TheFinalBossOfTests:
(329 Marks)
Question 1 (19 marks)
a) Name any three advantages of mathematical modelling.
(3)
b) What is the break-even-point (BEP)?
(1)
c)Which three parameters (variables) are needed to calculate BEP?
(3)
d) What are the four properties that all Linear Programming problems have in common? (4)
e) Name only two differences between Slack and Surplus.
(4)
f) Name any four basic assumptions of linear Programming.
(4)
Question 2
(18 marks)
Question 2.1 (9 marks)
You and your friends start up a business of producing sunglasses for students. Renting the
equipment to manufacturing these sunglasses is R300. Raw materials for manufacturing one
pair of sunglasses cost R6 and you can sell these sunglasses for R12 each.
a) What will be the total revenue be if you sell 200 pairs of sunglasses? (1)
b) What will be the variables cost for 50% of the sunglasses sold? (1)
c) How many sunglasses must you sell to break even? (2)
d) What is your total revenue at BEP? (1)
e) Calculate your profit or loss for 200 sunglasses. Please indicate profit or loss. (2)
f) How many pairs of sunglasses should you sell to have a profit of R4 500.00? (2)
Question 2.2 (9 marks)
You start a business of producing caps and sell them to students at the first football game. Fixed cost
for producing these caps is R400.00. Variable cost for each cap is R3:00. You want to sell these caps
for R5:00 each. Unfortunately, there is also an additional R1000.00 that must be paid to cover the rights
for selling these caps on the premises of the university.
a) What will be the fixed cost, selling price and variable cost for each cap?
(3)
b) How many caps must be sold to break even?
(2)
c) After the first game you realize that you will sell only 500 caps. If it was possible to increase your
selling price, what should the new price for selling only 500 caps be?
(4)
Question 3
(26 marks)
Question 3.1 (20 marks)
You are an olive producer and produce two types of olives, green and black olives. To produce
one barrel of green olives it takes 5 hours of labour with 1 acre of land. To produce one barrel of
black olives it takes 2 hours of labour with 2 acres of land. You have a maximum of 250 labour
hours and a total of 150 acres of land is available. A barrel of green olives sells for R20/barrel
and a barrel of black olives sells for R30/barrel. An economic constraint forced you not to
produce more than 40 barrels of green olives.
Do the following:
a) Formulate the problem into a LP problem. (11)
b) Show a graphical solution to solve this problem. (3)
c) Calculate the best combination of barrels of green and black olives using the corner point
method. The profit for each corner point must be shown. (5)
d) How many barrels green and black olives must be produced to optimize your profit? (1)
Question 3.2 (6 marks)
Consider the following formulation:
a) Solve the problem using the graphical method (corner point method). Clearly indicate
contraints and feasible regions.
(6)
Question 4
(22 marks)
Question 4.1 (10 Marks)
You want to manufacture furniture like sofas, tables and chairs in your company. You will use
three main resources to produce your furniture which are, wood, upholstery and labor. The
resource requirements for each piece of furniture and the total resources available per week are
as follows:
Resource Requirements
Furniture
Sofas
Tables
Chairs
Total available
resources
Wood (sqr
meters)
7
5
4
2250
Upholstery
(meters)
12
0
7
1000
Labor
(hours)
6
9
5
240
You want to produce furniture on a weekly basis, store it then ship it. Your warehouse has a total
capacity of only 650 pieces of furniture. Your profit for each piece of furniture is as follows:
-
Sofa = R400.00
Table = R275.00
Chair = R190.00
1. Please formulate a linear programming model to maximize your profit for the above problem.
(10)
Question 4.2 (12 Marks)
The Kleenglass Corporation makes a dishwasher that has excellent cleaning power. This
dishwasher uses less water than most competitors, and it is extremely quiet. Orders have
been received from several retail stores for delivery at the end of each of the next three
months, as shown below. Due to limited capacity, only 300 washers can be made each
month on regular time, and the cost is R2000 each. However, an additional 20 units can be
produced per month if overtime is used, but the cost per unit goes up to R2500 each. If any
washers are left over at the end of a month, there is an additional cost of R205 per washer in
warehouse fees. Kleenglass wants to keep their production costs as low as possible.
a) Formulate the problem as an LP model.
(12)
Question 5
(30 marks)
Question 5.1 (17 marks)
You are a dietician for the local soccer team. You want to determine a balanced nutritional
lunch for all players participating in soccer practice. The following is the national nutritional
accepted requirements for a proper lunch for athletes:
Calories
- between 900 and 1500
Iron
- at least 5 MG
Fat
- no more than 50 GM’s
Protein
- at least 26 GM’s
Carbohydrates - between 30 and 60 GM’s
-
You select a menu from 7 basic food items with corresponding national nutritional value for
each food item. It is summarized as follows:
Table for food values and associated cost
Food
Item
Milk
Meat
Chicken
Fish
Beans
Spinach
Potatoes
Calories /
Kg
295
1216
394
358
128
118
279
Iron
(MG/Kg)
0.2
0.2
4.3
3.2
3.2
14.1
2.2
Fat
(GM/Kg)
16
96
9
0.5
0.8
1.4
0.5
Protein
(GM/Kg)
16
81
74
83
7
14
8
Carbs
(GM/Kg)
22
0
0
0
28
19
63
Cost/Kg
(R)
0.60
2.35
1.15
2.25
0.58
1.17
0.33
You wants to optimize your menu by meeting the nutritional requirements, while minimizing the
total cost per serving.
1.
Please formulate a linear programming model for this blending problem.
(17)
QUESTION 5.2 (13 Marks)
You as a steel plant owner received a contract to produce steel sheets for an export
customer. There are strict quality control standards to adhere to. Each steel sheet
must have the following steel content requested by the customer as given by the
table below:
Material
Minimum %
Max %
Manganese
2.1
2.3
Silicon
4.3
4.6
Carbon
5.05
5.35
To produce these steel sheets, you need to mix batches of eight different materials
to produce one metric ton of steel from which the steel sheets will be produced
from. The table below gives the mix parameters for each metric ton of steel.
Material
available
Manganese
%
Silicon %
Carbon %
Kilograms
available
Cost per
Kg (Rand)
Alloy 1
70.0
15.0
3.0
No limit
0.12
Alloy 2
55.0
30.0
1.0
300
0.13
Alloy 3
12.0
26.0
0
No limit
0.15
Iron 1
1.0
10.0
3.0
No limit
0.09
Iron 2
5.0
2.5
0
No limit
0.07
Carbide 1
0
24.0
18.0
50
0.10
Carbide 2
0
25.0
20.0
200
0.12
Carbide 3
0
23.0
25.0
100
0.09
a)
Formulate a LP solution for the above blending problem to minimize your costs.
(13)
Question 6
(39 marks)
Question 6.1 (13 marks)
A Company manufactures three outdoor products, chairs, benches, and tables.
Each product must pass through four departments before it is shipped:
- Sawing department, sanding department, assembly department, and painting
department. The time requirements to produce each product (in hours) are
summarized in the tables below.
Hours Required
Product
Sawing Sanding
Unit
Assembly Painting Profit
Chairs
1.5
1.0
2.0
1.5
R15
Benches
1.5
1.5
2.0
2.0
R10
TABLES
2.0
2.0
2.5
2.0
R20
The production time available in each department each week and the minimum weekly
production requirement to fulfill contracts are as follows:
Minimum
Capacity
Department (In
Hours)
Sawing
450
Production
Product
Level
Chairs
100
Sanding
400
Benches
50
Assembly
625
Tables
50
Painting
550
a)
You are the production manager and have the responsibility of specifying the production
levels for each product for the coming week. Formulate the above scenario as a linear
programming problem to maximize profit.
(13)
Question 6.2 (26 marks)
You produce tables and chairs at your company. Each table yields a profit of R9.00 and
each chair yields a profit of R12.00. For a production period of one week, only 10 litres of
paint and 12 litres of glue are available. Each table requires 1 litre of paint and 1 litre of
glue. Each chair requires 1 litre of paint and 2 litres of glue.
Formulate the production mix for your company as a LP problem and then solve it using
the Simplex algorithm to determine how many tables and chairs should be produced
each week to optimize your profits. (Note: Let X1 = table and X2 = chairs)
a) Formulate the LP problem in standard form.
b) Solve this LP problem through the Simplex method.
c) State the profit for this problem.
Question 7
(4)
(21)
(1)
(65 marks)
Question 7.1 (26 Marks)
A LP problem is as follows:
Maximize profit : 10X1 + 8X2
Subject to :
4X1 + 2X2 <=80
X1 + 2X2 <=50
X1, X2 >= 0
a) Formulate the LP problem in standard form.
(4)
b) Solve this LP problem through the Simplex method. Show the associated tableau
with each iteration.
(21)
c) State the profit for this problem.
(1)
Question 7.2 (9 Marks)
1) Consider the following linear programming model:
a) Convert the constraints to equalities by adding the appropriate variables.
(6)
b) Add the new variables (in question 1.a) into the objective function with the appropriate
coefficients.
(3)
Question 7.3 (6 Marks)
The partial final simplex tableau for an LP maximization problem is shown below.
a) Complete the last two rows for this tableau.
b) Describe the situation encountered here.
Cj
Solution
Mix
5
-M
(3)
(3)
X2
A1
Zj
3
X1
1
-1
5+M
5
X2
1
0
5
0
S1
2
-2
10 + 2M
0
S2
0
-1
M
Cj – Z j
-2 - M
0
-10 - 2M
-M
-M
A1
0
1
Quantity
6
2
Question 7.4 (14 Marks)
Consider the following linear programming model:
a) Find the optimal solution for this model using the simplex method.
(14)
Question 7.5 (7 Marks)
Given the following dual formulation, reconstruct the original primal problem:
(7)
Question 7.6 (3 Marks)
Consider the LP model below together with the final simplex tableau with the optimal solution:
According to the tableau:
a) If the amount of resource A changed from 64 to 65, what will the total profit be? (1)
b) If the amount of resource B changed from 96 to 97, what will the total profit be? (1)
c) It is currently profitable to produce some units of 𝑥1 with a profit per unit of 20. What
is the lowest value that this could be to allow this variable to remain in the basis?
(1)
Question 8
(36 marks)
Question 8.1 (25 marks)
A LP problem is as follows:
Minimizing cost: R0.80 X1 + R0.40 X2 + R1.20 X3 - R0.10 X4
Subject to:
X1 + 2 X2 + X3 + 5 X4 <= 150
X2 - 4 X3 + 8 X4 = 70
6 X1 + 7 X2 + 2 X3 - X4 >=120
X1, X2, X3, X4 >= 0
Do the following:
1. Change the objective function by including the additional variables.
(4)
2. Convert the constraints to equalities by adding the appropriate slack, surplus,or artificial
variables.
(7)
3. Set up the complete initial (first) simplex tableau for this problem. Do not solve.
(14)
Question 8.2 (6 marks)
a) Convert the following linear program into the simplex form:
(6)
Question 8.3 (5 marks)
Convert the following constraints and objective function (by using slack, surplus and
artificial variables) into the proper form for use in the simplex method:
Question 9
(42 marks)
Question 9.1 (20 marks)
You are importing products into the USA. These products are shipped from European ports
(Hamburg, Marseilles, Liverpool) to warehouses in the USA (Norfolk, New York, Savannah) and
from there the product is send to distribution centres (Dallas, St Louis, Chicago) to be
distributed to various stores in the USA to be sold. Supply and demand is measured in Mt.
The shipping cost (R/Mt) and the supply (Mt) from Europe to warehouses in the USA are below:
European Ports
1. Hamburg
2. Marseilles
3. Liverpool
USA City
4. Norfolk
R420
R510
R450
5. New York
R390
R590
R360
6. Savannah
R610
R470
R480
Supply (Mt)
55
78
37
The transportation cost (R/Mt) and the demand (Mt) from each USA city to the three distribution
centres in the USA is below:
Ware Houses
4. North folk
5. New York
6. Savannah
Demand (Mt)
7. Dallas
R75
R125
R68
60
Distribution
center
8. St Louis
R63
R110
R82
45
9. Chicago
R81
R95
R95
50
(a) Draw the transhipment network
(9)
(b) Formulate this transhipment problem in a LP standard format.
(11)
Question 9.2 (10 marks)
Frosty Machines manufactures snow blowers in factories located in Toronto and Detroit.
These are shipped to regional distribution centres in Chicago and Buffalo, where they
are delivered to the supply houses in New York, Philadelphia and St Louis. The figure
below illustrates the basic network representation of the situation. The shipping costs
vary as shown in the table. Forecasted demands for New York, Philadelphia and St Louis
are also seen in this table, as are the available supplies of snow blowers at the two
factories. Notice that snow blowers may not be shipped directly from Toronto or Detroit
to any of the final destinations. Frosty would like to minimize the transportation costs
associated with shipping sufficient snow blowers to meet the demands at the three
destinations while not exceeding the supply of each factory.
a) Formulate the problem as a LP model
(10)
Question 9.3 (12 marks)
The Quicktrans Courier Company specializes in low cost courier services. They normally
use trucks, but have gotten special access to two cars on a freight train. They want to
know whether it would be more cost effective to transport their cargo by train than by
truck. The two cars they have access to have not been reserved for Quicktrans, and as a
result they have limited load capacity. One car is a flatbed car that is loaded according
to mass and has 150 kg capacity still available, while the other car is loaded by volume
and still has a capacity of 100 available. All available data is presented in the table
below. The company receives their loads in composite crates and the mass and volume
of the crates are only measured once.
The packers at the company have access to a wide range of packing material and
therefore the loads can be shared between the cars without affecting the costs.
Quicktrans wants to know what the maximum income is that they can get by
transporting their cargo by train.
a) Formulate the problem as an LP model.
(12)
Question 10 ( 32 marks)
Solution
Below is the solution for a alcohol producing company. They produce two types of whisky’s, rye and
bourbon.
The goal was to maximize the profit given the constraints for this operation, producing whisky’s.
The LP formulation for the problem is as follows:
x1 = no. of gallons of rye
x2 = no. of gallons of bourbon
Objective function: Maximize Z = 3x1 + 4x2
Subject to:
x1 + x2 ≥ 400
x1 ≥ .4(x1 + x2) {0.6x1 - .04x2 ≥ 0}
x2 ≤ 250
x1 = 2x2 {x1 - 2x2 = 0}
x1 + x2 ≤ 500
x1, x2 ≥ 0
Graphical Solution is:
Sensitivity analysis computer generated results:
Variable
Value
x1
333.333
x2
166.667
Shadow
Constraint Slack/Surplus Price
c1
100.000
0.000
c2
133.333
0.000
c3
83.333
0.000
c5
0.000
3.333
Objective Coefficient Ranges
Variables
Lower
Limit
Current
Values
Upper
Limit
Allowable
Increase
Allowable
Decrease
x1
−2.000
3.000
No limit
No limit
5.000
x2
−6.000
4.000
No limit
No limit
10.000
Right Hand Side Ranges
Lower Limit
Constraints
Current
Values
Upper
Limit
Allowable
Increase
Allowable
Decrease
c1
No limit
400.000
500.000
100.000
No limit
c2
No limit
0.000
133.333
133.333
No limit
c3
166.667
250.000
No limit
No limit
83.333
c4
−250.000
0.000
500.000
500.000
250.000
c5
400.000
500.000
750.000
250.000
100.000
Using all the provided information above, answer the following questions:
a)
Give the optimal solution point with the maximum associated profit.
(2)
b) Indicate the slack and surplus variables for each constraint at the optimal solution point
and explain their meaning by showing how it is calculated.
(9)
c)
By only observing the graphical solution without consulting the computer summary.
What increase / decrease in the objective function coefficients will change the optimal
solution point? Explain your answer.
(4)
d) Identify the sensitivity ranges for the objective function coefficients and explain what
the upper and lower limits mean.
e)
(6)
How much will it be worth to the company to obtain additional production capacity?
Explain how and which constraints will be affected.
(3)
f) Identify the upper and lower boundaries for the constraints and explain the effect on the
model.
(8)