Ratio Mathematica Vol. 35, 2018, pp. 29-45 ISSN: 1592-7415 eISSN: 2282-8214 Case Studies on the Application of Fuzzy Linear Programming in Decision-Making Miracle K. Eze∗and Babatunde O. Onasanya† Received: 10-09-2018. Accepted: 11-12-2018. Published: 31-12-2018. doi:10.23755/rm.v35i0.424 c©Miracle K. Eze et al. Abstract This study demonstrated the effectiveness of fuzzy method in decision-making and recommends the integration of fuzzy methods in decision-making in production, transportation, power production and distribution and utility main- tenance in Nigeria companies. Keywords: Fuzzy set, Fuzzy linear programming, Fuzzy constraints, Fuzzy optimization. 1 Introduction Linear Programming (LP), an important tool in operations research, has developed over the years in solving management problems [13]. It is in two forms: classical and fuzzy linear programming. It takes various linear inequalities relating to the situation being considered and finds the best value obtainable under that situation. ∗Centre for Petroleum Economics, Energy and Law (University of Ibadan, Ibadan, Nigeria), ezemiraclekasie@gmail.com. †Department of Mathematics, Faculty of Science (University of Ibadan, Ibadan, Nigeria), babtu2001@yahoo.com. 29 M.K. Eze and B.O. Onasanya Let A′is be the constraint functions and b ′ is the available resources. Generally, a linear programming problem can be written as Min(Max) z = cx (1) subject to Ai(x) ≤ bi, where x ≥ 0. In practice, all of the needed information such as c, A′is, bis are not completely available or determined; these parameters are uncertain and are said to be fuzzy variables [10]. A typical mathematical programming problem is to optimise an objective func- tion subject to some constraints. Usually, the classes of objects encountered in the real world do not have clearly defined criteria of membership. Hence, constraints and objective functions could be fuzzy [25]. In production processes, hardly does the firm utilize the exact resources avail- able to meet a proposed target. This may be due to waste in the cause of produc- tion and/or machine wear and tear over time or some other factors due to exigency. Thus, a firm is required to optimally plan around its available resources. Having recognized the shortcomings of traditional mathematical models in some areas of real life application, Zadeh (1965) proposed the notion of a fuzzy set. It began as an effort to use mathematics to define such concepts as “slightly” or “tall” or “fast” or “beautiful” or any other concept that has ambiguous bound- aries. The fuzzy set theory was developed to improve the over simplified model, thereby developing a more robust and flexible model in order to solve real-world complex systems involving human aspects. Fuzziness was modeled by membership functions which might be described as an extension of the usual characteristic function in the setting of mathemati- cal sets [16]. Fuzzification offers superior expressive power, greater generality and an improved capability to model complete problems at a low solution cost. The application of fuzzy set theory is claimed to be effective in decision making and coordinating multiple system requirements [18],[11]. Thus, it is an excellent method for planning and making decision under uncertainty. 2 Preliminaries Definition 2.1. [23] A fuzzy set A in X is a set of ordered pairs A = {(x,µA(x)) : x ∈ X}, where µA(x) is the grade of membership of x ∈ A and µA : X −→ [0,1]. Example 2.1. [25] Let X = {10,20,30,40,50,60,70,80,90,100,110} be pos- sible speeds(mph) at which cars can cruise over long distances. Then the fuzzy set A of “uncomfortable speeds for long distances” may be defined by a certain individual as: A = {(30,0.7),(40,0.75),(50,0.8),(60,0.8),(70,1.0),(80,1.0),(90,1.0)}, 30 Fuzzy linear programming in decision-making where 0.7, 0.75, 0.8 and 1.0 are the degree of uncomfortability, attaining ”cer- tainly uncomfortable” at ≥ 70 mph. Definition 2.2. [14] The support of a fuzzy set A, S(A) = {x ∈ A : µA(x) > 0}. Definition 2.3. [23] A fuzzy set A is empty if and only if µA(x) = 0,∀x ∈ X. Definition 2.4. [23] Two fuzzy sets A and B are equal if and only if µA(x) = µB(x), ∀x ∈ X. Definition 2.5. [23] A fuzzy set A is contained in a fuzzy set B, written as A ⊆ B, if and only if µA(x) ≤ µB(x). Definition 2.6. [23] The intersection of two fuzzy sets A and B is denoted by A∩B and is defined as the largest fuzzy set contained in both A and B. The membership function of A∩B is given by µA(x)∧µB(x) = min{µA(x),µB(x),∀x ∈ X}. Example 2.2. Consider the following set of cars, X = {Mercedez, Camry, Chevrolet, Accord}. Suppose A is the fuzzy subset of “durable cars” and B is the fuzzy subset of “fast cars”. A = {0.8/Me, 0.6/Ac, 0.5/Ca, 0.3/Ch} and B = {0.3/Me, 0.8/Ac, 0.6/Ca, 1.0/Ch}. The intersection of A and B, µA(x)∧µB(x) = {0.3/Me, 0.6/Ac, 0.5/Ca, 0.3/Ch}, is the fuzzy subset of the degree of compatibility of the quality of the cars being “durable and fast”. Definition 2.7. [23] The union of A and B, denoted as A ∪ B, is defined as the smallest fuzzy set containing both A and B. The membership function of A∪B is given by µA(x)∨µB(x) = max{µA(x),µB(x),∀x ∈ X}. Example 2.3. Consider the following set of cars, X = {Mercedez, Camry, Chevrolet, Accord}. Suppose A is the fuzzy subset of “durable cars” and B is the fuzzy subset of “fast cars”. Consider A and B as in Example 2.2. The union of A and B, µA(x)∨µB(x) = {0.8/Me, 0.8/Ac, 0.6/Ca, 1.0/Ch}, is the fuzzy subset of the degree of the quality of either “durable or fast or both”. 31 M.K. Eze and B.O. Onasanya Definition 2.8. [23] If A is a fuzzy subset of X, then an α-level set of A is a non- fuzzy set Aα which comprises all elements of X whose grade of membership is greater than or equal to α. It is denoted by Aα = {x ∈ X : µA(x) ≥ α ∀ x ∈ X}. Example 2.4. The intelligence quotient of students were tested and some were discovered to possess high intelligence quotient while some very low. Let FSIQ be fuzzy set of intelligence quotient. FSIQ = {(C,0.9),(M,0.7),(B,0.5),(S,0.4),(P,0.3)}. Then, A0.5 = (B,M,C). 3 Methodology The data were collected from two places: the production data for two products from a Water Venture and the value-added services provided by an Institute of Economic and Law, both in Oyo State, Nigeria. 3.1 Fuzzy Linear Programming Models In fuzzy linear programming, the fuzziness of available resources is charac- terised by the membership function over the tolerance range. The general model of linear programming with fuzzy resources is: Max(Min)z = cx, (2) subject to (s.t.) Ai(x) ≤ b̃i, i = 1,2, ...,m,x ≥ 0, where, for each i, Ai(x)′s are the m constraints, b̃i ∈ [bi,bi +pi] are the real numbers representing the quantities of each fuzzy resources and p′is are the tolerance levels of the decision-maker for each of the resources. The fuzzy linear programming may also be considered as: Max(Min) z = cx, (3) subject to (s.t.) Ai(x) / bi, i = 1,2, ...,m,x ≥ 0, where / is called “fuzzy less than or equal to”. If the tolerance pi is known for each fuzzy constraint, Ai(x) / bi could be seen as Ai(x) ≤ (bi + θpi), for all i, where θ ∈ [0,1]. 32 Fuzzy linear programming in decision-making 3.2 Verdegay’s Approach- A Nonsymmetric Model Verdegay [21] considered that if the membership functions of the fuzzy con- straints. µi(x) =   1, if Ai(x) < bi 1− Ai(x)−bi pi , bi ≤ Ai(x) ≤ bi + pi, i = 1, ...,m + 1 0, Ai(x) > bi + pi (4) are continuous and monotonic functions, and trade-off between those fuzzy con- straints are allowed, the general model of linear programming with fuzzy re- sources will be equivalent to: Max cx, s.t x ∈ Xα, (5) where Xα = {x : µ(x) ≥ α,x ≥ 0, for each α ∈ [0,1]}. The α-level concept is based on the work of [20]. It is indicated in the membership function that if Ai(x) ≤ bi then the i− th constraint is satisfied and µi(x) = 1. But, on the other hand, if Ai(x) ≥ bi + pi, where pi is the maximum tolerance from bi, (which is always determined by the decision-maker), then the i − th constraint is violated at this point and µi(x) = 0. Finally, if Ai(x) ∈ (bi,bi + pi), then the membership function is monotonically decreasing and, the less satisfied the decision-maker becomes. Using parametric programming, where α = 1 − θ, we can substitute membership function of Equation (4) into (5) and the problem below is obtained: Max cx, s.t (Ax)i ≤ bi + (1−α)pi, ∀i, (6) for x ≥ 0 and α ∈ [0,1]. 4 Result Analysis and Discussions In this section, fuzzy linear programming method is applied to some cases to optimize the decisions. These are the cases of a Water Venture and an Institute. 4.1 The Water Ventures The study was based on two different bottles of water which the Venture pro- duces : 75cl and 50cl. It makes 134.62NGN per carton of 50cl and 150.26NGN per carton of 75cl as profits. The firm employs machine for 7 hours in a day, with 33 M.K. Eze and B.O. Onasanya Basic Variables x1 x2 g1 g2 b x1 1 1.189 23.681 0 166.573 g2 0 0.032 -1.003 1 1.003 p 0 10.002 3,204.03 0 22,437.129 Table 1: Final Solution to Equation (7) by Simplex Method tolerance level of 2 hours and labor for 8 hours with tolerance level of 1 hour. The classical linear programming problem is constructed thus: Max p = 134.62x1 + 150.26x2, (7) s.t. g1(x) = 0.042x1 + 0.05x2 ≤ 7,g2(x) = 0.042x1 + 0.082x2 ≤ 8. where g1 is machine time, g2 is labour time, x1 is the 50cl bottle water and x2 is the 75cl bottle water. The final result of the simplex method is in Table 1. The fuzzy membership function of the machine time: µ1(x) =   1, if g1(x) ≤ 7 1− g1(x)−7 2 , 7 < g1(x) < 9 0, g1(x) ≥ 9 (8) The membership function of the labour time: µ2(x) =   1, if g2(x) ≤ 8 1− g2(x)−8 1 , 8 < g2(x) < 9 0, g2(x) ≥ 9 (9) The fuzzy linear programming problem associated with Equation (7) is Max p = 134.62x1 + 150.26x2, (10) s.t. µ1(x) ≥ α, µ2(x) ≥ α, where α ∈ [0,1] and x1,x2 ≥ 0. The fuzzy linear programming problem is expanded thus: Max p = 134.62x1 + 150.26x2, (11) s.t. g1 = 0.042x1 + 0.05x2 ≤ 7 + 2(1 −α), and g2(x) = 0.042x1 + 0.082x2 ≤ 8 + (1−α), where x1,x2 ≥ 0 and α ∈ [0,1]. 34 Fuzzy linear programming in decision-making Basic Variables x1 x2 g1 g2 b x1 1 1.189 23.681 0 166.573 + 47.362θ g2 0 0.032 -1.003 1 1.003 - 1.006θ p 0 10.002 3,204.03 0 22,437.129 + 6,408.06θ Table 2: Solution to the fuzzy linear programming Equation (12) θ p∗ x∗1 g1 g2 0.0 22,437.13 166.573 6.996 6.663 0.1 22,077.94 171.309 7.195 6.852 0.2 23,718.74 176.045 7.394 7.042 0.3 24,359.55 180.782 7.593 7.231 0.4 25,000.35 185.518 7.792 7.421 0.5 25,641.16 190.254 7.991 7.610 0.6 26,281.97 194.990 8.189 7.799 0.7 26,922.77 199.726 8.389 7.989 0.8 27,563.58 204.463 8.587 8.178 0.9 28,204.38 209.199 8.786 8.368 1.0 28,845.19 213.935 8.925 8.557 Table 3: Result of the Parametric Problem Setting θ = 1−α, the programming problem above becomes Max p = 134.62x1 + 150.26x2 (12) s.t. g1 = 0.042x1 + 0.05x2 ≤ 7 + 2θ, g2(x) = 0.042x1 + 0.082x2 ≤ 8 + θ, x1,x2 ≥ 0, where θ ∈ [0,1] is a parameter determining the tolerance level. Using the parametric technique and final result of simplex method, Table 2 was obtained. The optimal solution is (x∗1,x ∗ 2) = (166.573 + 47.362θ,0) and p∗ = 22,424.06 + 6,375.87θ. Therefore, the solution of the parametric programming problem is in Table 3. From the analysis above, it is observed that the Water Venture could make more profit by producing more of 50cl bottles than producing 75cl bottles. In essence, it will be more profitable for the firm to scale up its production of 75cl bottle water and cut down the production of 50cl. 35 M.K. Eze and B.O. Onasanya Basis E L g1 g2 g3 b g1 0 2 3 1 0 −1 1,440 710 3 g2 0 2 3 0 1 −1 1,440 4,496 3 E 1 1 3 0 0 1 1,440 4 3 p 0 259 3 0 0 235 1,440 940 3 Table 4: Final Result of the Simplex Method 4.2 The Institute of Energy Law and Energy Economics This section seeks to maximise profit and minimise cost in the sessional op- eration of the institute based on tuition alone. Annually, the institute admits Law and Energy Studies students. On each Law student, the institute makes a loss of approximately 8,000NGN and on each Energy Studies student, a profit of approximately 235,000NGN. For both Energy Law and Energy Economics, if the institute is willing to spend 238,000NGN with tolerance of 70,000NGN on internet, 1,500,000NGN with tol- erance of 500,000NGN on conference support, and 3 graduate assistant for Energy Study, 1 graduate assistant for Energy Law, with tolerance of 2 additional graduate assistants, the following will be the linear programming problem. Max p(E,L) = 235E −8L, (13) s.t. g1(E,L) = E+L ≤ 238(Internet), g2(E,L) = E+L ≤ 1,500(Coference Support) and g3(E,L) = 1,440E + 480L ≤ 1,920(GraduateAssistants). where E is Energy Studies, L is Energy Law, g1 is Internet, g2 is Conference Sup- port and g3 is Graduate Assistants. Using the Simplex method, Table 4 was obtained. The membership function of Internet µ1(E,L) =   1, if g1(E,L) ≤ 238 1− g1(E,L)−238 70 , 238 < g1(E,L) < 308 0, g1(E,L) ≥ 308 (14) 36 Fuzzy linear programming in decision-making Basis E L g1 g2 g3 b g1 0 2 3 1 0 −1 1,440 710 3 + 208θ 3 g2 0 2 3 0 1 −1 1,440 4,496 3 + 1,498θ 3 E 1 1 3 0 0 1 1,440 4 3 + 2θ 3 p 0 259 3 0 0 235 1,440 940 3 + 470θ 3 Table 5: Matrix Multiplication of the Simplex Method Solution and the Tolerance Level The membership function of Conference Support µ2(E,L) =   1, if g2(E,L) ≤ 1,500 1− g2(E,L)−1,500 500 , 1,500 < g2(E,L) < 2,000 0, g2(E,L) ≥ 2,000 (15) The membership function of Graduate Assistants µ3(E,L) =   1, if g3(E,L) ≤ 1,920 1− g3(E,L)−1,440 960 , 1,920 < g3(E,L) < 2,880 0, g3(E,L) ≥ 2,880 (16) The fuzzy linear programming is Max p(E,L) = 235E −8L, (17) s.t. g1(E,L) = E +L ≤ 238 + 70(1−α) g2(E,L) = E +L ≤ 1,500 + 500(1− α) and g3(E,L) = 1,440E + 480L ≤ 1,920 + 960(1−α). Setting θ = 1−α, the following is the parametric problem: Max p = 235E −8L, (18) s.t. g1(E,L) = E + L ≤ 238 + 70θ, g2(E,L) = E + L ≤ 1,500 + 500θ and g3(E,L) = 1,440E + 480L ≤ 1,920 + 960θ, where θ ∈ [0,1] is a parameter given the tolerance level. Using the parametric technique and final result of simplex method, Table 5 was obtained. 37 M.K. Eze and B.O. Onasanya θ E∗ p∗ Internet Conf. Supp. G.A 0.0 1.33 313.33 1.33 1.33 1,920.00 0.1 1.40 329.00 1.40 1.40 2,016.00 0.2 1.47 344.67 1.47 1.47 2,112.00 0.3 1.53 360.33 1.53 1.53 2,208.00 0.4 1.60 376.00 1.60 1.60 2,304.00 0.5 1.67 391.67 1.67 1.67 2,400.00 0.6 1.73 407.33 1.73 1.73 2,496.00 0.7 1.80 423.00 1.80 1.80 2,592.00 0.8 1.87 438.67 1.87 1.87 2,688.00 0.9 1.93 454.33 1.93 1.93 2,784.00 1.00 2.00 470.00 2.00 2.00 2,880.00 Table 6: Result of the Parametric Problem The optimal solution is p∗ = ( 940 3 + 470θ 3 )NGN and x∗ = (E∗,L∗) = (4 3 + 2θ 3 ,0). Therefore, the final result for the parametric problem is in Table 6. From the above analysis, it is observed that (under varying resources) the profit gotten by the institute comes from the Energy Study program. It is observed that the Energy Law program is not adding to the institute, instead they run at loss to keep the program. The researcher also observed that the random allocation of conference support to both program is not profiting the institute, but will rather jeopardise its continuity. 4.3 Minimisation of Cost Minimising the cost of operation of the institute, the classical linear program- ming problem becomes Min c = 125E + 368L, (19) s.t. g1(E,L) = E + L ≤ 238, g2(E,L) = E + L ≤ 1,500 and g3(E,L) = 1,440E + 480L ≤ 1,920. Using the Simplex method, Table 7 was obtained. 38 Fuzzy linear programming in decision-making Basis E L g1 g2 g3 b g1 -2 0 1 0 −1 480 234 g2 -2 0 0 1 −1 480 1,496 L 3 1 0 0 1 480 4 p 979 0 0 0 368 480 1,472 Table 7: Final Result of the Simplex Method The membership functions of the constraints, respectively Internet, con- ference support and Graduate Assistants are: µ1(E,L) =   1, if g1(E,L) ≤ 238 1− g1(E,L)−238 70 , 238 < g1(E,L) < 308 0, g1(E,L) ≥ 308 (20) µ2(E,L) =   1, if (g2(E,L)) ≤ 1,500 1− g2(E,L)−1,500 500 , 1,500 < g2(E,L) < 2,000 0, g2(E,L) ≥ 2,000 (21) µ3(E,L) =   1, if g3(E,L) ≤ 1,920 1− g3(E,L)−1,920 960 , 1,920 < g3(E,L) < 2,880 0, g3(E,L) ≥ 2,880 (22) The required fuzzy linear programming is Min c = 125E + 368L, (23) s.t. g1(E,L) = E +L ≤ 238+70(1−α), g2(E,L) = E +L ≤ 1,500+500(1− α) and g3(E,L) = 1,440E + 480L ≤ 1,920 + 960(1−α). Setting θ = 1−α, the following is the parametric programming problem: Min c = 125E + 368L, (24) s.t. g1(E,L) = E + L ≤ 238 + 70θ g2(E,L) = E + L ≤ 1,500 + 500θ and g3(E,L) = 1,440E + 480L ≤ 1,920 + 960θ, where θ ∈ [0,1] is a parameter. Using the parametric technique and final result of simplex method, Table 8 was obtained. 39 M.K. Eze and B.O. Onasanya Basis E L g1 g2 g3 b g1 -2 0 1 0 −1 480 234 + 68θ g2 -2 0 0 1 −1 480 1,496 + 498θ E 3 1 0 0 1 480 4 + 2θ C 979 0 0 0 368 480 1,472 + 736θ Table 8: Matrix Multiplication of the Simplex Method and the Tolerance Level θ C∗ Internet Conf. Supp. G.A Energy Law 0.0 1,472.00 4.00 4.00 1,920.00 4.00 0.1 1,545.60 4.20 4.20 2,016.00 4.20 0.2 1,619.20 4.40 4.40 2,112.00 4.40 0.3 1,692.80 4.60 4.60 2,208.00 4.60 0.4 1,766.40 4.80 4.80 2,304.00 4.80 0.5 1,840.00 5.00 5.00 2,400.00 5.00 0.6 1,913.60 5.20 5.20 2,496.00 5.20 0.7 1,987.20 5.40 5.40 2,592.00 5.40 0.8 2,060.80 5.60 5.60 2,688.00 5.60 0.9 2,134.40 5.80 5.80 2,784.00 5.80 1.0 2,208.00 6.00 6.00 2,880.00 6.00 Table 9: Result of the Parametric Problem The optimal solution is C∗ = (1,472 + 736θ)NGN and x∗ = (E∗,L∗) = (0,4 + 2θ). Therefore, the final result for the parametric problem is in Table 9. From the analysis above on cost minimisation, the Energy Law program viably increases the cost of running the institute. 4.4 Proposed Model From the results above, the institute is discovered not to be making optimal profit running both Energy Law and Energy Studies’ program. Therefore, the researcher proposes that the fees of the Energy Law should be increased in such a way that it contributes meaningfully to the institute. Suppose the Law Student and Energy Student contribute 230,000NGN and 235,000NGN respectively, and the conference support is given in the ratio 742 to 758 (from contribution made by 40 Fuzzy linear programming in decision-making Basis E L g1 g2 g3 b g1 0 0 1 −238 1,468 90202 1,056,960 0 L 0 1 0 3 1,468 −758 704,640 1 E 1 0 0 −1 1,468 2,226 2,113,920 1 p 0 0 0 455 1,468 1 23,488 1,395 3 Table 10: Final Result of the Simplex Method each program), then the new linear programming problem becomes: Max p(E,L) = 235E + 230L, (25) s.t. g1(E,L) = 119E + 119L ≤ 238 g2(E,L) = 758E + 742L ≤ 1,500 and g3(E,L) = 1,440E +480L ≤ 1,920, where E is Energy Studies and L is Energy Law. Using the Simplex method, Table 10 was obtained. The membership functions of the constraints, respectively Internet, con- ference support and Graduate Assistants are: µ1(E,L) =   1, if g1(E,L) ≤ 238 1− g1(E,L)−238 70 , 238 < g1(E,L) < 308 0, g1(E,L) ≥ 308 (26) µ2(E,L) =   1, if g2(E,L) ≤ 1,500 1− g2(E,L)−1,500 500 , 1,500 < g2(E,L) < 2,000 0, g2(E,L) ≥ 2,000 (27) µ3(E,L) =   1, if g3(E,L) ≤ 1,920 1− g3(E,L)−1,920 960 , 1,920 < g3(E,L) < 2,880 0, g3(E,L) ≥ 2,880 (28) 41 M.K. Eze and B.O. Onasanya Basis E L g1 g2 g3 b g1 0 0 1 −230 1,468 90202 1,056,960 74,901,120θ 1,056,960 L 0 1 0 3 1,468 −758 704,640 1 - 11,520θ 1,056,960 E 1 0 0 −1 1,468 2,226 2,113,920 1 + 708,480θ 1,056,960 p 0 0 0 455 1,468 1 23,488 1,395 3 + 163,939,200θ 1,056,960 Table 11: Matrix Multiplication of the Simplex Method and the Tolerance Level Hence, Max p = 235E + 230L, (29) s.t. g1(E,L) = E + L ≤ 238 + 70(1−α) g2(E,L) = 758E + 742L ≤ 1,500 + 500(1−α) and g3(E,L) = 1,440E + 480L ≤ 1,920 + 960(1−α). Setting θ = 1−α, the following is the parametric problem: Max p = 235E + 230L, (30) s.t. g1(E,L) = E + L ≤ 238 + 70θ g2(E,L) = 758E + 742L ≤ 1,500 + 500θ and g3(E,L) = 1,440E + 480L ≤ 1,920 + 960θ, where θ ∈ [0,1] is a parameter. Using the parametric technique and final result of simplex method, Table 11 was obtained. The optimal solution is p∗ = ( 1,395 3 + 163,939,200θ 1,056,960 )NGN = 465 + 155.10445NGN and x∗ = (E∗,L∗) = (1 + 708,480θ 1,056,960 ,1− 11,520θ 1,056,960 ). Therefore, the final result for the parametric problem is given in Table 12. From the analysis above, the profit of the institute increased greatly as a result of the viable contribution from both programs. 5 Conclusions The potency of fuzzy set theory, fuzzy logic and so on in decision-making cannot be over-emphasized. Its use has proved very efficient from the above anal- ysis, and gives the decision-maker the opportunity to make decision in a robust and flexible environment. 42 Fuzzy linear programming in decision-making θ E L Internet Conf. Supp. G.A. P 0.0 1.000 1.000 238.00 1500.00 1920.00 465.00 0.1 1.070 1.001 246.09 1551.53 2016.96 480.51 0.2 1.130 1.002 254.18 1603.06 2113.92 496.02 0.3 1.200 1.003 262.28 1654.58 2210.88 511.53 0.4 1.270 1.004 270.37 1706.11 2307.84 527.04 0.5 1.340 1.005 278.46 1757.64 2404.80 542.55 0.6 1.400 1.006 286.55 1809.17 2501.76 558.06 0.7 1.470 1.007 294.64 1860.70 2598.72 573.57 0.8 1.540 1.008 302.74 1912.22 2695.68 589.08 0.9 1.600 1.009 310.83 1963.75 2792.64 604.59 1.0 1.670 1.010 318.92 2015.28 2889.60 620.10 Table 12: Result of the Parametric Problem 6 Acknowledgements The authors acknowledge the support of the members of staff of the Centre for Petroleum Economics, Energy and Law and those of the Water Venture where the data used in this study were obtained. References [1] D. Ali, M. Yohanna, M.I. Puwu and B.M. Garkida, Long-term load fore- casting modelling using a fuzzy logic approach, Pacific Science Review A: Natural Science and Engineering 18 (2016), 123–127. [2] D. Anuradha and V.E. 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