An Introduction to Fuzzy Linear Programming Problems: by Jagdeep Kaur, Amit Kumar

By Jagdeep Kaur, Amit Kumar

The e-book offers a picture of the state-of-the-art within the box of absolutely fuzzy linear programming. the main target is on exhibiting present equipment for locating the bushy optimum resolution of totally fuzzy linear programming difficulties within which the entire parameters and choice variables are represented through non-negative fuzzy numbers. It provides new tools constructed through the authors, in addition to present equipment built via others, and their software to real-world difficulties, together with fuzzy transportation difficulties. in addition, it compares the results of the several tools and discusses their advantages/disadvantages. because the first paintings to gather at one position an important equipment for fixing fuzzy linear programming difficulties, the ebook represents an invaluable reference consultant for college kids and researchers, supplying them with the required theoretical and functional wisdom to accommodate linear programming difficulties lower than uncertainty.

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Extra resources for An Introduction to Fuzzy Linear Programming Problems: Theory, Methods and Applications

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4 Illustrative Examples 49 max{w1 , 4w1 } + max{0w2 , 3w2 } = 40 x1 ≥ 0, y1 − x1 ≥ 0, z 1 − y1 ≥ 0, w1 − z 1 ≥ 0 x2 ≥ 0, y2 − x2 ≥ 0, z 2 − y2 ≥ 0, w2 − z 2 ≥ 0 Step 5 Using Steps 7 and 8 of the method, presented the problem, obtained in Step 3, can be written as: 3x1 7x2 3x2 5y1 y1 7y2 1 5x1 −|− |+ −|− |+ −|− |+ − |− Maximize ( 4 2 2 2 2 2 2 2 y2 5z 1 z1 7z 2 z2 5w1 3w1 7z 2 3z 2 |+ +|− |+ +|− |+ +|− |+ +|− |) 2 2 2 2 2 2 2 2 2 subject to 3x1 3x1 5x2 3x2 −|− |+ −|− |=2 2 2 2 2 3y1 y1 5y2 y2 −|− |+ − | − | = 10 2 2 2 2 3z 1 z1 5z 2 z2 +|− |+ + | − | = 24 2 2 2 2 3w1 3w1 5w2 3w2 −|− |+ +|− | = 44 2 2 2 2 5x1 3x1 3x2 3x2 −|− |+ −|− |=1 2 2 2 2 y1 3y2 y2 5y1 −|− |+ −|− |=8 2 2 2 2 3z 1 z1 3z 2 z2 +|− |+ + | − | = 21 2 2 2 2 5w1 3w1 3w2 3w2 −|− |+ −|− | = 40 2 2 2 2 x1 ≥ 0, y1 − x1 ≥ 0, z 1 − y1 ≥ 0, w1 − z 1 ≥ 0 x2 ≥ 0, y2 − x2 ≥ 0, z 2 − y2 ≥ 0, w2 − z 2 ≥ 0 Step 6 Since, x1 ≥ 0, y1 ≥ 0, z 1 ≥ 0, w1 ≥ 0, x2 ≥ 0, y2 ≥ 0, z 2 ≥ 0 and w2 ≥ 0 so the problem, obtained in Step 5, can be written as: 1 Maximize (x1 + 2x2 + 2y1 + 3y2 + 3z 1 + 4z 2 + 4w1 + 5w2 ) 4 subject to 0x1 + x2 = 2 x1 + 0x2 = 1 y1 + 2y2 = 10 2y1 + y2 = 8 50 3 Fuzzy Optimal Solution of Fully Fuzzy Linear … 2z 1 + 3z 2 = 24 3z 1 + 2z 2 = 21 3w1 + 4w2 = 44 4w1 + 3w2 = 40 x1 ≥ 0, y1 − x1 ≥ 0, z 1 − y1 ≥ 0, w1 − z 1 ≥ 0 x2 ≥ 0, y2 − x2 ≥ 0, z 2 − y2 ≥ 0, w2 − z 2 ≥ 0 Step 7 The optimal solution of the crisp linear programming problem, obtained in Step 6, is x1 = 1, y1 = 2, z 1 = 3, w1 = 4, x2 = 2, y2 = 4, z 2 = 6 and w2 = 8.

M j=1 n bij = gi ∀ i = 1, 2, . . 13) j=1 n cij = hi ∀ i = 1, 2, . . , m j=1 n dij = ki ∀ i = 1, 2, . . , m j=1 xj ≥ 0, yj − xj ≥ 0, zj − yj ≥ 0, wj − zj ≥ 0 ∀ j = 1, 2, . . 14): n Maximize/Minimize ( (pj , qj , rj , sj )) j=1 subject to n aij = bi ∀ i = 1, 2, . . , m j=1 n bij = gi ∀ i = 1, 2, . . 14) j=1 n cij = hi ∀ i = 1, 2, . . , m j=1 n dij = ki ∀ i = 1, 2, . . , m j=1 xj ≥ 0, yj − xj ≥ 0, zj − yj ≥ 0, wj − zj ≥ 0 ∀ j = 1, 2, . . 15): 26 2 Non-negative Fuzzy Optimal Solution of Fully Fuzzy Linear Programming … n (pj , qj , rj , sj ) Maximize/Minimize j=1 subject to n aij = bi ∀ i = 1, 2, .

2) in which all the parameters are represented by unrestricted trapezoidal fuzzy numbers. 2) can be written as: n ( p j , q j , r j , s j ) ⊗ (x j , y j , z j , w j ) Maximize/Minimize j=1 subject to n (ai j , bi j , ci j , di j ) ⊗ (x j , y j , z j , w j ) = (bi , gi , h i , ki ) ∀ i = 1, 2, . . 3) where (x j , y j , z j , w j ) is an unrestricted trapezoidal fuzzy number. Step 2 Using the product of trapezoidal fuzzy numbers, presented in Sect. 4) n (min{ai j , ai j }, min{bi j , bi j }, max{ci j , ci j }, max{di j , di j }) j=1 = (bi , gi , h i , ki ) ∀ i = 1, 2, .

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