L P
formulation Questions
Q1
The Flair Furniture Company produces inexpensive tables and chairs. The
production process for each is similar in that both require a certain number of
labour hours in the carpentry department, and a certain number of labour hours
in the painting department. Each table takes 4 hours of carpentry work and 2
hours of painting work. Each chair requires 3 hours in carpentry time and 1 hour
in painting. During the current production period, 240 hours of carpentry time
and 100 hours of painting time are available. The marketing personnel are
confident that they can sell all the tables that are made. However, due to an
existing inventory of chairs, they want Flair to make no more than 60 new chairs.
Each table sold results in a profit contribution of $7, and each chair sold yields a
profit contribution of $5. Flair Furniture is to determine the best possible
combination of tables and chairs to manufacture in order to attain the maximum
profit. The firm would like this product mix situation formulated as an LP
problem.
a.
Formulate using LP
.
Q2
The Win Big Gambling Club promotes gambling junkets from a large midwestern city to casinos in the Bahamas. The club has budgeted up to
$8,000 per week for local advertising. The money is to be allocated among
four promotional media: tv spots, newspaper ads, and two types of radio
advertisements. Win Big’s goal is to reach the largest possible highpotential audience through the various media. The following table
presents the number of potential gamblers reached by making use of an
advertisement in each of the four media. It also provides the cost per
advertisement placed and the maximum number of ads that can be
purchased per week.
Medium
Audience
Reached
AD
Tv spot (1 minute)
5,000
Daily newspaper (full- 8500
page ad)
Radio
spot 2400
(30seconds,
primetime)
Radio spot (1minute 2800
afternoon)
Cost
Per Ad $
800
925
Per Maximum
Ads
per
Week
12
5
290
25
380
20
Win Big’s contractual arrangements require that at least 5 radio spots be placed
each week. To ensure a broad scoped promotional campaign, management also
insists that no more than $1,800 be spent on radio advertising every week.
a. Formulate using LP
Q3 Caricom Jewelers uses rubies and sapphires to produce two types of rings.
A"sweetheart" ring requires 2 rubies, 3 sapphires and 1 hour of jeweler's labour.
An engagement ring requires 3 rubies, 2 sapphires and 2 hours of jeweler's
labour. Each "sweetheart ring sells for $500, and each engagement ring sells for
$750. All rings produced by Caricom Jewelers can be sold. At present, Caricom
Jewelers has 100 rubies,120 sapphire and 70 hours of jeweler's labour. Extra
ruby can be purchased at a cost of$100 per ruby. Market demand requires that
the company produce at least 20
"sweetheart"rings and at least 25 engagement rings. The executives of Caricom
Jewelers are interested to find the combination of rings that will allow them to
maximize profits.
a What is the Objective of Caricom Jewelers?
b). List the decision variables.
c). List the constraints.
d). What is the Objective Function?
e) Formulate the model.
Q4
The Pamoch Aid Agency has to evacuate the residents of a volcanic island and
their
belongings by boat. The island’s small harbour can only handle small boats.
There are
two types of boat the agency can hire locally: the Lotka and the Soodna. A Lotka
can
take 25 passengers and 10 tons of cargo, and costs £800 per day to charter. A
Soodna
can take 40 passengers and 4 tons of cargo, and costs £1000 per day to charter.
The
agency needs capacity for at least 2000 passengers and 440 tons of cargo. How
many of each type of boat should be chartered to minimize the agency’s costs
per day, and what is the minimum daily cost?
Formulate using LP
Q5
Pianni Beverages produce two ready-mixed cocktail drinks; the
Zombie and the Skyjack. Each is a mixture of vodka, vermouth
and ginger. It takes 3 litres of vodka, 6 litres of vermouth and 1
litre of ginger to make 10 litres of Zombie, and 5 litres of vodka,
3 litres of vermouth and 2 litres of ginger to make 10 litres of
Skyjack. The company makes £15 profit per 10 litres of Zombie
and £20 profit per 10 litres of Skyjack. The maximum available
supplies per day are: 1500 litres of vodka, 1500 litres of vermouth
and 400 litres of ginger.
Formulate using LP
Q6
A company specializing in lubrication products for vintage
motors produce two blended oils, Smazka and Neftianikov.
They make a profit of £5 per litre of Smazka and £4 per litre of
Neftianikov. A litre of Smazka requires 0.4 litres of heavy oil
and 0.6 litres of light oil. A litre of Neftianikov requires 0.8
litres of heavy oil and 0.2 litres of light oil. The company has
100 litres of heavy oil and 80 litres of light oil. How many litres
of each product should they make to maximize profits and
what level of profit will they obtain?
Formulate using LP
Q7
Domar Properties plc have a site covering 20,000m2 on which
they intend to build a mixed estate of 2- and 4-bedroom
houses. The 2-bedroom houses will each occupy a plot of 60m2
and will be sold at a profit of £15,000. On average each will
house 2 people owning 1.2 cars. The 4-bedroom houses will
each occupy a plot of 300m2 and will be sold at a profit of
£50,000. On average each will house 5 people owning between
them 1.5 cars.
They anticipate that roads, verges, play areas and other communal
facilities will occupy 40% of the total site. Water and
sewage considerations mean that the total number of residents
on the estate should not exceed 250. The local authority has
told Domar that the road entrance is unsuitable for more than
120 cars to be based on the estate.
How many houses of each type should Domar build in order
to maximize their profit, and how much profit should they
expect if they did so?
Formulate using LP
Q8
The Ooze Haircraft Corporation make two brands of hair treatment:
Volossy, which is produced under licence, and its own
products Sedina. The company is in dispute with its supplier of
colourant and solidifier as a result of which the supplier is no
longer taking their orders. An alternative source of supply cannot
be arranged for a month and Ooze must plan production
for the month with their stock of 480 litres of colourant and
900 litres of solidifier.
A bottle of Volossy requires 3 millilitres of colourant and 9
millilitres of solidifier. A bottle of Sedina requires 4 millilitres
of colourant and 6 millilitres of solidifier. The licensing agreement
restricts production of Volossy to no more than 60,000
bottles a month and commitments to existing customers mean
that at least 20,000 bottles of Sedina must be produced.
The company makes £0.40 profit per bottle of Volossy and
£0.20 profit per bottle of Sedina.
Formulate using LP
Q9
Roo Satellite Systems manufacture two types of broadcast
receiving system, the ‘Soap Dish’ and the ‘Houston’. The production
process for each includes wiring, assembly and inspection,
the times in hours required in each section are:
Soap Dish Houston
Wiring
3
3
Assembly
1.2
3
Inspection
0.4
0.5
Each month the company has 4500 hours of wiring labour,
3000 hours of assembly labour and 600 hours of inspection
labour available. They make a profit of £16 from each Soap
Dish and £12 from each Houston. How many of each system
should they make per month in order to maximize their profit,
and what is the maximum profit?
Formulate using LP
Q10
Tapachki & Sons produce handmade clogs in Lancashire. They
produce two types of clog, the Nelson and the Oldham. The
profits per pair are £10 and £12 respectively. The company
employs 2 leather-cutters, 4 sole-makers and 3 stitchers. Each
works a 40-hour week. The amount of labour time in hours
required for a pair of each type of clog is:
Nelson Oldham
Leather-cutting
0.40
0.25
Sole-making
1.00
0.80
Stitching
0.80
0.50
Work out the optimal weekly production mix for the company
to make the highest level of profit possible and state the maximum
weekly profit they can expect.
Formulate using LP
Q11
The Chic Sheet Company have to plan production for the next
week. The firm produces two types of sheet, standard and luxury,
in packs that are sold to retailers for £80 and £145 respectively.
The costs of materials are £25 per pack of standard sheets
and £45 per pack of luxury sheets. These materials are available
in unlimited amounts.
There are three production departments, cutting, machining
and packing. The workforce includes 10 cutters, 150 machinists
and 40 packers. The labour required per pack for each product
and the labour charges for each department are:
The factory works a 37.5-hour week.
(a) Work out the profit per pack for each product.
(b) Find the production mix that will enable the company to
maximize its profit for the week, and determine the level
of profit that would result.
Formulate using LP
Q12
Wooffer & Tweeter make speakers for the specialist audio market.
Following the death of the founder of their rival, they have
acquired the assets of his company, the Croaker Can & Cab Co.
These assets consist of raw materials used to make the two types
of speaker produced by Croaker, the Cosmic and the Celestial.
The assets and their value are:
Wooffer & Tweeter plan to use these materials to make a final
consignment of Croaker speakers. The production requirements
for the products are:
An unlimited supply of labour is available at a cost of £10
per hour
Croaker had an outstanding order for 200 pairs of Cosmic
speakers for the Asteroid retail group, and have agreed to fulfill
this order as they are anxious to do further business with
Asteroid.
(a) If Cosmic speakers sell for £100 a pair and Celestial speakers
sell for £91 a pair, Work out the variable cost for each product and subtract it
from the selling price to find the profit from each product.
Formulate using LP
Q13
Kolbasnik the pig farmer needs to add 36 kg of protein and
10 kg of vitamin to the pig feed. There are two possible additives.
The cost and protein and vitamin content of each are:
Formulate using LP
Q14
Stroika Concrete makes bags of easy-use concrete for the DIY
market. The company mixes their concrete using two ingredients,
‘Great Grit’ and ‘A1 Aggregate’. Great Grit costs £1 per
kilogram and consists of 20% fine sand, 40% coarse sand and
40% gravel. A1 Aggregate costs £0.75 per kilogram and consists
of 10% fine sand, 50% coarse sand and 40% gravel. Each bag of
Stroika concrete must contain at least 2 kg of fine sand, 2 kg of
coarse sand and 6 kg of gravel.
Formulate using LP
Q15 Ruth Madoff has recently inherited one hundred thousand dollars. She
wishes to invest this money to ensure that she has future income. Sue Smith,
who is a stockbroker, has identified three stocks which are acceptable
investments for Ms. Madoff. The stock of the BAC Corporation is expected to
return 12% per year, the stock of the MPP Company will return 12% per year m
while the stock of JJJ Inc. will generate a 15% annual return. Ms Madoff
requires that her investments be in a diversified portfolio ( to minimize risk) and
as such requires that no more than fifty percent of the portfolio be invested in
one stock. Further, no more than eighty percent of the portfolio can be invested
in any two stocks. If Ms. Madoff wishes to invest all of her money, formulate this
as a LP problem
Q 16 (a) The CWD Brokerage firm has just been instructed that by one of its
clients to invest $250,000 for her – money recently obtained through the sale of
land holdings in Ohio. The client has a good deal of trust in the investment
house, but she also has her own ides about the distribution of the funds being
invested. In particular, she requests that the firm select whatever stocks and
bonds they believe are well rated, but within the following guidelines:
1. Municipal bonds should comprise at least 20% of the investment.
2. At least 40% of the funds should be placed in a combination of
electronic firms, aerospace firms and drugs manufacturers.
3. No more than 50% of the amount invested in municipal bonds
should be placed in a high-risk, high –yielding nursing home stock.
Subject to these restraints, the client’s goal is to maximise projected return on
investments. The analysts at CWD , aware of these guidelines, prepared a list of
quality stocks and bonds and their corresponding rates of return.
Investment
Los Angeles Municipal Bonds
Thompson Electronics, Inc.
United Aerospace Corp.
Palmer Drugs
Happy Days Nursing Home
Projected Rate of Return (%)
5.3
6.8
4.9
8.4
11.8
Formulate this portfolio selection problem using linear programming
Q17
The John Geer Gear Company is planning a production run for the next week.
The times and cost for each gear to go through each process (forming,
hardening, deburring) is shown. It is also possible to outsource the gears from
Gary’s Gears (John Geer pays Gary for gears made), located down the street.
Gary can supply a maximum of 300 units of each type of gear next week. They
anticipate the maximum demand for four types of gears, as shown in the table
below; John Geer is not making to contract so will not have any use for surplus
gears. What should the production and/or outsource plan be for the next week to
maximise the profits?
Table 5: Production, Demand, Revenue and Cost Data for Jon Geer’s Gears
Gear
Type
Deman
d
Revenue
per gear
Forming
Hours
Hardening
Hours
Deburrin
g Hours
Out Source
Cost
G3
400
$12.50
0.30
0.20
0.30
$7.50
G4
500
$15.60
0.40
0.30
0.30
$8.50
G5
450
$17.40
0.35
0.25
0.35
$7.75
G6
600
$19.30
0.45
0.23
0.25
$7.55
500
300
300
$9.00
$8.00
$7.50
Hours Available
Cost Per Hour
You are required to formulate a linear programming model.
a)
Define the decision variables
b)
Generate the objective function using your variables from above
c)
Generate the process constraints
d)
Generate the demand constraints
e)
Generate the outsource constraints
f)
Generate the non-negative constraint
Q18
Saddam Hussein Inc. produces two steak sauces, Spicy Diablo and Mild Red
Baron. These sauces are both mace by blending two ingredients, A and B. A
certain level of flexibility is permitted in the formulae for these products. The
allowable percentages, along with revenue and cost data, are given in the
following table. Up to 40 quarts of A and 30 quarts of B could be purchased.
Saddam can sell as much of these sauces as it produces. Formulate an LP
whose objective is to maximise the net profit from the sale of the sauces.
Sauce Ingredient A
Spicy
At least 25%
Diablo
Red
At most 75%
Baron
Cost
$1.60
per
Quart
Ingredient B
At least 50%
Sales Price per quart (s)
3.35
No explicit maximum
percentage
$2.59
2.85
Q19
(Media Selection) The advertising director for Diversy Paint and Supply, a
chain
of four retail stores on Chicago's North Side, is considering two media
possibilities. One
plan is for a series of half-page ads in the Sunday Chicago Tribune newspaper
and the
other is for advertising time on Chicago TV. The stores are expanding their line of
do-it yourself tools, and the advertising director is interested in an exposure level
of at least
40% within the city's neighbourhoods and 60% in northwest suburban areas.
The TV viewing time under consideration has an exposure rating per spot of 5%
in city
homes and 3% in the northwest suburbs. The Sunday newspaper has
corresponding
exposure rates of 4% and 3% per ad. The cost of a half-page Tribune ad is $925;
a
television spot costs $2,000. Diversy would like to select the least costly
advertising
strategy that would meet desired exposure levels.
a). What is the objective of Diversy Paint and Supply?
b). What are the decision variables?
c). What are the constraints?
d) Formulate the model using LP
Q20 A client of an Investment Broker has asked that the Broker make
recommendations concerning an investment portfolio for the client.
Specifically, the client wants a diversified portfolio and wishes to invest a total
amount of $300,000.
After some investigation, the Broker has generated a list of four (4) stocks
and four (4) bonds which he believes are consistent with the wishes of the
client. He has tabulated some of the data concerning these eight options as
shown below:
Investment Option
(measured in
Annual dividends
per $1000
Expected long-term
yield per $1000
Risk
dollars per
$1000
invested)
Stock
Stock
Stock
Stock
Bond
Bond
Bond
Bond
#1
#2
#3
#4
#1
#2
#3
#4
$70
$60
$75
$80
$30
$20
$10
$10
$300
$320
$260
$250
$200
$180
$150
$150
$14
$18
$16
$15
$6
$2
$1
$1
Assume that the eight variables (x1 through x8) have been defined as the
amount (in thousands of dollars) to invest in each of the eight options.
(a)
Write the appropriate objective function if the objective is to:
(i)
Minimise the total risk.
(ii)
Maximise the annual dividends.
(iii)
Maximise the long term yield.
(b)
Suppose that the investor wanted to ensure that the entire total of
$300,000 was invested in some combination of eight options. Write
the appropriate constraint.
(c)
Suppose that the client has specified that his portfolio should
produce at least $12,000 in annual dividends and at least $75,000 in longterm yield.
Write the appropriate constraints for these two
specifications.
(d)
Suppose that the investor wanted to ensure that the total amount
invested
did not exceed the $300,000 available for investment. Thus,
there was
no requirement that all the funds invested at this time;
however, it was possible to do so. Write the appropriate constraint.
Q 21
The Employee credit Union at Caribbean University is planning the allocation of
funds for the coming year. The credit union makes four types of loans to its
members. In addition, the
credit union invests in risk-free securities to
stabilise income. The various revenue-producing
investments to together
with annual rates of return are as follows:
Type of Loan/Investment
Annual Rate of Return
(%)
Automobile loans
8
Furniture loans
10
Other secured loans
11
Signature loans
12
Risk-free securities
9
The credit union will have $2000,000 available for investment during the coming
year. State laws and credit union policies impose the following restrictions on the
composition of loans and investments:
•
•
•
•
Risk-free securities may not exceed 30% of the total funds available for
investment.
Signature loans may not exceed 10% of the funds invested in all loans
(automobile, furniture, and other secured and signature loans)
Furniture loans plus other secured loans may often not exceed
automobile loans.
Other secured loans plus signature loans may not exceed the funds
invested in risk-free securities.
Required :
Formulate a linear programming model to show how the
$2,000,000 should be allocated to each of the loan investment alternatives
to maximise total annual return.
(10 marks)
Q 22
a. Lawns Unlimited is a lawn care and maintenance company. One of its
services is to seed new lawns as well as bare or damaged areas in
established lawns. The company uses three basic grass seed mixes it
calls Home 1, Home 2, and Commercial 3. It uses three kinds of grass
seedtall fescue, mustang fescue, and bluegrass. The requirements for
each grass mix are as follows:
Mix
Home 1
Mix Requirements
No more than 50% tall fescue
At least 20% mustang fescue
Home 2
At least 30% bluegrass
At least 30% mustang fescue
No more than 20% tall fescue
Commercial 3 At least 50% but no more than 70% tall fescue
At least 10% bluegrass
b.
c. The company believes it needs to have at least 1,200 pounds of Home 1
mix, 900 pounds of Home 2 mix, and 2,400 pounds of Commercial 3 seed
mix on hand. A pound of tall fescue costs the company $1.70, a pound of
mustang fescue costs $2.80, and a pound of bluegrass costs $3.25. The
company wants to know how many pounds of each type of grass seed to
purchase to minimize cost.
d. Formulate a linear programming model for this problem
e.
f. .
Solve this model by using the computer
Q 23
Fifth Avenue Industries, a nationally known manufacturer of menswear, produces
four varieties of ties. One is an expensive, all-silk tie, one is an all-polyester tie,
and two are blends of polyester and cotton. The following table illustrates the
cost and availability (per monthly production period) of the three materials used
in the production process:
Material
Cost Per Yards ($)
Silk
Polyester
Cotton
21
6
9
Material Available Per
Month Yards
800
3,000
1,600
The firm has fixed contracts with several major department store chains to supply
ties. The contracts require that Fifth Avenue Industries supply a minimum
quantity of each tie but allow for a larger demand if Fifth avenue chooses to meet
that demand. (Most of the ties are not shipped with the name Fifth Avenue on
their labels, incidentally, but with private stick” labels supplied by the stores.)
The table below summarizes the contract demand for each of the fur styles of
ties, the selling price per tie, and the fabric requirements of each variety.
Variety
Tie
of Selling
Monthly
Price Per Contract
Tie ($)
Minimum
6.70
6,000
3.55
10,000
All Silk
All
Polyester
Poly-cotton
blend 1
Poly-cotton
blend 1
Monthly
Demand
7,000
14,000
Material
Required per
Tie (yards)
0.125
0.08
4.31
13,000
16,000
0.10
4.81
6,000
8,500
0.10
Material
Requirements
100% silk
100%
polyester
50% polyester
– 50% cotton
30% polyester
– 70% cotton
Formulate the LP and Solve using Solver
Q 24
Wintel must decide on the quantity of each of 5 different modules to assemble to
maximize profit. These are made from Type A & Type B Chips and Circuit
Boards. The time required to assemble each module is given along with the per
unit requirements for each module, as well as the profit per module in the table
below. These additional considerations are included in the plan:
1. [ORDER] At least 200 regular modules and at least 100 small modules must
be made according to an order received.
2. Since the large and extra-large modules are usually ordered in groups, with on
average at least 2 large modules for every extra-large one, the firm wants this
condition to be ensured.
3. [MIXTURE] The miniature module quantity cannot exceed half of the combined
total of the other 4 modules.
Module
Type A
Profit $
Extra-large
28
58
Large
24
43
Regular
18
25
Small
12
17
Miniature
5
28
Available
10,000
Type B
52
Circuit Boards
Assembly Hours
25
48
1.50
15
1.25
40
10
1.00
60
5
0.75
75
25,000
1
1.50
50,000
2,000
You are required to formulate a linear programming model
Linear Programming Sensitivity Questions
Q25:
Anderson Electronics
Anderson Electronics is considering the production of four potential products:
VCRs, Stereos, TVs, and DVD players. Below is Solver’s Answer and
Sensitivity Report for Anderson Electronics’ optimal solution. Answer the
following questions based on these reports.
Answer Report
Target Cell
Cell
$F$8
Name
Profit
Original Value
$0
Adjustable Cells
Cells
Name
Original Value
$b$5
Solution Value VCR 0
$C$5
Solution
Value 0
Stereo
Final Value
$69,400
Final Value
0
380
$D$5
$E$5
Solution Value TV
Solution Value DVD
Constraints
Cell
Name
$F$10
Electronic comp
$F$11
Non-Electronic
comp
$F$12
Assembly Time
0
0
0
1060
Cell Value
4700
3940
Status
Binding
Not Binding
Slack
0
560
2500
Binding
0
Sensitivity Report
Adjustable Cells
Cell
Name
$b$5
Solution
Value
VCR
Solution
Value
Stereo
Solution
Value TV
Solution
Value
DVD
$C$5
$D$5
$E$5
Constraints
Cell
Name
Final
Value
0
Reduced
Cost
-1
Objective Allowable Allowable
Coefficient Increase Decrease
29
1
1E+20
380
0
32
40
1.67
0
-8
72
8
1E+30
1060
0
54
10
5
Final
Value
Electronic 4700
comp
Non3940
Electronic
comp
Assembly 2500
Time
$F$10
$F$11
$F$12
Shadow
Price
2
Constraints Allowable Allowable
R.H. Side
Increase Decrease
4700
2800
950
0
4500
1E+30
560
24
2500
466.67
1325
Questions
What is the Objective Function Value?
a.
What is the impact on profit if we could increase the supply of electronic
components by 400 units?
What would happen if we could increase the supply of electronic
components by 4,000 units?
What would happen if the supplier of these units wants $8 per unit, rather
than the current cost of $7?
Assume we have an opportunity to get 250 additional hours of assembly
time. However, this time will cost us time and a half (i.e $15 per hour
rather than the current $10 per hour). Should we take it?
If we force the production of VCRs, what would be the impact on total
profit? Alternatively, how profitable must VCRs become before Anderson
considers producing them?
b.
c.
d.
e.
f.
Assume that there is some uncertainty in the price for DVD players. For
what range of prices will the current production be optimal? If DVD players
sold for $106, what would be Anderson’s new total profit? (Assume the
original price for DVD was $110).
Q26:
Burn- Off Diet Drink
Burn off, a manufacturer of diet drinks, is planning to introduce a miracle
drink that magically burn the fat away. Below is Burn-Off’s optimal
solution for the mixed ingredients at a minimum cost per daily dose. Use
Burn-off Diet Drink’s Answer and Sensitivity Reports from the LP model to
answer the following questions.
Answer Report
Target Cell
Cell
$F$6
Name
Cost (cents)
Adjustable Cells
Cell
Name
$B$5
Number
of
Ounces
of
Ingredient A
$C$5
Number of ounces
of Ingredient B
$D$5
Number of ounces
of Ingredient C
Original Value
0
Final Value
130.625
Original Value
0
Final Value
10.250
0
0
0
4.125
$E$5
Number of ounces 0
of Ingredient D
Constraints
Cell
Name
$F$11
Chemical Z
$F$8
Daily
Dosage
$F$9
Chemical X
$F$10
Chemical Y
21.625
Cell Value
1050
36
Formula
$F$11<= H$11
$F$8>=$H$8
Status
Binding
Binding
Slack
0
0
280
205.750
$F$9>=$H$9
Binding
$F$10>=$H$10 Not
Binding
0
5.750
Sensitivity Report
Adjustable Cells
Cell
Name
$B$5
$C$5
$D$5
$E$5
Final
Value
Number of 10.250
Ounces of
Ingredient A
Number of 0
ounces of
Ingredient B
Number of 4.125
ounces of
Ingredient C
Number of 21.625
ounces of
Ingredient D
Constraints
Cell
Name
$F$11
$F$8
Chemical
Z
Daily
Dosage
Reduced Objective Allowable Allowable
Cost
Coefficient Increase Decrease
0
4
3.5
2.5
5.688
7
1E+30
5.688
0
6
15
2.333
0
3
3.8
1E+30
Final
Value
1050
Shadow
Price
-0.238
Constraints Allowable Allowable
R.H. Side
Increase Decrease
1050
47.143
346
36
3.750
36
16.5
1.278
$F$9
$F$10
Chemical
X
Chemical
Y
280
0.875
280
41
11
205.750
0
200
5.750
1E+30
Questions
a. What is the impact on cost if Burn-Off insists on using one ounce of ingredient
B to make the drink?
b. There is some uncertainty in the cost of ingredient C. How sensitive is the
current optimal solution to this cost?
c. What the shadow prices for Chemical X and Chemical Z imply in this
problem?
d. Burn-off can decrease the minimum requirement for chemical X by 5 (from
280 to 275) provided the maximum limit allowed for chemical Z is reduced to
1,000 units (that is, reduced by 50 units). Is this trade off cost-effective for
Burn-Off to implement?
Q27
(ii) Consider the following linear programming and computer solution:
Let Xi = Number of units of service i to provide
i = 1, 2, 3,4
Minimise cost 10X1+12X2+9X3+11X4
Subject to
2X1 + 4X2 + X3 + 5X4>= 30 (Req. A.)
5X1 + 6X2 + 2X3-X4 >=40 (Req. B)
12X1 + 7X3 +8X4 >=75 ( Req. C)
Answer report
Target Cell
(Max)
Cell
Name
$B$15
Total cost
Original Value Final Value
$
$
103.1356
Adjustable
Cells
Cell
$B$11
$C$11
$D$11
$E$1!
Original Value Final Value
0
5.063560
0
2.7436440
.00000
1.779661
Name
X1
X2
X3
X4
Constraints
Cell
$F$7
$F$8
$F$9
Name
Req A
ReQ B
Req C
Cell Value
Formula
30$F$7>=$H$7
40$F$8>=$H$8
75$F$9>=$H$9
Status
Binding
Binding
Binding
Slack
0
0
0
Sensitivity report
Adjustable
Cells
Cell
$B$11
$C$11
$D$11
$E$1!
Name
X1
X2
X3
X4
Final Reduced Objective AllowableAllowable
Value
Cost Coefficient Increase Decrease
5.063560
0
10 7.653543 3.05884
2.7436440
0
12 3.851852
7.2727
.000004.118644
9 infinity
4.118644
1.7796610.0000
116.66667 13.0000
Constraints
Cell
$F$7
$F$8
$F$9
Name
Req A
ReQ B
Req C
Final
Shadow Constraint AllowableAllowable
Value
Price R.H. Side Increase Decrease
30-1.983051
30 49.7916 11.66667
40-0.677966
40 17.5000 29.43182
75 -.220339
75 47.9629 70.29412
(a) Identify the active constraints.
(b) What is the minimum cost?
(c) Will there be a surplus of any of the three requirements? If so, which one(s),
and how much excess will be provided?
(d) Suppose that the manager of the firm were able to change the cost of service
#2 from $12/unit to $10/unit. What would happen to the optimal solution?
(e) What would happen to the minimum cost if the first decided to provide one
unit of service #3?
(f) If the cost of service #4 were increased from $11/unit to $18/unit, what would
happen to the optimal solution?
g) What would be the effect on the optimal solution if there is a one unit
increase in Requirement A.
h) Is the solution degenerate?
i) Do we have alternative optima?
Q 28
ii) Consider the following linear programming and computer solution:
Let Xi = number of level i staff personnel to hire a new facility i = 1, 2,3
Minimise hourly cost
Subject to
12X1 + 18X2 + 24X3
2X1-X2 ≥ 0 (Balance)
X1+X2+X3
≥1000 ( State
regulation)
X1 ≥ 400 (Type 1 needed)
X2 ≥ 400 (Type 2 needed)
X3 ≥ 100 (Type 3 needed)
X2 ≤500 (Type 2 available)
X1 X2, X3 ≥ 0
Target Cell
(Max)
Cell
$B$15
Adjustable
Cells
Cell
$B$11
$C$11
Constraints
Cell
$F$7
$F$8
$F$9
$F$10
$F$11
$F$12
Name
Total cost
Name
X1
X2
X3
Name
BALANCE
State Regulation
Type 1 needed
Type 2 needed
Type 3 needed
Type 2 available
Original Value Final Value
$
15600
Original Value Final Value
0
500
0
400
0
100
Cell Value
Formula
Status
$F$7>$H$7
$F$8>$H$8 binding
$F$9>$H$9
Slack/surplus
600
0
100
0
0
100
Sensitivity report
Adjustable Cells
Cell
$B$11
$C$11
$D$11
$E$1!
Name
X1
X2
X3
Final Reduced Objective Allowable Allowable
Value
Cost Coefficient Increase Decrease
500
0
12
0
12
400
0
18
infinity
6
100
24infinity
12
Constraints
Cell
$F$7
$F$8
$F$9
$F$10
$F$11
$F$12
Name
BALANCE
State Regulation
Type 1 needed
Type 2 needed
Type 3 needed
Type 2 available
Final
Value
Shadow Constraint Allowable Allowable
Price R.H. Side Increase Decrease
0
0
600
Infinity
0
-12 INFINITY
100
1000
0
0
100
Infinity
400
-6
100
400
400
-12
100
100
100
0
0
infinity
100
500
(a) Identify the active or binding constraints
(b) What is the minimum hourly cost?
(c) Will there be a surplus over and above the basic need of each level of
staff personnel? If so, which one(s) and how many excess personnel will
there be of this type(s)?
(d) Suppose that the manager of the firm were able to reduce the cost for
level 2 personnel from $18/hour to $12/hour. How many personnel of
each type will now be hired?
(e) What would happen to the minimum cost if State Regulation were to
change from 1000 to 999?
.
Q 29
The Monet Company produces four types of picture frames. The four types off
frames differ with respect to size, shape and material used. Each type requires a
certain amount of skilled labour, metal and glass (as shown in the Table
below). This table also lists the profit Monet makes from each type of frame,
where the profit is the selling price minus the cost of labour and materials.
During the coming week Monet has 4,000 hours of skilled labour, 6,000 ounces
of metal, and 10,000 ounces of glass (of the same thickness) available. Also,
market constraints are such that it is impossible to sell more than 1,000 type 1
frames, 2,000 type 2 frames, 500 type three frames, and 1000 type four frames.
The company wants to maximize its weekly profits.
Data for Monet Picture Frame
Frame 1
Frame 2
Frame 3
Frame 4
Skilled
Labour
2
1
3
2
Metal
Glass
4
2
1
2
Profit
6
2
1
2
$6
$2
$4
$3
Product Mix Problem with Optimal Solution
Input data
1
2
Frame Type
3
4
Labour hrs. 2
per frame
Metal (oz.) 4
per frame
Glass (oz.) 6
per frame
1
3
2
2
Frame
1
type
Profit Per $6.00
Frame
2
Total
used
4,000
≤
Total
Available
4,000
1
2
6,000
≤
6,000
1
2
8,000
≤
10,000
2
3
4
Total profit
$2.00
$4.00
$3.00
$9,200
Production Plan
Frame Type
Frames
1
1000
2
800
3
400
4
0
Produced
Maximum
sales
≤
1000
≤
2000
≤
500
≤
1000
THE MONET COMPANY
Microsoft Excel 9.0 Sensitivity Report
Adjustable Cells
Cell
$B$15
$C$15
$D$15
$E$15
Name
Frames Type 1
Produced
Frames Type 2
Produced
Frames Type 3
Produced
Frames Type 4
Produced
Final
Value
Reduced Objective Allowable Allowable
Cost Coefficient Increase Decrease
1000
0
6
1E+30
2
800
0
2
1
0.25
400
0
4
2
0.5
0
-0.2
3
0.2
1E+30
Constraints
Cell
$F$6
$F$7
$F$8
Name
Labour hrs per frame
total used
Metal (oz) per frame
total used
Glass (oz) per frame
total used
Final
Value
Shadow Constraint Allowable Allowable
Price R.H. Side
Increase Decrease
4,000
1.2
4,000
250
1,000
6,000
0.4
6,000
2000
500
8,000
0
10,000
1E+30
2,000
QUESTIONS
a) What is the optimal profit?(1 mark)
b) What is the optimal solution for the Monet Company?(1 mark)
c) If the price of Frame type 1 increases to $20, what would happen (in
totality)?(2 marks)
d) One of the company’s employees is a thief. He stole 1000 ounces (ozs.)
of metal. What would be the result of this act?(1mark)
e) Hurricane Ivan recently wreaked havoc on the company’s operations
resulting in a loss of 500 ounces (ozs.) of glass. What would be the
effect? Explain.(1mark)
f) A company that runs a similar operation elsewhere is closing down, and
has offered Monet 800 ounces of metal. Evaluate the overall effect.(1
mark)
g) The type 4 frames have suddenly become very fashionable. Under what
circumstances (conditions) should the Monet Company now produce this
type?(1mark)
h) If the profitability of the type two frames increases by $2, what effect would
it have on profits?(1mark)
i) There is a general strike at the plant resulting in the loss of 500 hours of
labour. What effect, if any will this have on the company’s operations?(1
mark)
j) Is the solution degenerate?
k) i) Do we have alternative optima?
l)
Q 30 . Lawless and Greedy (Course Guide) Lawless and Greedy (Activity 4.2 in
text. Pg. 42
Q 31 Antigua Aggregate see question below
a) What is the optimal profit'
b) what is the optimal solution?
c) A rival manufacturer is closing down its business and has offered Antigua
Aggregates2.000 kg of rocks free of cost Should the company acquire this?
Why?
d) The Government is unhappy with the amount of sand being stored at the
company's
warehouse, citing environmental concern. The new guidelines now stipulate that
the sand bin should be filled with 1,500 kg. What effect will this have on the
Company? Explain fully.
e) There is a boom on in the building industry. As a result the demand for Grade
A mixture has increased, resulting in a price increase of $500 per bag. What
effect will this have on the optimal solution?
f) What would be the effect of producing one bag of Grade B mixture?
g) Under what conditions would it be feasible to produce any Grade B?
h) Does the problem have alternative optima? Explain to receive credit.
i) Is the solution degenerate? Explain.