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Linear Programming and Integer ProgrammingAssignment 1October 4, 2016(Submit on or before the class of October 17, 2016, Monday)The first Mid-term exam will be on October 17, 2016, Monday1. Solve the following problem by showing the simplex tableau and row operations performed in each iteration:
maxx1,x2,x3∈Rn4x1+ 3x2+ 6x3s.t.3x1+x2+ 3x3≤302x1+ 2x2+ 3x3≤40x1, x2, x3≥0.2. Consider the following problem:maxx1,x2∈Rn2x1+ 3x2s.t.x1+ 2x2≤4x1+x2= 3x1, x2≥0.(a) Solve the above problem by graphical method. Show the feasible set, all corner-point feasible solutions,and their respective objective function values.(b) Solve the above problem by the Big M method. Show the simplex tableau and row operations performedin each iteration.3. The following is anoptimalLP tableau:zx1x2x3x4x5RHS100032z*0100-112001010600011-12for the following problemmaxx∈Rnc0xs.t.Ax≤bx≥0.wherex= (x1, x2) andx3,x4,x5in the tableau are slack variables.1
(a) Use matrix manipulations to find A, b and c.(b) What is the optimal objective function value, i.e.,z*, in the above tableau?4. Consider the following problem4.4-6.Consider the following problem.MaximizeZ3x15x26x3,subject to2x1x2x34x12x2x34x1x22x34x1x2x33andx10,x20,x30.2x1238andx10,x20,x30.D,I(a)Work through the simplex method step by step in algebraicform.D,I(b)Work through the simplex method step by step in tabularform.C(c)Use a software package based on the simplex method tosolve the problem.