Homework

Chapter 2 Modeling with Linear Programming res two products on three sequential 2-8. A company that operates 10 hrs a day manufactures two products on three processes. The following table summarizes the data of the problem: Minutes per unit Process 2 Product Process Process 3 Unit profit $20 $30 Determine the optimal mix of the two products. *2-9. A company produces two products, A and B. The sales volume for A is at least 80% of the total sales of both A and B. However, the company cannot sell more than 110 units of A per day. Both products use one raw material, of which the maximum daily availabilitu is 300 lb. The usage rates of the raw material are 2 lb per unit of A, and 4 lb per unit of R The profit units for A and B are $40 and $90, respectively. Determine the optimal product mix for the company. 2-10. Alumco manufactures aluminum sheets and aluminum bars. The maximum production capacity is estimated at either 800 sheets or 600 bars per day. The maximum daily demand is 550 sheets and 560 bars. The profit per ton is $40 per sheet and $35 per bar. Determine the optimal daily production mix. 2-11. An individual wishes to invest $5000 over the next year in two types of investment: Investment A yields 5%, and investment B yields 8%. Market research recommends an allocation of at least 25% in A and at most 50% in B. Moreover, investment in A should be at least half the investment in B. How should the fund be allocated to the two investments? 2-12. The Continuing Education Division at the Ozark Community College offers a total of 30 courses each semester. The courses offered are usually of two types: practical and humanistic. To satisfy the demands of the community, at least 10 courses of each type must be offered each semester. The division estimates that the revenues of offering prac- tical and humanistic courses are approximately $1500 and $1000 per course, respectively. (a) Devise an optimal course offering for the college. (b) Show that the worth per additional course is $1500, which is the same as the reve. nue per practical course. What does this result mean in terms of offering additional courses? 2-13. ChemLabs uses raw materials / and II to produce two domestic cleaning solutions, A and B. The daily availabilities of raw materials I and II are 150 and 145 units, respectively. One unit of solution A consumes 5 unit of raw material / and .6 unit of raw material Il. One unit of solution Buses 5 unit of raw material / and 4 unit of raw material Il. The profits per unit of solutions A and B are $8 and $10, respectively. The daily demand for solution A lies between 30 and 150 units, and that for solution B between 40 and 200 units. Find the optimal production amounts of A and B.

Answer

Answer of first question:

Let us first write the objective function where, 

Z= Total profit which we have to maximize

X1= quantity of product 1

X2= Quantity of product 2

Objective function will be: 

Z (total profit) = profit from each unit of product 1*product 1 quantity + profit from each unit of product 2*product 2 quantity= 20*X1+30X2

Let us write the constraint

10×1 + 5×2 <= 600
6×1 + 20×2 <= 600
8×1 + 10×2 <= 600
and x1,x2 >= 0 

Solution: 

X1= 900/17=53 units

X2=240/17= 14 units

Total profit (Max)= 20*53+30*14=1060+420=1480

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