}^{q}\right)$ are the multichoice cost, supply, and demand parameters for the t th objective function, respectively. In a objective function, respectively. In a transportation problem, the total demand should be less or equal to the total capacity of supply to get a feasible solution. In the present case for multichoice of supply and demands, the information of total capacity of supply in the origins and demands in the destinations is not precisely calculated. So we have selected here the maximum possible supply in the origins and consequently the minimum demand in the destinations and then formulated the feasibility condition as $\sum _{i=1}^{m}max\left\{\underset{i}{\overset{1}{\xe3}},{\xe3}_{i}^{2},\dots ,{\xe3}_{i}^{p}\right\}\ge \sum _{j=1}^{n}min\left\{\underset{j}{\overset{1}{\stackrel{~}{b}}},\underset{j}{\overset{2}{\stackrel{~}{b}}},\dots ,\underset{j}{\overset{q}{\stackrel{~}{b}}}\right\}$. This feasibility condition is the best possible wide range of feasible region of the MCMTP. However, the feasibility condition can be remodeled as per as decision maker’s choice.
Transformation technique for multichoice parameters like cost, supply, and demand to the equivalent form
When there are multiple choice of parameters such as cost, supply, and demand, we should select a single choice satisfying supply and demand restrictions. The selection of choices should be done in such a way that the whole problem is optimized. Introduction of binary variables is an important concept to select a choice in the problem.
If we have to choose one among t number of possibilities, then we use p number of binary variables where 2^{p−1}<t≤2^{p}. Let t=^{p}C_{0}+^{p}C_{1}+^{p}C_{2}+⋯+^{p}C_{ d }+k, for some d satisfying 1≤d≤p,0≤k<^{p}C_{d+1}. Let us take p binary variables ${z}_{j}^{1},{z}_{j}^{2},\dots ,{z}_{j}^{p}$ to deduce a formula which will select one among the t values ${c}_{j}^{1},{c}_{j}^{2},\dots ,{c}_{j}^{t}$. Let us form a function with p binary variables, ${f}_{0}\left(z\right)=\left({z}_{j}^{1}{z}_{j}^{2}\dots {z}_{j}^{p}\right){c}_{j}^{1}$ where $z=\left({z}_{j}^{1},{z}_{j}^{2},\dots ,{z}_{j}^{p}\right)$. When each ${z}_{j}^{i}=1$ for $i=1,2,\dots p,\phantom{\rule{1em}{0ex}}{f}_{0}\left(z\right)={c}_{j}^{1}$. Thus, ${f}_{0}\left(z\right)={c}_{j}^{1}$, when ${z}_{j}^{1}+{z}_{j}^{2}+\cdots +{z}_{j}^{p}=p$. Again, let us assume a function
When ${z}_{j}^{1}+{z}_{j}^{2}+\cdots +{z}_{j}^{p}=p1$, f_{1}(z) gives output one of the following ${c}_{j}^{t}$s: ${c}_{j}^{2},{c}_{j}^{3},\dots ,{c}_{j}^{1{+}^{p}{C}_{1}}$. Similarly, we consider
⋮
When ${z}_{j}^{1}+{z}_{j}^{2}+\cdots +{z}_{j}^{p}=p2$, the above function f_{2}(z) gives one among the following ${c}_{j}^{t}$s: ${c}_{j}^{1{+}^{p}{C}_{1}+1},{c}_{j}^{1{+}^{p}{C}_{1}+2},\dots ,{c}_{j}^{1{+}^{p}{C}_{1}{+}^{p}{C}_{2}}$.
Proceeding in the same manner, we find
⋮
When ${z}_{j}^{1}+{z}_{j}^{2}+\cdots +{z}_{j}^{p}=pd$, the above function gives one among the following${c}_{j}^{t}$s: ${c}_{j}^{1{+}^{p}{C}_{1}{+}^{p}{C}_{2}+{\cdots}^{p}{C}_{d1}+1},{c}_{j}^{1{+}^{p}{C}_{1}{+}^{p}{C}_{2}+{\cdots}^{p}{C}_{d1}+2},\dots ,{c}_{j}^{1{+}^{p}{C}_{1}{+}^{p}{C}_{2}+{\cdots}^{p}{C}_{d1}{+}^{p}{C}_{d}}$.
When k=0, the function f(z)=f_{0}(z)+f_{1}(z)+⋯+f_{ d }(z) gives one of the value ${c}_{j}^{t}$ for all z satisfying$pd\le {z}_{j}^{1}+{z}_{j}^{2}+\cdots +{z}_{j}^{p}\le \mathrm{p.}$
If k≠0, then k<^{p}C_{d+1} and we formulate the function
When ${z}_{j}^{1}+{z}_{j}^{2}+\cdots +{z}_{j}^{p}=p\left(d+1\right)$, f_{d+1}(z) can give one output among ^{p}C_{d+1} number of choices. Here we have used ^{p}C_{d+1}−k restrictions to restrict its possible outputs in k numbers. Let the k th term occur at ${i}_{1}={i}_{1}^{\prime},{i}_{2}={i}_{2}^{\prime},\dots ,{i}_{d+1}={i}_{d+1}^{\prime}$, then the restrictions are
${z}_{j}^{{i}_{1}}+{z}_{j}^{{i}_{2}}+\cdots +{z}_{j}^{{i}_{d+1}}\ge 1$, for all ${i}_{1}={i}_{1}^{\prime},{i}_{2}={i}_{2}^{\prime},\dots {i}_{d}={i}_{d}^{\prime},{i}_{p}\ge {i}_{d+1}>{i}_{d+1}^{\prime}$;
${z}_{j}^{{i}_{1}}+{z}_{j}^{{i}_{2}}+\cdots +{z}_{j}^{{i}_{d+1}}\ge 1$, for all ${i}_{1}={i}_{1}^{\prime},{i}_{2}={i}_{2}^{\prime},\dots ,{i}_{d1}={i}_{d1}^{\prime},{i}_{p1}\ge {i}_{d}>{i}_{d}^{\prime}$;
⋮
${z}_{j}^{{i}_{1}}+{z}_{j}^{{i}_{2}}+\cdots +{z}_{j}^{{i}_{d+1}}\ge 1$, for all ${i}_{pd1}\ge {i}_{1}>{i}_{1}^{\prime}$.
Thus, f(z)=f_{0}(z)+f_{1}(z)+⋯+f_{ d }(z)+f_{d+1}(z) gives the generalized selection function of the multichoice ${c}_{j}^{t}$s.
Without loss of any generality in treating the value of ${c}_{j}^{t}=1$ and using the product and summation notation, we have formulated the following formulae to select the crisp value of multichoice parameters:
where $pd\le {z}_{j}^{{i}_{1}}+{z}_{j}^{{i}_{2}}+\cdots +{z}_{j}^{{i}_{p}}\le p$ for all i_{1}<i_{2}<⋯<i_{ p }.
When k≠0, we add first k terms with the above function from the following formula:
Assuming that i_{1}<i_{2}<⋯·<i_{ p } and let k th term occurred at ${i}_{1}^{\prime},{i}_{2}^{\prime},\dots ,{i}_{d+1}^{\prime}$, then the restrictions are
${z}_{j}^{{i}_{1}}+{z}_{j}^{{i}_{2}}+\cdots +{z}_{j}^{{i}_{d+1}}\ge 1$, for all ${i}_{1}={i}_{1}^{\prime},{i}_{2}={i}_{2}^{\prime},\dots {i}_{d}={i}_{d}^{\prime},{i}_{p}\ge {i}_{d+1}>{i}_{d+1}^{\prime}$;
${z}_{j}^{{i}_{1}}+{z}_{j}^{{i}_{2}}+\cdots +{z}_{j}^{{i}_{d+1}}\ge 1$, for all ${i}_{1}={i}_{1}^{\prime},{i}_{2}={i}_{2}^{\prime},\dots ,{i}_{d1}={i}_{d1}^{\prime},{i}_{p1}\ge {i}_{d}>{i}_{d}^{\prime}$;
⋮
${z}_{j}^{{i}_{1}}+{z}_{j}^{{i}_{2}}+\cdots +{z}_{j}^{{i}_{d+1}}\ge 1$, for all ${i}_{pd1}\ge {i}_{1}>{i}_{1}^{\prime}$.
where (term)^{g} (for g=1,2,…, T) are the T number of terms in the functions of the binary variables mentioned in above. Similarly,
where (term)^{g} (for g=1,2,…, P) is the P number of terms in the functions of the binary variables mentioned above to reduce the P number of choices ${a}_{i}^{g}$ to single choice ${a}_{i}^{\prime}$, and (term)^{g} (for g=1,2,…, Q) is the Q number of terms in functions of binary variables mentioned above to reduce the Q number of choices ${b}_{j}^{g}$ to single choice ${b}_{j}^{\prime}$.
Reduction of MCMTP to MOTP
The MCMTP as given in the Model M1 transformed to a MOTP by transforming the multichoice parameters in the objective functions (5) and the multichoice supplies and demands in constraints (6) and (7) to singlevalued ones, using the technique described in subsection “Transformation technique for multichoice parameters like cost, supply, and demand to the equivalent form”. Thus, the equivalent MOTP of Model M1 is given in the following model:
Model M2
Here ${C}_{\mathit{\text{ij}}}^{\mathrm{\prime t}},{a}_{i}^{\prime},{b}_{j}^{\prime}$ are the reduced cost, supply, and demand parameters of t th objective function in MOTP, respectively, and $\sum _{i=1}^{m}max\left\{\underset{i}{\overset{1}{\xe3}},{\xe3}_{i}^{2},\dots ,{\xe3}_{i}^{p}\right\}\ge \sum _{j=1}^{n}$$min\left\{\underset{j}{\overset{1}{\stackrel{~}{b}}},{\stackrel{~}{b}}_{j}^{2},\dots ,{\stackrel{~}{b}}_{j}^{q}\right\}$ is the feasibility condition.
The transportation problem Model M2 is same as the problem described in Model 1. We can solve the Model M2 as the procedure described using the different techniques to solve Model 1.
Solution procedure
The approaches such as goal programming and revised multichoice goal programming are used to solve the MOTP, which are defined as follows: A. Goal programming approach
Let us briefly discuss the goal programming approach for solving MOTP (see Model 1A). If ${d}_{t}^{+}$ and ${d}_{t}^{}$ be positive and negative deviations corresponding to the t th goal of the objective function, then the mathematical model is defined as follows:
Model 1A
B. Revised multichoice goal programming approach
In the similar way, the RMCGP is introduced to solve the MOTP. Let us assume that the multiple goals are considered to the objective functions and this can be achieved by considering the following model (see Model 1B) as
Model 1B
where t th aspiration level is defined as y_{ t } which is the continuous variable that lies between the upper (g_{t,max}) and lower (g_{t,min}) bounds. Again, ${e}_{t}^{+}$ and ${e}_{t}^{}$ are positive and negative deviations attached to t th goal of y_{ t }−g_{t,max}, and α_{ t } is the weight attached to the sum of the deviations of y_{ t }−g_{t,max}.
Utility function approach to solve MOTP
Here, the concept of utility function has been addressed to solve MOTP. A short introduction is presented here and then we discuss the methodology for solving MOTP using utility function.
Utility function
In this paper, introduction of utility is taken to be correlative to ‘Desire’ or ‘Want’. It has been already argued that desire cannot be measured directly, but only indirectly, by the outward phenomena in which the context is presented.
Definition 1. The utility function describes a function U:X→ℜ which assigns a real number to every outcome in such a way that it captures DM’s preferences over the desired goals of the objectives, where X is the set of feasible points and ℜ is the set of real numbers.
The purpose of this study is to derive the achievement function of MOTP under the light of utility function for the DM according to the priority of goals. In our proposed approach, the DM wants to maximize his/her expected utility. For the sake of simplicity, two popular utility functions (linear and Sshaped) are considered as follows.
Linear utility function u_{ i }(y_{ i }) for decisionmaking (management) problems can be found in Lai and Hwang [14] and Sshaped utility function (for the same purpose) has been proposed by Chang [15]. The utility function is generally considered in three cases as follows: Case 1: Left linear utility function (LLUF)
Case 2: Right linear utility function (RLUF)
Case 3: Sshaped utility function
where g_{i,min} and g_{i,max} are lower and upper bounds corresponding to the i th goal respectively. The graphs of above utility functions are drawn in the following figures (see Figures 1,2 and 3).
Model formulation for case 1
The DM would like to increase the utility value u_{ t }(y_{ t }) as much as possible in the case of LLUF (Figure 1). In order to achieve this goal, the value of y_{ t } should be as close to the target value g_{t,min} as possible. The MOTP from Model 1A can be reformulated using the proposed LLUF as follows:
Model 2A
where β_{ t } is the weight attached to deviation ${f}_{t}^{}$. The role of weight β_{ t } can be seen as the preferential component for the utility value u_{ t }.
Proposition 1. Achievement of optimal utility in the LLUF (Figure 1) is equivalent to the optimal solution of Model 2A.
Proof. When u_{ t } approaches to the highest value 1, then the deviation ${f}_{t}^{}\to 0$ of the utility function (from Equation 29), because ${f}_{t}^{}$ should be minimized in the objective function to obtain the optimal solution of Model 2A. This represents y_{ t } approach to g_{t,min} (from Equation 28), and Z_{ t }(X) is also closer to g_{t,min} (from Equation 26) because ${d}_{t}^{+}$ and ${d}_{t}^{}$ should also be minimized in the objective function. It is obvious that the behavior of Model 2A and the level of utility are achieved. This completes the proof.
Model formulation for case 2
The DM would like to increase the utility value u_{ t }(y_{ t }) as much as possible in the case of RLUF (Figure 2). In order to achieve this goal, the value of y_{ t } should be as close to the target value g_{t,max} as possible. The MOTP from Model 1A can be reformulated using the proposed RLUF as follows:
where β_{ t } is the weight attached to the deviation ${f}_{t}^{}$. The role of weight β_{ t } can be seen as a preferential component for the utility value u_{ t }.
Proposition 2. Achievement of optimal utility in the RLUF (Figure 2) is equivalent to the optimal solution of Model 2B.
Proof. Similar way can be followed as we have done in Proposition 1.
The advantages of the use of LLUF and RLUF in the decisionmaking problems are as follows:

(1)
The DM can easily formulate their MOTP by taking into account their preference mappings with utility functions in real situation.

(2)
The two linear utility models represented as linear form which can be easily solved using software.
Due to variation of deviation variables ${d}_{t}^{+}$, ${d}_{t}^{}$, and f^{−} in different ranges, biasness may occur towards the objective functions with larger magnitude. Normalization technique may help to remove this biasness. Several normalization approaches such as percentage, Euclidean, summation, and zeroone notarizations (Tamiz et al. [12]; Kettani et al. [16]) are available to execute this. According to the normalization technique proposed by Tamiz et al. [12], Model 2A can be redesigned as follows:
where ϕ_{ t } is the normalization constant for t th goal.
In order to solve this problem, utility normalization concept is introduced as follows: Let ${d}_{t}^{+},{d}_{t}^{}\in \phantom{\rule{2.77626pt}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,{\u016b}_{t}]$ and ${f}_{t}^{}\in \phantom{\rule{2.77626pt}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,1]$ where ${\u016b}_{t}$ is the upper bound of ${d}_{t}^{+}$ and ${d}_{t}^{}$. The normalized weights w_{ t } and β_{ t } can be easily obtained as ${w}_{t}=\frac{1}{1+{\u016b}_{t}}$ and ${\beta}_{t}=\frac{{\u016b}_{t}}{1+{\u016b}_{t}}$. This technique of normalization ensures that deviation variables ${d}_{t}^{+}$, ${d}_{t}^{}$, and ${f}_{t}^{}$ approximated the same magnitude. Similarly, the same methodology can be applied to the Model 2B.
The utility value for Sshaped utility function can be expressed as a sum of linear utility functions (RLUF or LLUF) by introducing binary variables [17]. But Chang [15] proposed in his paper that the utility value for Sshaped utility function can be considered without using the binary variables and this is shown in the following model (see Model 2C):
Model 2C
MCMTP which occurred in many reallife situations can be reduced to MOTP and then the problem can be reduced to the models such as 2A, 2B, and 2C, with interval goals under the consideration of utility functions related to these goals. Solving the formulated problem, the DM obtained the satisfactory solution.
Numerical examples
Here we have presented two numerical examples; the first one explores the applicability of MOTP and the second one represents the applicability of MCMTP.
Case 1
Let us consider the following MOTP (see Model 3) with three objectives:
Model 3
Goal 1: Z^{1}=7x_{11}+8x_{12}+7.5x_{13}+8x_{21}+7.2x_{22}+8.4x_{23}+9x_{31}+8x_{32}+7.7x_{33} with goal as [170, 220], more is better, follows RLUF.
Goal 2: Z^{2}=50x_{11}+65x_{12}+62x_{13}+60x_{21}+55x_{22}+58x_{23}+65x_{31}+60x_{32}+58x_{33} with goal as [1,550; 1,800], less is better, follows LLUF.
Goal 3: Z^{3}=10x_{11}+8x_{12}+9x_{13}+8.5x_{21}+9.5x_{22}+8.5x_{23}+9.5x_{31}+8.8x_{32}+9x_{33} with goal as [200, 290], more is better, follows Sshaped utility function as given in Figure 4.
To achieve the goals in the proposed problem (see Model 3), we may formulate the following models.
In the proposed problem, the deviations of goals 1, 2, 3 are 50, 250, 90, respectively. By considering the weights ${w}_{1}=\frac{1}{50},{w}_{2}=\frac{1}{250},{w}_{3}=\frac{1}{90}$ for the Model 1A, the above Model 3 reduces to the following model (see Model 3A) as
Model 3A
Again, considering the same weights w_{ t } as used in Model 3A for all t=1,2,3 and setting α_{ t }=w_{ t } for t=1,2,3 for deviation of goals and using Model 1B, Model 3 reduces to the following model (see Model 3B) as
Model 3B
Using the concept of utility function described in section “Transformation technique for multichoice parameters like cost, supply, and demand to the equivalent form”, Model 3 can be reformulated as follows.
The consideration of utility function depends on the DM. Here, we assume that goals 1, 2, and 3 follow the utility functions LLUF (Figure 1), RLUF (Figure 2), and Sshaped utility function as given in Figure 4, respectively. In the given example, the upper bound of variations ${d}_{1}^{+},{d}_{1}^{},{d}_{2}^{+},{d}_{2}^{},{d}_{3}^{+},{d}_{3}^{}$ are 50, 50, 250, 250, 90, 90, respectively, and the upper bounds of ${f}_{1}^{},{f}_{2}^{},{f}_{3}^{}$ are 1. We find the weights as described in Section “Mathematical model” as follows: ${w}_{1}=\frac{1}{50}$, ${w}_{2}=\frac{1}{250}$, ${w}_{3}=\frac{1}{90},{\beta}_{1}=\frac{50}{51}$, ${\beta}_{2}=\frac{250}{251}$, ${\beta}_{3}=\frac{90}{91}$.
With these supplied data, Model 3 can be reformulated as follows (see Model 3C):
Model 3C
Results and discussion for problem given in case 1
Using LINGO software, we solved Models 3A, 3B, and 3C and reported the solution as follows: The optimal solution of Model 3A is reported as
x_{11}=0, x_{12}=9, x_{13}=1, x_{21}=0, x_{22}=0, x_{23}=9, x_{31}=6, x_{32}=0, x_{33}=0; Z^{1}=209.1, Z^{2}=1559, Z^{3}=214.5.
The optimal solution of Model 3B is as follows:
x_{11}=10, x_{12}=0, x_{13}=0, x_{21}=0, x_{22}=9, x_{23}=0, x_{31}=0, x_{32}=0, x_{33}=11; Z^{1}=219.50, Z^{2}=1633, Z^{3}=284.5.
The optimal solution of Model 3C is also as follows:
x_{11}=10, x_{12}=0, x_{13}=0, x_{21}=0, x_{22}=5, x_{23}=4, x_{31}=0, x_{32}=3, x_{33}=7; Z^{1}=217.5, Z^{2}=1593, Z^{3}=270.9.
Here, the solution obtained in Model 3B is better compared with the solution of Model 3A, but the DM is not satisfied because in the proposed problem, satisfying the goal is not only the important notion but is also a utility factor to the DM which is important for the decisionmaking (management) problem. When the utility value is more important rather than the benefit, then the solutions obtained in Model 3A or in Model 3B are not satisfactory to the DM to make an appropriate decision. The marketing survey indicates that the higher utility value of goal 3 will increase the number of customers to the network service provider company. The solution obtained in Model 3C demonstrated the high utility value of goal 3, whenever the other two models failed to give satisfactory results (Table 1). In this context, we may suggest that the utility function approach provided better result compared with other results obtained in classical techniques like GP and RMCGP.
Case 2
Let us consider the following MCMTP (see Model 3) with two objectives:
Model 4 Goal 1: z_{1}={5 or 7}x_{11}+8x_{12}+{7 or 6 or 10}x_{13}+{6 or 8}x_{21}+8x_{22}+10x_{23} with goal as [150,200], more is better, but follows Sshape utility function (Figure 5). Goal 2: z_{2}=15x_{11}+{18 or 16}x_{12}+17x_{13}+16x_{21}+{18 or 20}x_{22}+20x_{23} with goal as [400,500], less is better, follows LLUF:
Model 4 is equivalent to the following model (see Model 5).
Model 5
Goal 1: ${Z}_{1}=\left\{5{z}_{11}^{11}+7\left(1{z}_{11}^{11}\right)\right\}{x}_{11}+8{x}_{12}+\left\{7{z}_{11}^{12}{z}_{11}^{13}+6{z}_{11}^{12}\left(1{z}_{11}^{13}\right)\right.$
with goal as [150,200], more is better, but follows Sshape utility function (Figure 5).
Goal 2: ${Z}_{2}=15{x}_{11}+\left\{18{z}_{12}^{11}+16\left(1{z}_{12}^{11}\right)\right\}{x}_{12}+17{x}_{13}+16{x}_{21}$
with goal as [400,500], less is better, follows LLUF:
In the given problem in Model 5, the deviations of goal 1, goal 2 are 50, 100 respectively. By considering the weights ${w}_{1}=\frac{1}{50},{w}_{2}=\frac{1}{100}$ for the Model 1A, Model 5 reduces to Model 5A as
Model 5A
Again, considering the same weights w_{ t } as used in Model 3A for all t=1,2 and the weights α_{ t }=w_{ t } for t=1,2 for deviation of goals and using the Model 1B, Model 5 reduces to Model 5B as
Model Ex. 5B
Let us solve the proposed problem (see Model 3) using the concept of utility function. The consideration of utility function depends on the DM. Here we assume that goal 1 and goal 2 follow the Sshaped utility function given in Figure 5 and the utility functions LLUF (Figure 1), respectively. In the given example, the upper bound of variations ${d}_{1}^{+},{d}_{1}^{};{d}_{2}^{+},{d}_{2}^{}$ are 50, 100 respectively, and the upper bounds of ${f}_{1}^{},{f}_{2}^{}$ are 1. We find the weights as suggested in Section “Mathematical model” as follows: ${w}_{1}=\frac{1}{50}$, ${\beta}_{1}=\frac{50}{51}$, ${w}_{2}=\frac{1}{100}$, ${\beta}_{2}=\frac{100}{101}$.
With these supplied data, Model 5 can be formulated as follows:
Model 5C
Results and discussion for problem given in case 2
Solving the model presented in Model 5A, the optimal solution of the Model 5A is reported as x_{11}=7, x_{12}=5, x_{13}=0, x_{21}=0, x_{22}=1, x_{23}=10, and the values of the objective functions are Z^{1}=197, Z^{2}=405.
The selection of the choices corresponding to the optimal solution is as follows:
a_{1}=16, a_{2}=14, b_{1}=7, b_{2}=6, b_{3}=9
Solving the model presented in Model 5B, we have listed the following solution:
x_{11}=7, x_{12}=6, x_{13}=0, x_{21}=0, x_{22}=0, x_{23}=10, and the values of the objective functions are Z^{1}=197, Z^{2}=401.
The selection of the choices corresponding to the optimal solution is as follows:
a_{1}=16, a_{2}=14, b_{1}=7, b_{2}=8, b_{3}=9
Solving the model presented in Model 5C, we obtained the solution listed below:
x_{11}=6, x_{12}=3, x_{13}=0, x_{21}=0, x_{22}=3, x_{23}=10, and the values of the objective functions are Z^{1}=190, Z^{2}=404.
The selection of the choices corresponding the optimal solution is as follows:
a_{1}=12, a_{2}=14, b_{1}=7, b_{2}=6, b_{3}=9
Table 2 helps us to conclude that the solution of the MCMTP obtained in Model 5B is better compared with the solution of Model 5A, but the DM is not satisfied because in the proposed problem satisfying the goal is not only the important notion, but is also a utility factor to the DM which is important for the decisionmaking (management) problem. When the utility value is more important rather than the benefit, then the solutions obtained in Model 5A or in Model 5B are not satisfactory to the DM to make the appropriate decision. The solution obtained in the Model 5C demonstrated the high utility value of goal 2, whenever the other two models failed to give satisfactory results. In this context, we may suggest that the utility function approach provided better result compared with other results.
Conclusions
In this paper, we have considered MCMTP where the cost, demand, and supply coefficients are multichoice type. Another important notion of this study is to give an impression of goal preferences of the DM. The approach of utility function is the most useful skill for representing the DM’s preferences. The concept of utility in this paper proposes a new approach for extending the utilization of reallife MCMTP. The MCMTP gives a new direction to handle the reallife transportation problems when the transportation parameters are multichoice in nature. The numerical examples presented in this paper explored the applicability and suitability for solving MOTP and MCMTP and also for representing the DM’s preferences. In addition, the proposed method can be used as a decisionmaking aid for multichoice multiobjective decisionmaking problem that occurred in the reallife purposes, like economical, agricultural, industrial management, and military. In particular, in case of incomplete information, the DM can use the proposed method to set the goals according to their own utility functions, while the proposed method can easily find the better solution than the previous methods (GP, RMCGP) used to solve MCMTP which is shown by the proposed examples in this paper.
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Keywords
 Transportation problem
 Multichoice programming
 Multiobjective decision making
 Goal programming
 Multichoice programming
 Utility function