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S-6 : Chapter 2 - Introduction to Optimization & Linear Programming

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Chapter 2 - Introduction to Optimization & Linear Programming : S-11

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Chapter 2

Introduction to Optimization & Linear Programming

1.

If an LP model has more than one optimal solution it has an infinite number of alternate optimal solutions. In Figure 2.8, the two extreme points at (122, 78) and (174, 0) are alternate optimal solutions, but there are an infinite number of alternate optimal solutions along the edge connecting these extreme points. This is true of all LP models with alternate optimal solutions.

2.

There is no guarantee that the optimal solution to an LP problem will occur at an integer-valued extreme point of the feasible region. (An exception to this general rule is discussed in Chapter 5 on networks).

3.

We can graph an inequality as if they were an equality because the condition imposed by the equality corresponds to the boundary line (or most extreme case) of the inequality.

4.

The objectives are equivalent. For any values of X1 and X2, the absolute value of the objectives are the same. Thus, maximizing the value of the first objective is equivalent to minimizing the value of the second objective.

5.

a.

linear

b.

nonlinear

c.

linear, can be re-written as: 4 X1 - .3333 X2 = 75

d.

linear, can be re-written as: 2.1 X1 + 1.1 X2 - 3.9 X3 SYMBOL 163 \f "Symbol" 0

e.

nonlinear

6.

7.

8.

9.

10.

11.

12.

13.

X1 = number of His chairs to produce, X2 = number of Hers chairs to produce

MAX

10 X1 + 12 X2

ST

4 X1 + 8 X2 SYMBOL 163 \f "Symbol" 1200

8 X1 + 4 X2 SYMBOL 163 \f "Symbol" 1056

2 X1 + 2 X2 SYMBOL 163 \f "Symbol" 400

4 X1 + 4 X2 SYMBOL 163 \f "Symbol" 900

1 X1 + 0.5 X2 ≥ 0

X1, X2 ≥ 0

14.

X1 = number of softballs to produce, X2 = number of baseballs to produce

MAX

6 X1 + 4.5 X2

ST

5X1 + 4 X2 SYMBOL 163 \f "Symbol" 6000

6 X1 + 3 X2 SYMBOL 163 \f "Symbol" 5400

4 X1 + 2 X2 SYMBOL 163 \f "Symbol" 4000

2.5 X1 + 2 X2 SYMBOL 163 \f "Symbol" 3500

1 X1 + 1 X2 SYMBOL 163 \f "Symbol" 1500

X1, X2 SYMBOL 179 \f "Symbol" 0

15.

X1 = proportion of beef in the mix, X2 = proportion of pork in the mix

MIN

.85 X1 + .65 X2

ST

1X1 + 1 X2 = 1

0.2 X1 + 0.3 X2 SYMBOL 163 \f "Symbol" 0.25

X1, X2 SYMBOL 179 \f "Symbol" 0

16.

X1 = number of generators, X2 = number of alternators

MAX

250 X1 + 150 X2

ST

2 X1 + 3 X2 SYMBOL 163 \f "Symbol" 260

1 X1 + 2 X2 SYMBOL 163 \f "Symbol" 140

X1, X2 SYMBOL 179 \f "Symbol" 0

17.

X1 = number of generators, X2 = number of alternators

MAX

250 X1 + 150 X2

ST

2 X1 + 3 X2 SYMBOL 163 \f "Symbol" 260

1 X1 + 2 X2 SYMBOL 163 \f "Symbol" 140

X1SYMBOL 179 \f "Symbol" 20

X2 SYMBOL 179 \f "Symbol" 20

d. No, the feasible region would not increase so the solution would not change -- you'd just have extra (unused) wiring capacity.

18.

X1 = number of propane grills to produce, X2 = number of electric grills to produce

MAX

100 X1 + 80 X2

ST

2 X1 + 1 X2 SYMBOL 163 \f "Symbol" 2400

4 X...

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