Operations research questions.

Define the term “Break Even Point” as used in Management Science and briefly explain its economic importance to the agribusiness manager.

Machinga makes handcrafted dolls, which he sells at craft fairs. He is considering mass production of dolls to sell in big outlets. He estimates that the initial investment for plant and equipment will be Tshs 25,000, whereas labor, material, packaging, and shipping will be about Tshs 10 per doll. If the dolls are sold for Tshs 30 each, what sales volume is necessary for machinga to break even?

The Morogoro Textile Mill produces bed sheets. The fixed monthly cost is Tshs 21,000, and the variable cost per pair of bed sheets is Tshs 0.45. The mill sells a pair of bed sheet at Tshs 1.30.

1. For a monthly volume of 18,000 pair of bed sheets, determine the total cost, total revenue, and profit
2. Determine the annual break-even volume for Morogoro textile Mill.

Why is the term linear is used in linear programming?

The Kalo fertilizer company makes a fertilizer using two chemicals that provide nitrogen, phosphate, and potassium. A kilogram of ingredient 1 contributes 10 grams of nitrogen and 6 grams of phosphate, while a kilogram of ingredient 2 contributes 2 grams of nitrogen, 6 grams of phosphate, and 1 gram of potassium. Ingredient 1 costs $3 per kilogram, and ingredient 2 costs $5 per kilogram. The company wants to know how many kilogram of each chemical ingredient to put into a bag of fertilizer to meet the minimum requirements of 20 grams of nitrogen, 36 grams of phosphate, and 2 grams of potassium while minimizing  cost.

1. Formulate a linear programming model for this problem
2. Write the model into standard form for solving by simplex method.

Using some economic theories, briefly explain the following

1. The coefficient of slack variables equal zero in the objective function
2. The basic variables in the initial simplex tableau are the slack variables
3. In the simplex method of solving L.P problem, the variable with the largest positive value is the entering variable. Give its economic rationale.

The HC & AC Ltd makes two kinds of tables-end tables(X1) and coffee tables(X2). The manufacture is restricted by material and labor constraints, as shown in the following linear programming formulation.

Maximize

Subject to

The final optimal simplex tableau for this problem is as follows:

			200	300	0	0
	Basic variables	Quantity	X1	X2	S1	S2
300	X2	30	0	1	1/3	-2/9
200	X1	15	1	0	-1/3	5/9
		12,000	200	300	100/3	400/9
	-		0	0	-100/3	-400/9

1. Formulate the dual for this problem
2. Prove that the shadow price for labor hour is $100/3
3. What is the importance of the duality theory to the agribusiness managers?

1. Why do waiting lines form at a service facility even though there may be more than enough service capacity to meet normal demand in the long run?
2. What are the assumptions and limitations of the Queuing theory?

1. Using vivid examples , distinguish between a transportation model and assignment model.
2. A concrete company transports concrete from three plants to three construction sites. The supply capacities of the three plants, the demand requirements at the three sites and the transportation costs per ton are as follows.

Determine the linear programming model formulation for this problem and find the initial optimal solution using the Minimum Cost (MCC) method.

Briefly describe the management science approach to problem solving.