BASIC PROBLEMS on LINEAR PROGRAMMING-FORMULATION

1. A plant produces 4 types of chemicals for industrial use. Each of these products undergoes three
processes. The daily capacities for each of the processes together with the usage rates of each product are
presented below. Also presented are the daily demands/committed volume for each product as well as their unit
contribution margins. As Chemical A and Chemical B are considered restricted products by the government, the
plant is only allowed to produce a total volume of 120 units per day for the two products. Formulate an LP model
(objective function & constraints) that can be employed to solve for the optimal ingredient mix.

Product A B C D Capacity
Contribution per unit 37 42 36 48 n/a
Process 1 3 6 5 7 2,000
Process 2 8 9 7 10 3,000
Process 3 17 19 18
22 5,000 Demand <=80 <=90
=>90 <=60 n/a

2. A manufacturer of microcomputers produces four models: Portable, Studen, Office and Network. The
profit per unit of each of these four models is $500, $350, $700, and 1000, respectively. More information is
presented below:

_ Item Portable Student Office Network Total _

Labor (hrs/unit) 5 5 6 8 4,000 hrs./week
Chassis (unit/unit) 1 1 1 1 400 units/week
Disk Drive (unit/unit) 2 1 2 1 300 units/week
Hard Disk (unit/unit) 0 0 0 1 20 units/week
Memory Chip (unit/unit) 16 8 32 64 22,000 units/week
Circuit Bds (unit/unit) 1 1 2 4 10,000 units/week

Formulate LP to maximize profit. How many of each model should be produced per week ?

3. St. Michael Beer has two (2) brewing plants that need to deliver to three (3) Warehouses. The capacities of the
said plants & requirements of the said warehouses together with the cost of delivering each truckload of beer
are presented below. Formulate an LP transportation model that would minimize the cost of transporting beer
from the plants to the warehouses.

Cost of transport of 1 truckload of beer from each Plant to each Warehouse:

Warehouse A Warehouse B Warehouse C

Plant 1 50 75 90
Plant 2 95 80 60

Capacities: Plant 1 = 80 truckloads/day , Plant 2 = 70 truckloads/day

Demand: Project A = 50 truckloads/day, Project B = 60 truckloads/day, Project C = 55
truckloads/day Let xij = delivery from plant i to project j in cubic meter
4. A fund manager has P 10 million to invest in treasury bills, bonds, private placements and in risky venture.
Relevant information about these various investments are presented below:

Investment Yield rate p.a. Maturity Risk rating

Treasury Bill 6% 3 months 1
Bonds 8% 5 years 2
Private Placement 12% 6 months 4
Risky Venture 20% 3 years 8

Guidelines given by the fund manager were:

a. 50% of the investment must have a maturity below one year
b. Investment in Risky venture should not exceed 20% of funds invested.
c. At least 30% should be invested in treasury bills.
d. Total investments in items risk rating above 3 should not exceed 40% of total funds invested.

Formulate an LP model that would optimize return for this portfolio where X1 = investment in Treasury bills, X2 =
investment in bonds, X3 = investment in private placement, X4 = investment in risky venture

5. A company manufactures products A, B, C, D, & E. Contribution margin per unit for each product are 12, 18, 9,
11, and 4, respectively. Consumption rates of these Products for Raw Material 1 are 2, 3, 2, 2, and 1 while
those for Raw Material 2 are 3, 5, 4, 3, and 3, also respectively. There are 1,000 units of Raw Material 1 and
2,000 units of Raw Material 2. Volume demands for products A, B , C, and D are 400, 500, 300, and 280 units
while committed delivery for Product E is at least 200 units. Product A & B both undergo a special
manufacturing Process A that has a gross capacity of 450 units (i.e. A & B Units put together). By the same
token, Products C & D go through a Process B that has a gross capacity of 290 units.

Formulate a product-mix LP model that will optimize contribution margin for the company

6. A transport company has two types of trucks, Type A and Type B. Type A has a refrigerated capacity of 20 m3
and a non-refrigerated capacity of 40 m3 while Type B has the same overall volume with equal sections for
refrigerated and non-refrigerated stock. A grocer needs to hire trucks for the transport of 3,000 m3 of
refrigerated stock and 4,000 m3 of non-refrigerated stock. The cost per kilometer of a Type A is $30 and $40
for Type B. How many trucks of each type should the grocer rent to achieve the minimum total cost?

7. A school is preparing a trip for 400 students. The company who is providing the transportation has 10 buses of
50 seats each and 8 buses of 40 seats, but only has 9 drivers available. The rental cost for a large bus is $800
and $600 for the small bus. Calculate how many buses of each type should be used for the trip for the least
possible cost.

8. A store wants to liquidate 200 of its shirts and 100 pairs of pants from last season. They have decided to put
together two offers, A and B. Offer A is a package of one shirt and a pair of pants which will sell for $30. Offer
B is a package of three shirts and a pair of pants, which will sell for $50. The store does not want to sell less
 than 20 packages of Offer A and less than 10 of Offer B. How many packages of each do they have to sell to
maximize the money generated from the promotion?

9. A farmer has 10 acres to plant in wheat and rye. He has to plant at least 7 acres. However, he has only $1,200
to spend and each acre of wheat costs $200 to plant and each acre of rye costs $100 to plant. Moreover, the
farmer has to get the planting done in 12 hours and it takes an hour to plan an acre of wheat and 2 hours to
plan an acre of rye. If the profit is $500 per acre of wheat and $300 per acre of rye how many acres of each
should be planted to maximize profits?

10. A publisher has orders for 600 copies of a certain text from San Francisco and 400 copies from Sacramento.
The company has 700 copies in a warehouse in Novato and 800 copies in a warehouse in Lodi. It costs $5 to
ship a text from Novato to San Francisco, but it costs $10 to ship it to Sacramento. It costs $15 to ship a text
from Lodi to San Francisco, but it costs $4 to ship it from Lodi to Sacramento. How many copies should the
company ship from each warehouse to San Francisco and Sacramento to fill the order at the least cost?