Question

1. Consider the following LP: Max z 5x1 X2 st. 2x1 xS6 6x1X2S 12 Plot the constraints on the graph and identify the feasible region and determine the optimal value of the objective function and the values of the decision variables. 2. Priceler manufactures sedans and wagons. The number of vehides that can be sold each of the next three months are listed to Table 1. Each sedan sells for $10000 and each wagon sells for $11000. It cost $7000 to produce a sedan and $8000 to produce a wagon. To hold a vehide inventory for one month cost $160 per sedan and $210 per wagon. During each month, at most 1600 vehicles can be produced. Production line restrictions dictate that during month 1, at least two thirds of all cars produced must be sedans. At the beginning of the first month, 400 sedans and 300 wagons are available. Formulate an LP that can be used to maximize Pricesler's profit during the next three months. Wagons Month 1 1200 1600 3. Truckco manufactures two types of trucks: 1 and 2. Each truck must go through the painting shop and assembly shop. If the painting shop were completely devoted to painting Type 1 trucks, then 800 per day could be painted; if the painting shop were completely devoted to painting Type 2 trucks, then 700 per day could be painted. If the assembly shop were completely devoted to assembling truck 1 engines, then 1,500 per day could be assembled; if the assembly shop were completely devoted to assembling truck 2 engines, then 1,200 per day could be assembled. Each Type 1 truck contributes $300 to profit; each Type 2 truck contributes $500. Formulate an LP that will maximize Truckco's profit 4. You have decided to enter the candy business. You are considering producing two types of candies: Slugger candy and Easy out candy, both of which consist of sugar, nuts and chocolate. At present, you have in stock 100 oz of sugar, 20 oz of nuts and 30 oz of chocolate. The mixture used to make easy out candy must contain at least 20% nuts. The mixture used to make slugger candy must contain at least 12% nuts and 10% chocolate. Each ounce of easy out candy can be sold for 25 cents and each ounce of slugger candy for 22 cents. Formulate an LP that wll enable you to maximize your revenue from candy sales. 5. Each year, Paynothing shoes faces demands (which must be met on time) for pairs of shoes as shown in Table. Workers work three consecutive quarters and then receive one quarter off. For example, a worker may work during quarters 3 and 4 of one year and quarter 1 of the next year. During a quarter in which a worker works, he or she can produce up to 50 pairs of shoes. Each worker is paid $500 per quarter. At the end of each quarter, a holding cost of $50 per pair of shoes is assessed. Formulate and LP that can be used to minimize the cost per year (labor +holding) of meeting the demands of shoes. To simplify the matters, assume that at the end of each year, the ending inventory is zero. (Hint: it is allowable to assume that a given worker will get the same quarter off during each year.) Quarter 1 Quarter 2 Quarter 3 Quarter 4 100

Answer 1

1.

We need to first draw the graph. The draw the constraint lines with equality. Next, determine the feasible region.

Finally use the corner points on the objective function to determine the optimal corner point. That will be the solution.

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The corner points are (0,0), (0,6), Crossing point of 2X1 + X2 = 6 and 6X1 + X2 = 12, and crossing point of 6X1 + X2 = 12 and X1 – X2 = 0

Now, crossing point between 2X1 + X2 = 6 and 6X1 + X2 = 12 is at (1.5, 3)

Crossing between 6X1 + X2 = 12 and X1 – X2 = 0 are at (1.71, 1.71)

The value of objective function at

(0,0) = 0

(0,6) = 5*0 + 6 = 6

(1.5,3) = 5*1.5 + 3 = 10.5

(1.71, 1.71) = 5*1.71 + 1.71 = 10.26

The optimal solution is (1.5, 3) and the objective function value is 10.5