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, Solution Manual – Optimization Modelling
CONTENT
Page#
Chapter 1 5
Chapter 2 8
Chapter 3 10
Chapter 4 19
Chapter 5 32
Chapter 6 41
Chapter 7 45
Chapter 10 49
Chapter 11 58
Chapter 12 62
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, Solution Manual – Optimization Modelling
Chapter 1
Solution to Exercises
1.1 Jenny will run an ice cream stand in the coming week-long multicultural
event. She believes the fixed cost per day of running the stand is $60.
Her best guess is that she can sell up to 250 ice creams per day at $1.50
per ice cream. The cost of each ice cream is $0.85. Find an expression
for the daily profit, and hence find the breakeven point (no profit–no
loss point).
Solution:
Suppose x the number of ice creams Jenny can sell in a day.
The cost of x ice creams ($) = 0.85x
Jenny’s cost per day ($) = 60 + 0.85x
Daily revenue from ice cream sale ($) = 1.50x
Expression for daily profit ($) P = 1.50x – (60 + 0.85x) = 0.65x – 60
At breakeven point, 0.65x – 60 = 0
So, x = 60/0.65 = 92.31 ice creams
1.2 The total cost of producing x items per day is 45x + 27 dollars, and the
price per item at which each may be sold is 60 – 0.5x dollars. Find an
expression for the daily profit, and hence find the maximum possible
profit.
Solution:
Daily revenue = x(60 – 0.5x) = 60x – 0.5x2
The expression for daily profit, P = 60x – 0.5x2 – (45x + 27)
= 15x – 0.5x2 – 27
Differentiating the profit function, we get:
dP
15 x 0, that means x = 15. So, the optimal profit is $85.5.
dx
The profit function looks like as follows:
95
85
75
65
55
45
35
25
4 9 14 19 24
Va l ue of X
1.3 A stone is thrown upwards so that at any time x seconds after throwing,
the height of the stone is y = 100 + 10x – 5x2 meters. Find the maximum
height reached.
@@SSeeisismmi4cicisisoolalatitoion
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, Solution Manual – Optimization Modelling
Solution:
Differentiating the expression of height with respect to time, we get:
dy
10 10x
that means x = 1.
0,
dx
So the corresponding /optimal height is (100 + 10 – 5) = 105 meters.
You can draw the function y in Excel to check the result.
1.4 A manufacturer finds that the cost C(x) = 2x2 – 8x + 15, where x is the
number of machines operating. Find how many machines he should
operate in order to minimize the total cost of production. What is the
optimal cost of production?
Solution:
Differentiating the cost function with respect to x, we get:
dC(x)
4x 8 0, that means x = 2.
dx
So the optimal cost = 8 – 16 + 15 = $7
1.5 A string 72 cm long is to be cut into two pieces. One piece is used to
form a circle and the other a square. What should be the perimeter of
the square in order to minimize the sum of two areas?
Solution:
Let us assume that each side of the square is x cm long.
The perimeter of the square is 4x.
The circumference of the circle will be 72 – 4x = 2 r, where r is the radius
of the circle. So, r = (72 – 4x)/(2 .
The area of the square = x2
The area of the circle = r2
The sum of two areas, A(x) = x2 + r2 = x2 + (72 – 4x)/(2 ]2
2
=x + (36 – 2x)/( ]2
= x2 + (1/ )(36 – 2x) 2
= x2 + (4/ )(18 – x) 2
= (4/ )(182) –(4/ )36x + ((4/ )x2
Differentiating A(x) with respect to x, we get:
dA(x) 4 4
( )36 ( 1)2x 0,
dx
4 4
or ( 1)2x ( )36,
or (4 )x 72,
or x 72
,
(4 )
or x 10.08
So the perimeter is 40.32 cm.
@@SSeeisismmi5cicisisoolalatitoion
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, Solution Manual – Optimization Modelling
The areas of the square and the circle and combined square and circle
with the length (integer value) of square are shown below.
400
A(Square)
350 A(Circle)
A(Square) + A(Circle)
300
Area (sq. cm)
250
200
150
100
50
0
1 3 5 7 9 11 13 15 17
Side of Square (x cm)
1.6 Find the maximum or minimum values of the following quadratic
functions, and the values of x for which they occur:
(a) f(x) = x2 – 4x + 7
(b) f(x) = 3 + 8x – x2
Solution:
(a) df (x) 2x 4 0, or x = 2
dx
2
d f (x)
2, so x = 2 is the minimum value.
2
dx
df (x)
8 2x 0, or x = 4
(b) dx
2
d f (x)
2, so x = 4 is the maximum value.
dx2
@@SSeeisismmi6cicisisoolalatitoion
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, Solution Manual – Optimization Modelling
Chapter 2
Solution to Exercises
2.1 The lift users of a multistoried building have complained about the delay
in getting an elevator. Being the property manager, how do you define
the problem in order to solve it? In other words, what is your problem
precisely which you intend to solve?
The problem definition may vary from person to person for such a
situation. If you cannot define the problem appropriately it is unlikely
that it will be solved. For example, the problem may be thought as:
1. Minimizing the waiting time by using better and efficient
elevator which would require an expensive re-engineering of
the elevator system.
2. Minimizing the people movement by studying the reasons for
frequent elevator usage and reduce them taking appropriate
action. For example, having laundry in each floor instead of a
common laundry at the basement.
3. Minimizing or eliminating the complaints using simple but
innovative means such as putting mirrors on the walls around the
lobby of the building. This would not change the waiting time of
the elevators and people movement, but will change the
perception, because people became occupied with another
activity. So the complaints will be disappeared.
4. You may add another option!
Most people would choose the first one as the problem definition
(minimizing waiting time) and suggest an expensive re-engineering as
the solution. Which one you would choose and why?
Solution: The minimization or elimination of the complaints, as in option
(3), could be an appropriate problem definition for some property
managers. The corresponding solution would be least expensive.
2.2 A manufacturing company produces several products in its shopfloor and
sells them, directly to their customers, through its retail section. Although
the production capacity is fixed and known, the demand of each product
varies from period to period. As a result, few products are experiencing
shortages in some periods whereas some other products are having
excess stocks. The retail manager knows that the overall performance of
the company can be improved by applying optimization techniques. The
company is currently performing very well financially. The top
management is neither familiar with optimization techniques nor
intended to make any changes in its current production schedule. As the
retail manager, how would you convenience the top management to
study the current system using optimization techniques?
Solution: There is no fixed answer for this question. You may try as
follows: perform a pilot study on your own and show the financial
@@SSeeisismmi7cicisisoolalatitoion
n
, Solution Manual – Optimization Modelling
benefits, due to application of optimization techniques, from this pilot
study. The top management will need to be convinced that the proposed
solution performs
better than the present alternative before deciding on whether to carry
out the proposed study.
2.3 Consider the problem in Exercise (2.2). As you are a key personnel in the
retail team, after your repeated requests, suppose the top management
has agreed to do the study. Although the study shows significant
improvement in company’s performance, the top management has no
idea how this result was obtained. As a consequence, the top
management is hesitant to implement the resulting production schedule.
What would you do now?
Solution: By giving presentations and making demonstrations of the
possible improvements that can be achieved by implementing the
derived solution may help substantially the analyst in convincing the
management to implement the proposed changes.
It may also be necessary to educate the management team (with the
help of external experts) in the proper application of the findings and
help them introduce the changes required to take them from the present
situation to the new desired mode of operations, and support them in
establishing control mechanisms to maintain and update the solution.
2.4 In most major airports, it is always a complaint that it takes too long to
get the arriving baggage. Being a key member of the airport baggage
handling team, how would you define the problem in order to solve it? In
other words, what is your problem precisely which you intend to solve?
Solution: The following options may be considered:
1. Study the bottlenecks and rectify them using better equipments
and more personnel.
2. Use of modern (and expensive) baggage handling system.
3. A long walk way from the aircraft to the baggage claim section (or
long immigration process) provides the baggage handlers a good
amount of time to ship all the baggage to the baggage claim
section. It would eliminate the baggage delay complaint. However
it may create some other complaints such as long walk and long
immigration process.
It is possible to improve the performance of baggage handling system, up
to certain level, by performing options 1 and 2. However it is unlikely that
it would be able to eliminate the complaint entirely. Option 3 –
minimizing or eliminating the complaint could be a good problem
definition. Alternatively, a combination of the above options could be
used.
@@SSeeisismmi8cicisisoolalatitoion
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, Solution Manual – Optimization Modelling
Chapter 3
Solution to Exercises
3.1 A furniture manufacturer employs 6 skilled and 11 semi-skilled
workers and produces two products: study table and computer table.
A study table requires 2 hours of a skilled worker and 2 hours of an un-
skilled worker. A computer table requires 2 hours of a skilled worker
and 5 hours of an un- skilled worker. As per the industrial laws, no one
is allowed to work more 38 hours a week. The manufacturer can sell
as many tables as he can produce. If the profit for a study table is
$100 and for a computer table
$160, how many study and computer tables should the manufacturer
produce in a week in order to maximize the overall profit? Formulate
a linear programming model.
Solution:
X1 the number of study table to be produced in a week
X2 the number of computer table to be produced in a week
Objective function: maximizing the total profit per week
Maximize Z = 100 X1 + 160 X2
Constraints:
(i) Skilled worker time availability in a week (6x38 = 228
hours) 2 X1 + 2 X2 <= 228
(ii) Un-skilled worker time availability in a week (11x38 = 418 hours)
2 X1 + 5 X2 <= 418
(iii) Non-
negativity X1,
X2 >= 0
The overall linear programming
model: Maximize Z = 100 X1 + 160 X2
Subject to
2 X1 + 2 X2 <= 228
2 X1 + 5 X2 <= 418 X1,
X2 >= 0
3.2 Consider the problem in (3.1). Suppose the demands of the study and
computer tables are at least 40 and 45 respectively. The manufacturer
pays
$900 and $600 per week for each skilled and unskilled worker
respectively. If the manufacturer intends to fulfil the demand in full,
what objective function would you suggest to the manufacturer’s
production planning problem? Justify your suggestion and formulate
the problem as a linear programming model.
Solution:
The total cost of skilled worker per week = 900x6 = $5,400
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, Solution Manual – Optimization Modelling
The total cost of un-skilled worker week = 600x11 = $6,600
The total cost of worker in a week = $5,400 + $6,600 = $12,000
As per the data and conditions given, the cost of worker is fixed in a
week irrespective of the number of tables produced. So the
consideration of cost
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, Solution Manual – Optimization Modelling
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SOLUTIONS + PowerPoint Slides
, Solution Manual – Optimization Modelling
CONTENT
Page#
Chapter 1 5
Chapter 2 8
Chapter 3 10
Chapter 4 19
Chapter 5 32
Chapter 6 41
Chapter 7 45
Chapter 10 49
Chapter 11 58
Chapter 12 62
@@SSeeisismmi3cicisisoolalatitoion
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, Solution Manual – Optimization Modelling
Chapter 1
Solution to Exercises
1.1 Jenny will run an ice cream stand in the coming week-long multicultural
event. She believes the fixed cost per day of running the stand is $60.
Her best guess is that she can sell up to 250 ice creams per day at $1.50
per ice cream. The cost of each ice cream is $0.85. Find an expression
for the daily profit, and hence find the breakeven point (no profit–no
loss point).
Solution:
Suppose x the number of ice creams Jenny can sell in a day.
The cost of x ice creams ($) = 0.85x
Jenny’s cost per day ($) = 60 + 0.85x
Daily revenue from ice cream sale ($) = 1.50x
Expression for daily profit ($) P = 1.50x – (60 + 0.85x) = 0.65x – 60
At breakeven point, 0.65x – 60 = 0
So, x = 60/0.65 = 92.31 ice creams
1.2 The total cost of producing x items per day is 45x + 27 dollars, and the
price per item at which each may be sold is 60 – 0.5x dollars. Find an
expression for the daily profit, and hence find the maximum possible
profit.
Solution:
Daily revenue = x(60 – 0.5x) = 60x – 0.5x2
The expression for daily profit, P = 60x – 0.5x2 – (45x + 27)
= 15x – 0.5x2 – 27
Differentiating the profit function, we get:
dP
15 x 0, that means x = 15. So, the optimal profit is $85.5.
dx
The profit function looks like as follows:
95
85
75
65
55
45
35
25
4 9 14 19 24
Va l ue of X
1.3 A stone is thrown upwards so that at any time x seconds after throwing,
the height of the stone is y = 100 + 10x – 5x2 meters. Find the maximum
height reached.
@@SSeeisismmi4cicisisoolalatitoion
n
, Solution Manual – Optimization Modelling
Solution:
Differentiating the expression of height with respect to time, we get:
dy
10 10x
that means x = 1.
0,
dx
So the corresponding /optimal height is (100 + 10 – 5) = 105 meters.
You can draw the function y in Excel to check the result.
1.4 A manufacturer finds that the cost C(x) = 2x2 – 8x + 15, where x is the
number of machines operating. Find how many machines he should
operate in order to minimize the total cost of production. What is the
optimal cost of production?
Solution:
Differentiating the cost function with respect to x, we get:
dC(x)
4x 8 0, that means x = 2.
dx
So the optimal cost = 8 – 16 + 15 = $7
1.5 A string 72 cm long is to be cut into two pieces. One piece is used to
form a circle and the other a square. What should be the perimeter of
the square in order to minimize the sum of two areas?
Solution:
Let us assume that each side of the square is x cm long.
The perimeter of the square is 4x.
The circumference of the circle will be 72 – 4x = 2 r, where r is the radius
of the circle. So, r = (72 – 4x)/(2 .
The area of the square = x2
The area of the circle = r2
The sum of two areas, A(x) = x2 + r2 = x2 + (72 – 4x)/(2 ]2
2
=x + (36 – 2x)/( ]2
= x2 + (1/ )(36 – 2x) 2
= x2 + (4/ )(18 – x) 2
= (4/ )(182) –(4/ )36x + ((4/ )x2
Differentiating A(x) with respect to x, we get:
dA(x) 4 4
( )36 ( 1)2x 0,
dx
4 4
or ( 1)2x ( )36,
or (4 )x 72,
or x 72
,
(4 )
or x 10.08
So the perimeter is 40.32 cm.
@@SSeeisismmi5cicisisoolalatitoion
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, Solution Manual – Optimization Modelling
The areas of the square and the circle and combined square and circle
with the length (integer value) of square are shown below.
400
A(Square)
350 A(Circle)
A(Square) + A(Circle)
300
Area (sq. cm)
250
200
150
100
50
0
1 3 5 7 9 11 13 15 17
Side of Square (x cm)
1.6 Find the maximum or minimum values of the following quadratic
functions, and the values of x for which they occur:
(a) f(x) = x2 – 4x + 7
(b) f(x) = 3 + 8x – x2
Solution:
(a) df (x) 2x 4 0, or x = 2
dx
2
d f (x)
2, so x = 2 is the minimum value.
2
dx
df (x)
8 2x 0, or x = 4
(b) dx
2
d f (x)
2, so x = 4 is the maximum value.
dx2
@@SSeeisismmi6cicisisoolalatitoion
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, Solution Manual – Optimization Modelling
Chapter 2
Solution to Exercises
2.1 The lift users of a multistoried building have complained about the delay
in getting an elevator. Being the property manager, how do you define
the problem in order to solve it? In other words, what is your problem
precisely which you intend to solve?
The problem definition may vary from person to person for such a
situation. If you cannot define the problem appropriately it is unlikely
that it will be solved. For example, the problem may be thought as:
1. Minimizing the waiting time by using better and efficient
elevator which would require an expensive re-engineering of
the elevator system.
2. Minimizing the people movement by studying the reasons for
frequent elevator usage and reduce them taking appropriate
action. For example, having laundry in each floor instead of a
common laundry at the basement.
3. Minimizing or eliminating the complaints using simple but
innovative means such as putting mirrors on the walls around the
lobby of the building. This would not change the waiting time of
the elevators and people movement, but will change the
perception, because people became occupied with another
activity. So the complaints will be disappeared.
4. You may add another option!
Most people would choose the first one as the problem definition
(minimizing waiting time) and suggest an expensive re-engineering as
the solution. Which one you would choose and why?
Solution: The minimization or elimination of the complaints, as in option
(3), could be an appropriate problem definition for some property
managers. The corresponding solution would be least expensive.
2.2 A manufacturing company produces several products in its shopfloor and
sells them, directly to their customers, through its retail section. Although
the production capacity is fixed and known, the demand of each product
varies from period to period. As a result, few products are experiencing
shortages in some periods whereas some other products are having
excess stocks. The retail manager knows that the overall performance of
the company can be improved by applying optimization techniques. The
company is currently performing very well financially. The top
management is neither familiar with optimization techniques nor
intended to make any changes in its current production schedule. As the
retail manager, how would you convenience the top management to
study the current system using optimization techniques?
Solution: There is no fixed answer for this question. You may try as
follows: perform a pilot study on your own and show the financial
@@SSeeisismmi7cicisisoolalatitoion
n
, Solution Manual – Optimization Modelling
benefits, due to application of optimization techniques, from this pilot
study. The top management will need to be convinced that the proposed
solution performs
better than the present alternative before deciding on whether to carry
out the proposed study.
2.3 Consider the problem in Exercise (2.2). As you are a key personnel in the
retail team, after your repeated requests, suppose the top management
has agreed to do the study. Although the study shows significant
improvement in company’s performance, the top management has no
idea how this result was obtained. As a consequence, the top
management is hesitant to implement the resulting production schedule.
What would you do now?
Solution: By giving presentations and making demonstrations of the
possible improvements that can be achieved by implementing the
derived solution may help substantially the analyst in convincing the
management to implement the proposed changes.
It may also be necessary to educate the management team (with the
help of external experts) in the proper application of the findings and
help them introduce the changes required to take them from the present
situation to the new desired mode of operations, and support them in
establishing control mechanisms to maintain and update the solution.
2.4 In most major airports, it is always a complaint that it takes too long to
get the arriving baggage. Being a key member of the airport baggage
handling team, how would you define the problem in order to solve it? In
other words, what is your problem precisely which you intend to solve?
Solution: The following options may be considered:
1. Study the bottlenecks and rectify them using better equipments
and more personnel.
2. Use of modern (and expensive) baggage handling system.
3. A long walk way from the aircraft to the baggage claim section (or
long immigration process) provides the baggage handlers a good
amount of time to ship all the baggage to the baggage claim
section. It would eliminate the baggage delay complaint. However
it may create some other complaints such as long walk and long
immigration process.
It is possible to improve the performance of baggage handling system, up
to certain level, by performing options 1 and 2. However it is unlikely that
it would be able to eliminate the complaint entirely. Option 3 –
minimizing or eliminating the complaint could be a good problem
definition. Alternatively, a combination of the above options could be
used.
@@SSeeisismmi8cicisisoolalatitoion
n
, Solution Manual – Optimization Modelling
Chapter 3
Solution to Exercises
3.1 A furniture manufacturer employs 6 skilled and 11 semi-skilled
workers and produces two products: study table and computer table.
A study table requires 2 hours of a skilled worker and 2 hours of an un-
skilled worker. A computer table requires 2 hours of a skilled worker
and 5 hours of an un- skilled worker. As per the industrial laws, no one
is allowed to work more 38 hours a week. The manufacturer can sell
as many tables as he can produce. If the profit for a study table is
$100 and for a computer table
$160, how many study and computer tables should the manufacturer
produce in a week in order to maximize the overall profit? Formulate
a linear programming model.
Solution:
X1 the number of study table to be produced in a week
X2 the number of computer table to be produced in a week
Objective function: maximizing the total profit per week
Maximize Z = 100 X1 + 160 X2
Constraints:
(i) Skilled worker time availability in a week (6x38 = 228
hours) 2 X1 + 2 X2 <= 228
(ii) Un-skilled worker time availability in a week (11x38 = 418 hours)
2 X1 + 5 X2 <= 418
(iii) Non-
negativity X1,
X2 >= 0
The overall linear programming
model: Maximize Z = 100 X1 + 160 X2
Subject to
2 X1 + 2 X2 <= 228
2 X1 + 5 X2 <= 418 X1,
X2 >= 0
3.2 Consider the problem in (3.1). Suppose the demands of the study and
computer tables are at least 40 and 45 respectively. The manufacturer
pays
$900 and $600 per week for each skilled and unskilled worker
respectively. If the manufacturer intends to fulfil the demand in full,
what objective function would you suggest to the manufacturer’s
production planning problem? Justify your suggestion and formulate
the problem as a linear programming model.
Solution:
The total cost of skilled worker per week = 900x6 = $5,400
@
@SSeeisismm
1ic0icisisoo
lalatitoionn
, Solution Manual – Optimization Modelling
The total cost of un-skilled worker week = 600x11 = $6,600
The total cost of worker in a week = $5,400 + $6,600 = $12,000
As per the data and conditions given, the cost of worker is fixed in a
week irrespective of the number of tables produced. So the
consideration of cost
@
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1ic0icisisoo
lalatitoionn
, Solution Manual – Optimization Modelling
THOSE WERE PREVIEW PAGES
TO DOWNLOAD THE FULL PDF
CLICK ON THE L.I.N.K
ON THE NEXT PAGE
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