Operations Research Answers

The Eastern Iron and Steel Company makes nails, bolts, and washers from leftover steel and coats them with zinc. The company has 24 tons of steel and 30 tons of zinc. The following linear pro- gramming model has been developed for determining the number of batches of nails (x1), bolts (x2), and washers (x3) to produce to maximize profit:

maximize Z = 6x1+2x2+12x3(profit,$1,000s) subject to

2x1+ 6x2 + 3x3 smaller than or equal to 30 (zinc, tons)

4X1+ x2+3x3 smaller than or equal to24(steel, tons)

x1, x2, x3 bigger than or equal to 0

Solve this model using the simplex method

The Cookie Monster Store at South Acres Mall makes three types of cookies—chocolate chip,

pecan chip, and pecan sandies. Three primary ingredients are chocolate chips, pecans, and sugar.

The store has 120 pounds of chocolate chips, 40 pounds of pecans, and 300 pounds of sugar. The

following linear programming model has been developed for determining the number of batches

of chocolate chip cookies pecan chip cookies and pecan sandies to make to maximize profit:

maximize Z = 10x1 + 12x2 + 7x3 (profit, $)

subject to:

x1 + 2x3 smaller than or equal to 40 (pecans, lb.)

10x1 + 5x2 smaller than or equal to 120 (chocolate chips, lb.)

20x1 + 15x2 + 10x3 smaller than or equal to 300(sugar, lb.)

x1, x2, x3 bigger than or equal to 0

Solve this model using the simplex method.

A baby products firm produces a strained baby food containing liver and milk, each of which con- tribute protein and iron to the baby food. Each jar of baby food must have 36 milligrams of protein and 50 milligrams of iron. The company has developed the following linear programming model to determine the number of ounces of liver (x1) and milk (x2) to include in each jar of baby food to meet the requirements for protein and iron at the minimum cost.

minimize Z = 0.05x1 + 0.10x2 (cost, $) subject to

6x1+ 2x2 > or equal to 36 (protein, mg)

5x1 + 5x2 >or equal to 50 (iron, mg)

X1, X2> or equal to 0

Solve this model using the simplex method.

A wood products firm in Oregon plants three types of trees - white pines, spruce, and ponderosa pines—to produce pulp for paper products and wood for lumber. The company wants to plant enough acres of each type of tree to produce at least 27 tons of pulp and 30 tons of lumber. The company has developed the following linear programming model to determine the number of acres of white pines (x), spruce (x), and ponderosa pines (xz) to plant to minimize cost.

minimize Z = 120x, + 40x2 + 240x3(cost, $) subject to

4x1 + x2 + 3x3 bigger than or equal to 27 (pulp, tons)

2x1 + 6x2 + 3x3 bigger than or equal to 30 (lumber, tons)

Solve this model using the simplex method.

1. An appliance dealer wants to purchase a combined total of no more than 100 refrigerators, and dishwashers for inventory. Refrigerators weigh 200 pound each, and dishwashers weigh 100 pounds each. The dealer is limited to a total of 12,000 pounds for these two items. A profit of $35 for each refrigerator and $20 on each dishwasher is projected.

(a) Write out the linear programming model by identifying the constraints and the objective function from the description above.

(b) Using a scale of 2 cm to 20 pounds on both axes, construct and shade the region R in which every point satisfies all the constraints.

(c) Based on the graph obtained in (b), determine the corner points and find out the maximum number of refrigerators and dishwashers that the dealer can purchase and sold to make the profit.

Solve the following linear programming model using the simplex method:

maximize Z = 100x1 + 20x2 + 60x3

subject to

x3 smaller than or equal to 40

2x1 + 2x2 + 2x3 smaller than or equal to 100

3x1 + 5x2 smaller than or equal to 60

x1, x2, x3 bigger than or equal to 0

A clothing shop makes suits and blazers. Three main resources are used: material, rack space, and labor. The shop has developed this linear programming model for determining the number of suits

and blazers to make ( and ) to maximize profits

maximize Z = 100x1 + 150x2 (profit, $)

Subject to

10x1 + 20x2 smaller than or equal to 300 (material, yd.2

)

x1 + x2 smaller than or equal to 20 (rack space)

10x1 + 4x2 smaller than or equal to160 (labor, hr.)

x1, x2 bigger than or equal to 0

A jewelry store makes both necklaces and bracelets from gold and platinum.

The store has developed the following linear programming model for determining the number of necklaces and bracelets

(x1 and x2) that it needs to make to maximize profit: maximize Z = 300x1+ 400x2(profit, S) subject to 3x1 + 2x2 less than equal to 18 (gold, oz)

2x1 + 4x2 less than equal to 20 (platinum,oz.)

X2 less than equal to 4 (demand, bracelets) X1,x2 greater than equal to 0

Solve this model using the simplex method.

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 produces 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.

Solve the following model using the simplex method:

minimize

Z = 0.06x1 + 0.10x2

subject to:

 4x1 + 3x2>=12
 3x1 + 6x2 >= 12
 5x1 + 2x2 >= 10
 x1,  x2 >= 0