IE 400: HW 3 Part II Due By November 28 2013, Thursday 17.30 PM

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1) Use the branch-and-bound method to find the optimal solution to the following IP:

Min z = 4x1 + 5x2 s.t. x1 + 4x2 ≥ 5

3x1 + 2x2 ≥ 7

x1, x2 ≥ 0; x1, x2 integer

2) ABC Company has six production plants. In those plants, 4 different models of TV sets are produced. The fixed cost of operating each plant for a year and the variable cost of producing a TV set of each model are given in the table below.

Variable Costs ($)

Plant Fixed Plant Costs ($)

Model 1 Model 2 Model 3 Model 4

1 2 million 1200 1400 1450 1450

2 3 million 1100 1300 1400 1400

3 4 million 950 1250 1100 1350

4 5 million 800 950 1000 -

5 2 million - 1300 1400 1375

6 3 million 1150 - 1350 1400

Pay attention that not all plants can produce all models!

a) Each plant can produce only one model of TV sets.

b) The total production of each TV set model must be at a single plant. In other words, if Model 2 is

produced in Plant 5, all Model 2 TV sets must be produced there.

c) If Plant 3 is used, then Plant 1 must also be used.

d) If Plants 3 and 5 are used, then Plant 4 must also be used.

e) If Model 2 is produced in Plant 3, then Model 3 must be produced in Plant 6.

f) If Model 1 is produced in Plant 2, then Model 4 cannot be produced in Plant 6.

g) If either Model 1 or Model 4 is produced in Plant 1, then Plant 3 must be used to produce either

Model 3 or Model 2.

According to the production planning requirements, BestWhite should produce 16000 of each TV set models during a year. Formulate an IP to minimize annual cost of producing TV sets.

IE 400: HW 3 Part II Due By November 28 2013, Thursday 17.30 PM

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3) Use the Branch-and-Bound method to find the optimal solution to the following IP:

a) Min z = 4X1 + 5X2 - 2X3 s.t. X1 + 4X2 + X3 ≥ 5

3X1 + 2X2 - 2X3 ≥ 7

X1, X2 ≥ 0; X1, X2 integer

X3ϵ{0,1}

(Hint: Without solving P0, start solving by branching on X3).

b) How would your answer change to the above question if there is no integrality requirement

on X1? (in other words, X1 is a nonnegative real number)