A company manufactures 3 types of parts which use precious metals, platinum and gold. Due to the shortage of these metals, the government regulates the amount that may be used per day. The relevant data with respect to supply, requirements and profit are summarized in the table below.
Product
Platinum
required/unit (gms)
Gold required/Unit
(gms)
Profit/unit
A
2
3
500
B
4
2
600
C
6
4
1200
Daily allotment of platinum and gold are 160 gm and 120 gm respectively. How should the company divide the supply of scarce precious metals? Formulate the linear programming model.
A dietician wishes to mix two types of food in such a way that the vitamin contends of the mixture contains at least 8 units of vitamin A and 10 units of vitamin B. Food I contains 2 units per Kg of vitamin A and 1 unit per Kg of vitamin B while food II contains 1 unit per Kg of vitamin A and 2 units of vitamin B. It costs 5 birr per Kg to purchase food I and birr 8 per Kg to purchase food II. Prepare the linear programming model for the problem.
A farmer has 1000 acres of land on which he can grow corn, wheat or soya beans. Each acre of corn costs birr 100 for preparation, requires 7 man days of work and yields a profit of birr 30. An acre of wheat costs birr 120 for preparation, requires 10 man days of work and yields a profit of birr 40. Soya beans cost birr 70 to prepare require 8 man days of work and yields a profit of birr 20. If the farmer has birr 100,000 and can count on 80 man days work, formulate the linear programming model.
Manufacturers produce tow products A and B each of which gives a net profit of birr 4 and 3 respectively per unit. These are made either in plant I or plan II. Plant I and II have capacities of 72 and 48 hrs a day. Plant I take 2hrs and 1hr to produce A and B respectively. The corresponding figures of plant II are 1 and 2hrs. Formulate linear programming model.
A firm makes two types of furniture: chairs and tables. The contribution for each product as calculated by the accounting department is 20 birr per chair and 30 birr per table. Both products are processed on three machines M1, M2 and M3. the time required by each product and total time available per week on each machine are as follows;
Machines
Chair
Table
Available hrs
M1
3
3
36
M2
5
2
50
M3
2
6
60
How should the manufacturer schedule his production in order to maximize contribution? Formulate the above problem as a linear programming model.
Suppose that a machine shop has two different types of machines; machine 1 and machine 2, which can be used to make a single product .These machines vary in the amount of product produced per hr., in the amount of labor used and in the cost of operation. Assume that at least a certain amount of product must be produced and that we would like to utilize at least the regular labor force. How much should we utilize each machine in order to utilize total costs and still meets the requirement?
Resources used
Machine 1
(X1)
Machine 2
(X2)
Minimum Required hrs.
Product produced/Hr
20
15
100
Labor/Hr
2
3
15
Operation Cost
$25
$30
Solve the following Linear programming models using graphical method.
Maximize Z= 7x1+ 3x2
Subject to: 2 x1+ 6 x2 ≤ 24
6 x1 + 2 x2 ≤ 24
x1, x2 ≥ 0
Minimize C= 4 x1 + 5 x2
Subject to: 2 x1 + 7 x2 ≥ 31
5 x1 + 3 x2 ≥ 34
x1, x2 ≥ 0
A company owns two flour mills (A and B) which have different production capacities for HIGH, MEDIUM and LOW grade flour. This company has entered contract supply flour to a firm every week with 12, 8, and 24 quintals of HIGH, MEDIUM and LOW grade respectively. It costs the Co. $1000 and $800 per day to run mill A and mill B respectively. On a day, Mill A produces 6, 2, and 4 quintals of HIGH, MEDIUM and LOW grade flour respectively. Mill B produces 2, 2 and 12 quintals of HIGH, MEDIUM and LOW grade flour respectively. How many days per week should each mill be operated in order to meet the contract order most economically standardize? Solve graphically.
A manufacturer of light weight mountain tents makes two types of tents, REGULAR tent and SUPER tent. Each REGULAR tent requires 1 labor-hour from the cutting department and 3labor-hours from the assembly department. Each SUPER tent requires 2 labor-hours from the cutting department and 4 labor-hours from the assembly department. The maximum labor hours available per week in the cutting department and the assembly department are 32 and 84 respectively. Moreover, the distributor, because of demand, will not take more than 12 SUPER tents per week. The manufacturer sales each REGULAR tents for $160 and costs$110 per tent to make. Whereas SUPER tent ales for $210 per tent and costs $130 per tent to make.
Required:
Formulate the mathematical model of the problem
Using the graphic method, determine how many of each tent the company should manufacture each tent the company should manufacture each week so as to maximize its profit?
What is this maximum profit assuming that all the tents manufactured in each week are sold in that week?
What rate of interest let a sum of money to double in 5 years?
How long does it take for a sum of money to triple at a 5% simple interest rate?
A saving account opened 3 months ago now has a balance of Birr 20,400. If the bank pays 8% simple interest, how much money was deposited?
Find the amount that an investor should deposit in a bank today if he needs Birr 20,000 in 3 months at a simple interest rate of 9%.
What percentage of simple interest return must a firm get on their deposit if it wants its Birr 25,000 to grow to birr 26,500 in 6 months?
Mr. Bitew wanted to buy a new car for his family. The cost of the car was Birr 250,000. He was in short of cash and went to his local bank and borrowed Birr 150,000 for 6 years at an annual interest rate of 12%. Find the total simple interest and the maturity value of the loan.
How much money should be deposited in a bank paying interest at a rate of 6% compounded monthly so that at the end of 3 years accumulated amount is Birr 20,000?
How long will it take for Birr 40,000 to amount to Birr 70,000 if it is invested at 7% compounded quarterly?
A firm wishing to have enough money for a retiring manager 17 years from now decided to buy a Birr 30,000 face value zero coupon bond. If the money is worth 10% compounded annually, what should they pay for the bond?
Find the effective rate of interest corresponding to a nominal rate of 9% per year compounded;
A) Semi annually B) Quarterly C) Monthly
Debebe wants to settle a debt of Birr 12,000 to be paid 10 years from now. If deposit earns interest at the rate of 5% compounded annually, how much money should he deposit at the end of each year for the coming 10 years?
A loan of 15,000 with interest rate of 10% compounded semi annually must be paid off in one payment after 3 years. In order to settle this amount, quarterly payments are to be placed in sinking fund which pays 16% compounded quarterly how much should each quarterly payment be?
Mrs. Hiwot has planned to save Birr 25,000 which she would like to reach 10 years from now. During the first 5 years, she is financially able to deposit only Birr 100 each month into the saving account. What must her monthly deposit over the last 5 years be if she is to reach her goals? The account pays 12% interest compounded monthly.
An employee deposits Birr 150 at the end of each month in a credit union saving plan that pays an interest rate of 12% compounded monthly. If the monthly periodic payments have increased to Birr 200 after the 8th payment,
What is the balance after one year?
What is the amount of interest that is earned?
What is the present value of an annuity of seven payments of Birr 1000 each made at the end of each quarter with an interest rate of 12% compounded monthly?
Mrs. Almaz purchased a house for Birr 50,000. She made an amount of down payment and pay monthly Birr 600 to retire the mortgage for 20 years at an annual interest rate of 24% compounded monthly.
Required: Find the mortgage, down payment, interest charged, and the percentage of the down payment to the selling price.
To divide the supply of scarce precious metals, the company needs to determine how much of each metal should be allocated to each product in order to maximize profit while staying within the allotted amount of platinum and gold.
Let:
- X1 = quantity of product A
- X2 = quantity of product B
- X3 = quantity of product C
The objective is to maximize profit, so the objective function is:
Maximize: 500X1 + 600X2 + 1200X3
Subject to the following constraints:
- Platinum constraint: 2X1 + 4X2 + 6X3 ≤ 160 (daily allotment of platinum)
- Gold constraint: 3X1 + 2X2 + 4X3 ≤ 120 (daily allotment of gold)
- Non-negativity constraint: X1, X2, X3 ≥ 0