Linear Programming and Integer Programming
Assignment 1
October 4, 2016
(Submit on or before the class of October 17, 2016, Monday)
The first Mid-term exam will be on October 17, 2016, Monday
1. 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.
2
x
1
2
3
8
and
x
1
0,
x
2
0,
x
3
0.
D,I
(a)
Work through the simplex method step by step in algebraic
form.
D,I
(b)
Work through the simplex method step by step in tabular
form.
C
(c)
Use a software package based on the simplex method to
solve the problem.