1. A firm consists of 600 acres of land, of which 500 acres will be planted with corn, soybeans, and wheat according to this condition:

-At least half of the planted acreage should be in corn

- No more than 200 acres should be soybeans

-the ratio of corn to wheat planted should be 2:1

 It costs birr 20 an acre to plant corn, 15 an acre to plant soybeans and birr 12 an acre to plant wheat. Formulate this problem as an LP model that will minimize planting costs.

2. A marketing manager of large savings and loan association that has branches throughout the state would like to mail a promotional  piece to prospective borrowers. The manager has selected three regions in which to do a mailing. A marketing agency can supply lists of prospective borrowers in each region, as shown below.

          Region              list size                       cost/ name

            1                      10,000             .11

            2                      20,000             .12

            3                      30,000             .13

The manger would like to have a pool of 15,000 and would like the mail to contain at least 10 percent of the people on each list. Formulate an LP model that will accomplish these requirements for ____. Assume there is no overlap among lists, and that any quantity can be ordered from each list.

3. a company sells two different products A and B, making a profit of 40 birr and 30 birr per unit on them respectively. They are produced in a common production process and are sold in two different markets. The production process has a total capacity of 30,000 man-hours. It takes three hours to produce a unit of A and an hour to produce a unit of B. the market has been surveyed and company officials fee that the maximum number of unit A that can be sold is 8,000 units and that of B is 12,000 units. Subject to these limitations, products can be sold in any combination. Formulate this problem as an LP model to maximize profit.

4. The manager of an electric company is engaged in the production of two components C1 and C2 used in the radio sets. Each unit of C1 costs the company birr 5 in wages and birr 5 in material, while each of  C2 costs the company birr 25 in wages and birr 15 in the material. The company sells both products on one-period credit terms, but the company’s labors and material expenses must be paid in cash. The selling price of C1 is birr 30 per unit and C2 is birr 70 per unit. Because of the strong monopoly of the company for these components, it is assumed that the company can sell at the prevailing prices as many units as it produces. The company’s production capacity is, however, limited by two considerations. First at the beginning of period one the company has an initial balance of birr 4,000(cash +bank credit+ collection from past credit sales). Second, the company has available in each period 2,000 hours of machine time and 1,400 hours of assembly time. The production of each C1 requires 3 hours of machine and 2 hours of assembly time, where as the production of each C2 requires 2 hours of machine time and 3 hours for assembly time. Formulate this problem as an Lp model so as to maximize the total profit to the company.

5. An advertising company wishes to pan an advertising campaign in three different media: television( prime day, prime time), radio, and magazine. The purpose of the advertising is to reach as many potential customers as possible. Following are the results of the market study:

                                                                       Television                            radio             magazine

                                                            Prime day        prime time      

Cost of an advertising unit                 40,000             75,000                30,000                  15,000

No of potential customers                 

reached per unit                                  400,000           900,000           500,000                    200,000

no of women customers

 reached per unit                                 300,000           400,000           200,000                    100,000

the company does not want to spend more than birr 800,000 on advertising. It is further required that 

i) at least 2 million exposures take place among women

ii) advertising on television be limited to birr 500,000

iii)at least 3 advertising units be bought on prime day and two units during prime time and

iv)the number of advertising units on radio and magazine should each be between  5 and 10.  Formulate this problem as an LP model to maximize potential customer reach. 

6. The manger of an oil refinery must decide on the optimal mix of two possible blending processes of which the input and output per production run are given below.

Process (units)                                    input                                        output

                                    Crude A                      crude B                       gasoline X      gasoline Y

1                                       5                     3                        5                     8

2                                       4                     5                        4                     4

 The maximum amount available of crude A and B is 200 units and  150 units respectively. Market requirements show that at least 100 gasoline X and 80 units of gasoline Y must be produced. The profit per production run from processes 1 and processes 2 are birr 300 and birr 400 respectively. Formulate this problem as the Lp model to maximize profit.    

7.a wine maker  has a stock of three different  wines with the following characteristics

      Wine         proofs              acids(%)         specific gravity                       stock(gallons)

          A                          27                   .32                        1.70                             20

         B               33                   .20                        1.08                             34

        C                32                   .30                        1.04                             22          

A dry table wine should be between 30 and 31 degree proof, it should contain at least 0.25% acid and should have a specific gravity of at least 1.06. The wine maker wishes to blend the three types of wine to produce as large  a quantity as possible of satisfactory dry table wine. However, his stock of wine A must be completely used in the blend because further storage  would cause it to deteriorate. What quantities of wines of B and C should be made in the blend. Formulate this problem as an LP model.

8. a pharmaceutical company produces two pharmaceutical products: A and B. production of both products require s the same process I and II. The production of B  results also by-product C at no extra cost. The product A can be sold at profit of birr 3  per unit  and B at  a profit of birr 8 per unit. Some of this by-product can be sold at a unit profit of birr 2, the remainder has  to be destroyed  and destruction cost is  birr 1 per unit. Forecast show only up to  5 units of C can be sold. The company gets 3 units of C  for each unit of B produced. The manufacturing times are  3 hours per unit of A on process I and II respectively, and  4 and 5 hours per unit of B on the process I and II respectively. Because product C results from producing B, no time is used in producing C. the available times are  18 and 21 hours of process I and II respectively. Formulate this problem as LP model  to  determine the quantity of  A and B, which  should be produce , keeping C in mind  the highest profit to the company.

 

Solve the following problems using the graphical method

 

1. A company produces two types of belts, A and B. belt A is of superior quality and B is inferior. Profit on the two belts is 1.50 birr and 1 birr per belt respectively. Each type of belt A requires twice as much time as required by the belt of type B. if all the belts were of type B, the company could produce 1000 belts per day. But the supply of leather is sufficient only for 800 belts per day. Belt A requires a fancy buckle and only 400 of them are available per day. For belt B only 700 buckles are available per day. How should the company manufacture the two types of belts in order to have maximum overall profit?

2.  a company possesses two manufacturing plants, each of which can produce three products: X, Y, and Z  from common raw material. However, the proportions in which the products are produced are different in each plant and so are the plant’s operating costs per hour. Following are the data on the production per hour and costs together with current orders in hand for each product.

Plant                                        product                                                operating cost per hour(birr)

                                    X         Y         Z

A                                 2          4          3                                              9

B                                 4          3          2                                              10

Orders on hand           54        24        60

You are required to use graphical method to find out the number of production hours needed to fulfill the orders on hand at minimum cost.

3. a company produces two types of presentation goods, A and B that require gold and silver. Each unit of type requires 3 g of silver and 1 g of gold while B requires 1 g of silver and 2 g of gold. The company can produce 9 g of silver and 8 g of gold if each unit of type A brings a profit of  40 birrs and that of type B birr 50, determine the number of units of each type that should be produced to maximize the profit?

4.  a firm plans to purchase at least 200 quintals of scarp containing high-quality metal X and low-quality metal Y. it decides that the scrap to be purchased must contain at least 100 quintals of X metal and not more than 35  quintals of Y metal.  The firm can purchase the scrap from two suppliers (A and B) in unlimited quantities. The percentage of X and Y metals in terms of weight in the scarp supplied by A and B is given below.

            Metals             supplier A                   supplier B

                 X                      25%                           75%                

                  Y                      10%                           20%

The price of A’s scrap is birr 200 per quintal and that of B is birr 400 per quintal. The firm wants to determine the quantities that it should buy from the two suppliers so that total cost is minimized.