A goal geometric programming problem (G2P2) with logarithmic deviational variables and its applications on two industrial problems

A very useful multi-objective technique is goal programming. There are many methodologies of goal programming such as weighted goal programming, min-max goal programming, and lexicographic goal programming. In this paper, weighted goal programming is reformulated as goal programming with logarithmic deviation variables. Here, a comparison of the proposed method and goal programming with weighted sum method is presented. A numerical example and applications on two industrial problems have also enriched this paper.


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The concept of taking multiplicative deviational variables as an objective function is not new. Previously, Verma (1990) and a paper entitled 'Goal geometric programming problem (G 2 P 2 ) with product method' by Ghosh and Roy (2012) used this concept. In this paper, we have started with additive deviational variables as the objective function which were then converted into multiplicative deviational variables as objective function using the logarithmic concept. The method of conversion is given in the form of 'Result 1'.
The arrangement of the paper is as follows: the background of the study followed by the goal programming model are presented. A result is presented together with its proof (Result 1), and the model of weighted goal programming with logarithmic deviational variables is then presented. The sections for goal geometric programming model with logarithmic deviational variables and its solution procedure are followed by a theorem on the model of weighted goal programming with logarithmic deviational variables and its proof (Result 2). Next, a numerical example and applications on lightly loaded bearing problem, optimal production, and marketing planning are presented. Finally, the conclusions of the study is presented.
C j0i and C ri are positive real numbers ∀ j, r, i, and a k0i , a ki are real numbers ∀ k, i. P j0 = Number of terms present in j0th objective function, P r = Number of terms present in rth constraint, C r = Boundary value of rth constraint, The multi-objective programming model contains m, the number of minimizing objective functions; q, the number of inequality type constraints; and n, the number of strictly positive decision variables.
Result 1. As mentioned, the goal programming model may be reduced to the following form:

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Proof. In the multi-objective programming model (1), objective functions are minimized and have target values, e.g., minimize f j0 (X) with target value C j0 , i.e., minimize log(f j0 (X)) with target value log(C j0 ).
According to the method of goal formulation, positive deviation should be minimized. Similarly, in model (1), constraints are of ≤ type. Thus, positive deviations should also be minimized. Therefore, when The goal formulation is as follows: , we can turn model (2) into the following problem: , which is obviously equivalent to the following goal programming form with logarithmic deviational variables:

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x k > 0, k = 1, 2, ..., n; The goal programming formulation where the constraints are in inequality form the following: 2, ..., n; u + j0 , v + r > 1, hence the result.

Results and discussion
Weighted goal programming with logarithmic deviational variables According to model (1), all of the objective functions are minimized. If the decision maker wants to get a much more minimized value for any particular objective function or wants to satisfy strictly the constraints, then weight factors (priorities) are introduced. In goal programming formulation with logarithmic deviational variables, weights (priorities) are given with the deviational variable. Hence, the weighted goal programming formulation becomes the following: f r (X)/v + r ≤ C r , r = 1, 2, ..., q, x k > 0, k = 1, 2, ..., n; u + j0 , v + r > 1.
Here, W j0 values are the weights for objective functions and W r values are the weights for the constraints.
Solutions of goal programming (Romero 1991), even those of weighted goal programming and lexicographic goal programming (Miettinen 1999), are pareto optimal. Here, we prove a result which also shows that goal programming with logarithmic deviation gives pareto optimal solutions.
Result 2. The following is the solution of weighted goal programming with logarithmic deviation: .., k, which comes from the following goal programming model: is pareto optimal if u + i for each function f i (X) to be minimized has a value greater than 1 at the optimum.
Proof. If x * ∈ S with a positive deviation vector, then let (u + i ) * (> 1) be the solution of the following weighted goal programming problem: .., k. If possible, let x * be not pareto optimal, then there exists a vector x 0 with a positive deviational variable ≤ C i as x * be the solution of (7), i.e.,
Goal geometric programming model with logarithmic deviational variables and its solution procedure Linear goal programming is a very commonly used tool of the MCDM problem. However, nonlinear goal programming is very rare in this context. In many engineering problems, as well as problems of science, there are nonlinear equations to optimize. To solve that type of nonlinear goal programming problem, the geometric programming method can be used. Hence, we can turn model (6) into a goal geometric programming form as in the following: The corresponding dual geometric programming of model (8) can be written as follows: δ ri , r = 1, 2, ..., q.

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Numerical example: a multi-objective goal programming problem Minimize f 2 (x 1 , x 2 ) = 2x −2 1 x −3 2 with target value 50, subject to x 1 + x 2 ≤ 1, x 1 , x 2 > 0.
In goal geometric programming model with logarithmic deviational variables, model (8) can be written as follows: Minimize
From primal dual relation Solving from primal dual relation for different values of weights, we get the optimal values of the decision variables which are given in Table 1. From the table, we see that each deviation (u i , v i ) has values greater than 1 when minimized. Thus, according to our theorem, the solutions are pareto optimal.

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Again, we have solved the mentioned example in goal geometric programming with weighted sum method. Here, we have compared the results of the mentioned example in equal weights solved in two different methods: goal geometric programming with weighted sum method and goal geometric programming with logarithmic deviational variables which are given in Table 2.
From the comparison, it is clear that in both methods, the optimum values of the first and second objectives are almost the same. We have solved the same example in both processes where we have used geometric programming to solve a nonlinear goal programming problem. The advantage of the pro- Table 2 Comparison of optimal solutions in two different methods Method W 1 W 2 First objective (f * 1 ) Second objective (f * 2 ) Goal geometric programming 0.5 0.5 6.919487 57.88240 with weighted sum Goal geometric programming 0.5 0.5 6.826667 58.25422 with logarithmic deviational variables posed method lies in the method of solution, i.e., geometric programming where degree of difficulty is less than the degree of difficulty of the previous process (goal geometric programming with weighted sum method). For this reason, the solution procedure of this process becomes easier than that of the previous.

Application on lightly loaded bearing problem
A lightly loaded bearing is to be designed to minimize the linear combination of frictional moment and angle of twist of the shaft and the temperature rise of the oil while carrying a load of 1,000 lb, and the angular velocity of the shaft is to be greater than 100 rad s −1 . Assume that 1 in-lb of frictional moment in the bearing is equal to 0.0025 rad of the angle of twist. The following are the goals: Priority 1: Linear combination of frictional moment, angel of twist of the shaft, and temperature rise of the oil should be minimized and near 10.
Priority 2: Angular velocity of the shaft per 100 rad s −1 should be minimized and near 0.2.
In formulating the mentioned goal programming problem and finding the dimension of the bearing that is to be built for this purpose, it should be done in such a way that it can carry the maximum load.
Solution Let R (in.) be the radius of the journal and L (in.) be the half length of the bearing, T be the temperature rise of the oil, and frictional moment of the bearing (M ) = 8πµωR 2 L √ 1−n 2 c where ω is the angular velocity of the shaft, µ is the viscosity of the oil (lubricant), n is the eccentricity ratio, and c is the radial clearance.
The angle of twist of the shaft (ϕ) = Sel GR , where S e is the shear stress, l is the length between the driving point and rotating mass, and G is the shear modulus. The temperature rise of the oil in the bearing is given by T = 0.045µωR 2 c 2 n √ 1−n 2 . For the given data, c R = 0.0015, n = 0.9, µ = 10 −6 lb s in. −2 , l = 10 in., S e = 30, 000 psi, and G = 12 × 10 6 psi. Hence, linear combination of frictional moment, angle of twist of the shaft, and temperature rise of the oil equals 0.038ωR 2 L + 0.025R −1 + 0.592RL −3 with target value 10 (11.1) and angular velocity ω ≥ 100 rad s −1 . (11.2) From the given data in the chart of 'Dimensionless performance parameters for full journal bearing' ωR −1 L 3 = 11.6, i.e., ω = 11.6R/L 3 .
As per the assumption that 1 in. lb of frictional moment in bearing is equal to 0.0025 rad angle of twist, Equation 11.1 becomes Z 1 = 0.44R 3 L −2 + 10R −1 + 0.592RL −3 with the target value of 10.

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Equation 11.2 becomes Z 2 = 8.62R −1 L 3 with the target value of 0.2. Hence, the model of lightly loaded bearing problem in G 2 P 2 with logarithmic deviational variable is as follows: In solving with the use of geometric programming method where the degree of difficulty is 5−(4+1) = 0, we get the optimal values of the radius of the journal (R) and half length of the bearing (L) which are given in Table 3. From the table, we have seen that each deviation (u, v) has values greater than 1. Thus, the solution is pareto optimal. Application on optimal production and marketing planning Consider a manufacturer who produces a single product where the demand is affected by the selling price. Let P be the selling price per unit, α be the price elasticity to the demand, M be the marketing expenditure per unit, and γ be the marketing expenditure elasticity to the demand (Sadjadi et al. 2005). Assume that demand D = KP −α M γ , where K is the predetermined constant and production cost C, which is inversely related to production lot size (units) Q, i.e., C = rQ −β , where r is the predefined constant for unit production cost and β is the lot size elasticity of production unit cost. Again, let µ and a be the production rate and the setup cost of production, respectively. We assume the production rate µ to vary with the demand D proportionally. Hence, µ = uD where u > 1. There are some restrictions on variables such as α, γ, and β. The equation α > 1 indicates that D increases at a diminishing rate as P decreases. The equation 0 < β < 1 is almost the same as α and 0 < γ < 1.
We want to minimize the equation (Marketing cost + Production cost + Setup cost + Holding cost), which is subject to some constraint that total revenue should be bigger. These are the following goals: Priority 1: Total revenue should be greater than 0. Let u = 1 − 1 u , and from assumptions and consideration, the above model becomes the following: Transforming the model (12.2) into G 2 P 2 with logarithmic deviation variables, we get the following: Solving with the use of geometric programming method where the degree of difficulty is 6 − (5 + 1) = 0, we get the optimal values of decision variables, e.g., price per unit (P ), production lot size (Q), and marketing expenditure per unit (M ), which are given in Table 4.