Problems

A-47

x 1 , x 2 Ú 0 5 x 1 + 2 x 3 x 1 + 6 x 4 x 1 2 2 + 3 x 2 Ú 1 2 Ú 12 Ú 10

14.

The Copperfield Mining Company owns two mines, both of which produce three grades of ore— high, medium, and low. The company has a contract to supply a smelting company with at least 12 tons of high-grade ore, 8 tons of medium-grade ore, and 24 tons of low-grade ore. Each mine pro- duces a certain amount of each type of ore each hour it is in operation. Mine 1 produces 6 tons of high-grade, 2 tons of medium-grade, and 4 tons of low-grade ore per hour. Mine 2 produces 2 tons of high-grade, 2 tons of medium-grade, and 12 tons of low-grade ore per hour. It costs $200 per hour to mine each ton of ore from mine 1, and it costs $160 per hour to mine a ton of ore from mine 2. The company wants to determine the number of hours it needs to operate each mine so that contractual obligations can be met at the lowest cost. Formulate a linear programming model for this problem and solve using the simplex method.

15.

A marketing firm has contracted to do a survey on a political issue for a Spokane television station. The firm conducts interviews during the day and at night, by telephone and in person. Each hour an interviewer works at each type of interview results in an average number of interviews. In order to have a representative survey, the firm has determined that there must be at least 400 day inter- views, 100 personal interviews, and 1,200 interviews overall. The company has developed the following linear programming model to determine the number of hours of telephone interviews during the day (x

_{1}), telephone interviews at night (x_{2}), personal interviews during the day (x_{3}), and personal interviews at night (x_{4}) that should be conducted to minimize cost:

minimize Z = 2x_{1 }+ 3x_{2 }+ 5x_{3 }+ 7x_{4 }(cost, $)

subject to

( d a y i n t e r v i e w s ) ( p e r s o n a l i n t e r v i e w s ) ( t o t a l i n t e r v i e w s ) x 1 , x 2 , x 3 , x 4 Ú 0 x 1 + x 2 + x 3 + x 4 Ú 1 , 2 0 0 4 x 3 + 5 x 4 Ú 1 0 0 1 0 x 1 + 4 x 3 Ú 4 0 0

Solve this model using the simplex method.

16. A jewelry store makes both necklaces and bracelets from gold and platinum. The store has devel- oped the following linear programming model for determining the number of necklaces and bracelets (x_{1 }and x_{2}) that it needs to make to maximize profit:

maximize Z = 300x_{1 }+ 400x_{2 }(profit, $)

subject to

( g o l d , o z . ) ( p l a t i n u m , o z . ) ( d e m a n d , b r a c e l e t s ) x 1 , x 2 Ú 0 x 2 … 4 2 x 1 + 4 x 2 … 2 0 3 x 1 + 2 x 2 … 1 8

Solve this model using the simplex method.

17. A sporting goods company makes baseballs and softballs on a daily basis from leather and yarn. The company has developed the following linear programming model for determining the number of baseballs and softballs to produce (x_{1 }and x_{2}) to maximize profits: