Part I: Formulate the following problems
1. a firm consists of 600 acres of land, of which 500 acres will be planted with corn, soybeans
and wheat according to these 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 LP model that will minimize planting cost.
2. a marketing manger of large savings and loan association that has branches throughout the
state would like to mail a promotional piece to prospective borrowers. Manger 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 produce in 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 LP model to maximize profit.
4. The manger 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 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 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 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 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 are 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 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 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 units 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.
Part I: Solve the following problems using 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 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
A
X
2
product
Y
Z
4
3
operating cost per hour(birr)
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 birr 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.