Maximize Z = -x1 + 2x2 + x3
Subject To
3x2 + x3 =< 120
x1 - x2 - 4x3 =< 80
-3x1 + x2 + 2x3 =< 100
(no nongeative constraints)
A) Reformulate this problem so tha all variables have nonnegative constraints
B) Work through the simplex method step by step to solve the problem

The final table of the LP relaxation of an integer linear programming problem is as follows:
Cost Basic 2 1 1 0 0 Solution
variables x₁ x₂ x₃ x₄ x₅
0 x₄ 0 -1/2 1/2 1 -1/2 3/2
2 x₁ 1 3/2 1/2 0 1/2 7/2
0 -2 0 0 -1 7
To use the branch and bound method to find an optimal solution to the original integer linear programming problem.

To write down the dual of the LPP given by:
Maximise 20x₁+30x₂ subject to
x₁+2x₂≤20
x₁+x₂≤12
5x₁+x₂≤40
x₁, x₂≥0
To solve the primal LPP graphically.
To use the optimal solution to primal LPP and complementary slackness condition to identify the dual variable that will have zero value in the optimal solution to the dual.

Four companies viz. W, X, Y and Z supply the requirements of three warehouses viz. A,
B and C respectively. The companies’ availability, warehouses requirements and the unit
cost of transportation are given in the following table. Find an initial basic feasible
solution using a. North West Corner Method b. Least Cost Method c. Vogel
approximation Method (VAM) d. Find optimal solution using stepping stone algorithm
(20 Marks)
Company Warehouses
A B C Supply
W 10 8 9 15
X 5 2 3 20
Y 6 7 4 30
Z 7 6 9 35
Requirement 25 26 49 100

Given below are 2 consecutive simplex tables for an LPP in the maximisation form.Determine the values of the unknowns a,b,c,d,e,f,g.Also check if the solution corresponding to the second table is optimal.
Variables in
the basis A_1 A_2 A_3 A_4 Solu
x_3 1 2 1 0 8
x_4 2 1 0 1 10
∆_j -4 a 0 0 0
Variables in
the basis A_1 A_2 A_3 A_4 Solu
x_2 e 1 1/2 0 b
x_4 f 0 -1/2 1 c
∆_j g 0 3 0 d

show that for a transportation problem if the availability and requirement are in integer units then any basic feasible solution obtained by the transportation algorithm is also and integer factor.

The advertising director a large retail store in Columbus, Ohio, is considering three advertising media possibilities: (1) ads in the Sunday Columbus dispatch newspaper, (2) ads in a local trade magazine that is distributed free to all houses in the city and northwest suburbs, and (3) ads on Columbus’ WCC-TV station. She wishes to obtain a new customer exposure level of at least 50% within the city and 60% in the northwest suburbs. Each TV ad has a new-customer exposure level of 5% in the city and 3% in the northwest suburbs. The dispatch ads have corresponding exposure levels per ad of 3.5% and 3%, respectively, while the trade magazine has exposure levels per ad of 0.5% and 1%, respectively. The relevant costs are $1,000 per dispatch ad, $300 per trade magazine ad, and $2,000 per TV ad. The advertising policy is that no single media type should consume more than 45% of the total amount spent. Find the advertising strategy that will meet the store’s objective at minimum cost.

Solve the following linear program using SIMPLEX algorithm: (10)
Minimize z = a + b + c
Subject to: 1. a - b - c >= 0
2. a + b + c ≥ 4
3. a + b - c = 2
4. a, b >= 0

1.4 Given the constraints (10)
A+B + C <= 24, B +C >=8 and A >= 0, B >= 0, C>= 0.
Maximize 24-A-B - C
A: amount of time spent on school work
B: amount of time spent on fun
C: amount of time spent on pay work

Maximize z=3a+b+2c
Subject to: a + b+ 3c <=30, a>=0, b>=0, c>=0.