Operations Research Answers

A firm makes items A and B and the total number of items it can make in a day is 24 It takes one hour to make an item of A and only half an hour to make an item of B. The maximum time available per day is 16 hours. The profit on an item of A is Rs. 300 and on one item of B is Rs.

A tractor operator has only one tractor and he operates it on the job order from farmers. The requestors for the jobs arrive with Poisson distribution having interval time of 0.7 day. The average time to do a job is distributed exponentially with mean 0.5 day. Assuming that the tractor can take up the next job immediately on completion of the previous job, determine the following: i. Will a queue be formed? Explain ii. If queue is formed will it statistically stabilize? Explain your answer iii. What is the utilization factor of the tractor? iv. What is the idle time in daily duty of 7 hours? v. What is the mean number of job orders waiting? vi. What is the mean waiting time for job orders in the system? vii. What is the mean waiting time in the queue?

Use Simplex method to

Maximize Ζ = 4_x + 10_x2

Subject 2_X+_X2 ≤ 50

2_X + 5_X2 ≤ 100

2_X + 3_X2 ≤ 100

_{X, X2} ≥ 0

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 available. The requirements for ingredients ( in tons per batch) and the profit for each product (in thousand dollars per batch) are given in the chart. The company wants to determine how many batches of each product should be produced to maximize profit.

For an objective function, Min Z = 10x + 20y; if the corner points are (0,18), (2,6), (4,2), (12,0); the optimal value of Z is 

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.