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Parametrical approach for studying and solving bilinear programming problem 169and find m points y 1 , y 2 , . . . , y m , which correspond to m basic solutions of the system (3), i.e.y 1 , y 2 , . . . , y m represent neighboring basic solutions for y 0 . If the system (6) is compatible withrespect to u for each y = y 1 , y = y 2 , . . . , y = y m , then go to step 3; otherwise the system (3) isnot compatible for every y satisfying (3) and STOP.Step 3.For each s = 1, m, solve the following problem:maximize ton subject⎧⎪⎨⎪⎩−A 1 T u ≤ Cy + d 1 ;b 1 u ≤ d 2 y − h;u ≥ 0y = y 0 + b s t, t ≥ 0and find m solutions t ′ 1, t ′ 2, . . . , t ′ m. Then fix m points y s = y 0 + b s t ′ s, s = 1, m.Step 4.Find the hyperplanem∑a ′ jy j + a ′ 0 = 0,j=1which passes through the points y 1 , y 2 , . . . , y m . Consider that the basic solution y 0 = (y1 0, y0 2 , . . . , y0 m)satisfies the following conditionm∑a ′ jyj 0 + a′ 0 ≤ 0.j=1Then add to the system (3) the inequality−m∑a ′ jy j − a ′ 0 ≤ 0.j=1If after that the obtained system is not compatible, then conclude that the system (6) is compatiblefor every y satisfying the initial system (3) and STOP; otherwise change the initial systemwith the obtained one and go to step 1.Note that in [3] it is proposed an algorithm for the establishment of the compatibility of theparametrical problem in the case when the system (3) has the following form⎧⎪⎨⎪⎩y 1 ≤ b 2 1 ;y 2 ≤ b 2 2 ;. . . . . . . . . . . . . . . . . . . . . . . .y m ≤ b 2 m ;y 1 ≥ 0, y 2 ≥ 0, . . . , y m ≥ 0i.e. the set of solutions of the system (3) is a m-dimensional cube. Even in this case the compatibilityproblem remains NP-hard, but the approach from [3], based on Theorems 1,2, allows topropose suitable algorithms.