Problems

A-51

minimize Z = 40x_{1 }+ 55x_{2 }+ 30x_{3 }

subject to

2 x_{1 }+ 2x + 3x

3

2 2x_{1 }+ x + x

3

2 x_{1 }+ 3x + x

3

5x_{2 }+ 3x

3

1 2 x ,x ,x

3

…

60

= 40 Ú 50 Ú 100 Ú0

31. A manufacturing firm produces two products using labor and material. The company has a con- tract to produce 5 of product 1 and 12 of product 2. The company has developed the following lin- ear programming model to determine the number of units of product 1 (x_{1}) and product 2 (x_{2}) to produce to maximize profit:

maximize Z = 40x_{1 }+ 60x_{2 }(profit, $)

subject to

( m a t e r i a l , l b . ) ( l a b o r , h r . ) ( c o n t r a c t , p r o d u c t 1 ) ( c o n t r a c t , p r o d u c t 2 ) x 1 , x 2 Ú 0 x 2 Ú 1 2 x 1 Ú 5 4 x 1 + 4 x 2 … 7 2 x 1 + 2 x 2 … 3 0

Solve this model using the simplex method.

32. A custom tailor makes pants and jackets from imported Irish wool cloth. To get any cloth at all, the tailor must purchase at least 25 square feet each week. Each pair of pants and each jacket requires 5 square feet of material. The tailor has 16 hours available each week to make pants and jackets. The demand for pants is never more than 5 pairs per week. The tailor has developed the following lin- ear programming model to determine the number of pants (x_{1}) and jackets (x_{2}) to make each week to maximize profit:

maximize Z = x_{1 }+ 5x_{2 }(profit, $100s)

subject to

( l a b o r , h r . ) ( d e m a n d , p a n t s ) x 1 , x 2 Ú 0 x 1 … 5 2 x 1 + 4 x 2 … 1 6 5 x 1 + 5 x 2 Ú 2 5 ( w o o l , f t . 2 )

Solve this model using the simplex method.

33. A sawmill in Tennessee produces cherry and oak boards for a large furniture manufacturer. Each month the sawmill must deliver at least 5 tons of wood to the manufacturer. It takes the sawmill 3 days to produce a ton of cherry and 2 days to produce a ton of oak, and the sawmill can allocate 18 days out of a month for this contract. The sawmill can get enough cherry to make 4 tons of wood and enough oak to make 7 tons of wood. The sawmill owner has developed the following linear programming model to determine the number of tons of cherry (x_{1}) and oak (x_{2}) to produce to minimize cost: