Needing help with a Written Optimization Setup in Finite Math

Needing someone to do a written optimization setup for the problem I have attached in the problem file. Please attach the completed problem in a word document for me.

I have attached 2 example pdfs of how they should be setup. As well as instructions below for the assignment.

Thanks for the help in advance!

————

————————————————-

———————————————–

 

You have some work to do before you ever open Microsoft Excel or download Geogebra, and this is to make sure you are on the right track.  

 

Please review the set up of the example problems as outlined in the pdf files included in this acticity.  The included files will cover the thought process or “how to” for setting up these types of problem.

————

As an assignment you will need to describe how you are thinking about the assigned problem, any and all relationships you have established, and define your variables, set up the objective function, constraint inequalities…..and describe what it all means.

 

You DEFINITELY need to have these correct before moving forward with the technology to get an answer.

 You are free to discuss the topics here, but be careful about copying your classmates without thinking for yourself.   I will give feedback on Friday to make sure you are on the right track.

 

Include the following in your post (including any follow up posts you make)

————————————————-

Problem1

The variable x represents:

The variable y represents:

The objective function is:

The constraint inequalities are:

  

Problem2

The variable x1 represents:

The variable x2 represents:

The variable x3 represents:

The objective function is:The constraint inequalities are:

To: Software Development Team Leaders
From: Robert Hofmann
Date: 12 December
Re: Written Optimization Analysis

M e s s a g e
The state has created new legislation about “crunch-time”, and as such requires us to adjust our
projects for the next quarter. So for the next 13 weeks we will assume that there will be no
scheduled overtime. Our adjusted budget for the quarter gives us 520 hours to schedule per
employee (13 weeks x 40 hours per week = 520 total hours).

We have to determine how many projects we can complete during this time period, making
adjustments based on the man-hours worked on previous projects. Please figure out the
number of game we can make of each type that yields the highest profit.

Additional Information and the projected profit and estimated man-hours based on previous
projects are presented in the following pages pages.

Q u a r t e r l y P r o j e c t i o n
U p d a t e

Written Optimization Report:
Problem 1

To create a single console game, we previously required 10,920 man-hours of development,
13,000 man-hours on art, 3,120 man-hours for design, and 2,080 man-hours for production
management. The projected profit for console game titles is $3.6 million each.

The work requirements for a mobile game are quite a bit different: 7,280 man-hours in
development, 2,600 man-hours art, 9,360man-hours in design, and 2,600 man-hours in
production management. The projected profit for handheld game titles is $2 million each.

Our current staff consists of 238 programmers, 225 artists, 180 designers, and 57 production
managers.

This should give us:

§ 123,760 man-hours in the development pool (238 x 520 = 123,760)
§ 117,000 man-hours in the art pool (225 x 520 = 117,000),
§ 93,600 man-hours in the design pool (180 x 520 = 93,600), and
§ 29,640 man-hours in management (57 x 520 = 29,640)

to allocate to projects,

Department
  Console
 Game
  Mobile
 Game
 
Development
  10,920
 man-­‐hours
  7,280
 man-­‐hours
 
Art
  13,000
 man-­‐hours
  2,600
 man-­‐hours
 
Management
  2,080
 man-­‐hours
  2,600
 man-­‐hours
 
Design
  3,120
 man-­‐hours
  9,360
 man-­‐hours
 

 
 
 Profit
 
Projection
 
 $3,600,000
 
 
 $2,000,000
 
 

Figure out how many console and mobile games can be made this quarter to maximize
our profit. In addition, report what pools (development, artists, designers, and
managers) have some unutilized employees, and which pools could be expanded to
increase profits.

Note: If you want to analyze this problem in terms of number of people on each project, you will
need to convert from man-hours to number of people. Divide the number of man-hours by 520
hours to determine the number of people needed to complete the work in 13 weeks.

Written Optimization Report:
Problem 2

We have decided to expand and create a new PC games department. Out projections indicate
PC game titles will make $2.8 million in profit each. The work requirements are 9,360 man-
hours for development, 8,840 man-hours on artwork, 5,720 man-hours for design, and 1,560
man-hours for production management.

Department
  Console
 Game
  Mobile
 Game
  PC
 Game
 
Development
  10,920
 man-­‐hours
  7,280
 man-­‐hours
  9,360
 man-­‐hours
 
Art
  13,000
 man-­‐hours
  2,600
 man-­‐hours
  8,840
 man-­‐hours
 
Management
  2,080
 man-­‐hours
  2,600
 man-­‐hours
  1,560
 man-­‐hours
 
Design
  3,120
 man-­‐hours
  9,360
 man-­‐hours
  5,720
 man-­‐hours
 

 
 
 
 Profit
 
Projection
 
 $3,600,000
 
 
 $2,000,000
 
 
 $2,800,000
 
 

To help staff this department, we hired 44 more programmers for the development team, 58
more artists for the art team, and 2 more managers for the management team. Please adjust
the total man-hours available based on this new staff before calculating the new estimations.

Figure out how many console, PC, and mobile games can be made this quarter to
maximize our profit. In addition, report what pools (development, artists, designers, and
managers) have some unutilized employees, and which pools could be expanded to
increase profits.

2

 

Variable
 Example
 Problem

 

Part
 2:
 Analysis
 and
 set-­‐up

 

 

Analysis
 

 
There
 is
 quite
 a
 bit
 of
 information
 to
 sort
 through
 in
 these
 optimization
 problems,
 
and
 this
 example
 is
 no
 different.
 
 
 In
 this
 part
 of
 the
 problem
 we
 are
 only
 trying
 to
 
mathematically
 define
 the
 relationships
 for
 all
 the
 information
 we
 are
 given.
 
 
 
 
Do
 not
 try
 to
 “solve”
 anything
 yet
 at
 this
 point
 in
 the
 process!
 

 

 
Define
 the
 Variables
 

 
The
 most
 important
 step
 is
 to
 define
 the
 variables
 for
 the
 problem,
 which
 is
 tricky
 
because
 there
 is
 nothing
 to
 really
 point
 to
 in
 the
 presentation
 of
 the
 problem.
 
 
 The
 
main
 criteria
 you
 are
 looking
 for
 in
 a
 variable
 are:
 

 

• There
 is
 an
 unknown
 or
 unstated
 number
 or
 amount
 of
 objects/people
 
• The
 provided
 numbers
 in
 the
 problem
 relate
 directly
 to
 the
 number
 or
 

amount
 of
 the
 unknown
 objects.
 

 
The
 second
 question
 might
 be
 the
 easier
 of
 the
 two
 to
 figure
 out.
 
 
 Consider
 the
 
following
 statements
 from
 the
 presentation
 of
 the
 problem
 in
 part
 1:
 

 
-­‐“What
 is
 the
 cheapest
 diet
 that
 will
 fulfill
 the
 dietary
 requirements?”
 
-­‐“An
 uncooked
 cup
 of
 rice
 contains
 15
 grams
 of
 protein,
 810
 calories,
 and
 1/9
 of
 a
 
milligram
 of
 B2,
 all
 at
 a
 price
 of
 21
 cents.”
 
-­‐“An
 uncooked
 cup
 of
 soybeans
 costs
 14
 cents…”
 

 
Since
 we
 are
 trying
 to
 figure
 out
 the
 cost
 of
 the
 diet,
 we
 want
 to
 look
 for
 price
 
information
 in
 the
 problem.
 
 
 We
 have
 information
 for
 the
 price
 per
 cup
 for
 both
 
soybeans
 and
 rice,
 but
 do
 not
 know
 how
 many
 cups
 of
 each
 type
 of
 food
 we
 will
 
require!
 
 
 These
 unknown
 quantities
 will
 be
 our
 variables:
 

 
x :
 Number
 of
 cups
 of
 rice
 per
 day
 
y :
 Number
 of
 cups
 of
 soybeans
 per
 day
 

 

 
Establish
 the
 Objective
 Function
 

 
In
 case
 you’re
 not
 sure
 how
 to
 create
 a
 mathematical
 expression
 to
 represent
 the
 
cost,
 consider
 the
 situations
 below
 

 

• If
 we
 have
 zero
 cups
 of
 rice
 and
 zero
 cups
 of
 soybeans,
 it
 should
 be
 obvious
 
that
 the
 cost
 of
 that
 diet
 will
 be
 zero
 cents.
 
 

• If
 we
 have
 one
 cup
 of
 rice
 only,
 the
 diet
 will
 cost
 21
 cents.
 

2
 Variable
 Example
 Problem
 
Part
 2:
 Analysis
 and
 set-­‐up
 

 

• If
 we
 have
 two
 cups
 of
 rice,
 the
 diet
 will
 cost
 42
 cents.
 
 
 We
 calculate
 this
 by
 
multiplying
 the
 cost
 per
 cup
 by
 the
 number
 of
 cups:
 
cost
 =
 21×2
 =
 42
 cents
 

• If
 we
 add
 a
 cup
 of
 soybeans
 (so
 two
 cups
 of
 rice,
 one
 cup
 of
 soybeans)
 we
 will
 
add
 the
 cost
 of
 a
 cup
 soybeans
 to
 the
 cost
 of
 the
 rice:
 
cost
 =
 21×2
 +
 14
 =
 56
 cents
 

 

 
Since
 we
 do
 not
 know
 the
 amount
 of
 cups
 of
 rice
 and
 soybeans,
 but
 instead
 have
 
variables
 to
 stand
 in
 place
 of
 them,
 we
 can
 create
 a
 general
 function
 to
 describe
 the
 
cost
 for
 any
 amount
 of
 rice
 and
 soybeans.
 

Cost = 21x+14y
 

 

 

 
Defining
 the
 Constraint
 Inequalities
 
 

 
Now
 that
 we
 know
 what
 we
 are
 trying
 to
 optimize
 (minimal
 cost),
 have
 defined
 our
 
variables
 (number
 of
 cups
 of
 rice,
 number
 of
 cups
 of
 soybeans),
 and
 have
 created
 an
 
objective
 function
 describing
 the
 value
 we
 are
 trying
 to
 optimize
 we
 can
 now
 take
 a
 
look
 at
 our
 constraints.
 

 
While
 it
 should
 be
 obvious
 that
 the
 fewer
 cups
 of
 “food”
 (rice
 and
 soybeans)
 we
 
have
 in
 the
 diet,
 the
 cheaper
 it
 is
 going
 to
 be.
 
 
 However,
 we
 have
 to
 make
 sure
 me
 
meet
 some
 nutritional
 requirement.
 

 
In
 our
 original
 problem
 we
 stated:
 
“We
 want
 to
 insure
 that
 people
 on
 this
 diet
 receive
 at
 least
 
 

§ 90g
 of
 protein,
 
 
§ 1620
 calories,
 
 
§ 1
 milligram
 of
 Vitamin
 B2.”
 

 
With
 that
 in
 mind,
 we
 have
 to
 create
 expressions
 that
 define
 how
 much
 protein,
 
how
 many
 calories,
 and
 how
 much
 vitamin
 B2
 is
 in
 the
 diet.
 
 
 This
 follows
 the
 
same
 pattern
 as
 our
 objective
 function.
 

 
NOTE:
 
 The
 amount
 of
 protein,
 calories,
 and
 vitamin
 B2
 all
 vary
 by
 different
 
amounts
 per
 cup
 of
 soybeans
 or
 rice.
 
 We
 will
 treat
 each
 separately,
 each
 with
 
their
 own
 constraint
 inequality.
 

 
Protein
 
A
 cup
 of
 rice
 contains
 15
 grams
 of
 protein.
 
 
 15x
 will
 represent
 the
 amount
 of
 protein
 
from
 x
 cups
 of
 rice
 (remember
 our
 variables?)
 
 

2
 Variable
 Example
 Problem
 
Part
 2:
 Analysis
 and
 set-­‐up
 

 
A
 cup
 of
 soybeans
 contains
 22.5
 grams
 of
 protein.
 
 
 22.5y
 will
 represent
 the
 amount
 
of
 protein
 from
 y
 cups
 of
 soybeans
 

 
The
 amount
 of
 protein
 in
 our
 diet
 is
 expressed
 by:
 

15x+22.5y
 
Now
 since
 this
 amount
 has
 to
 be
 greater
 than
 or
 equal
 to
 90
 grams
 (the
 phrase
 “at
 
least”
 means
 we
 cannot
 have
 a
 value
 below
 90)
 we
 define
 the
 following
 inequality
 as
 
a
 requirement
 for
 the
 diet
 

15x+22.5y ! 90
 
The
 other
 constrains
 will
 follow
 a
 similar
 pattern.
 

 
Calories
 
A
 cup
 of
 rice
 contains
 810
 calories.
 
 
 180x
 will
 represent
 the
 amount
 of
 calories
 from
 
x
 cups
 of
 rice.
 
 
A
 cup
 of
 soybeans
 contains
 270
 calories.
 
 270y
 will
 represent
 the
 amount
 of
 protein
 
from
 y
 cups
 of
 soybeans.
 

 
The
 amount
 of
 calories
 in
 our
 diet
 is
 expressed
 by:
 

810x+270y
 
Now
 since
 this
 amount
 has
 to
 be
 greater
 than
 or
 equal
 to
 1620
 (the
 phrase
 “at
 least”
 
means
 we
 cannot
 have
 a
 value
 below
 1620)
 we
 define
 the
 following
 inequality
 as
 a
 
requirement
 for
 the
 diet
 

810x+270y !1620
 

 
Vitamin
 B2
 
 
A
 cup
 of
 rice
 contains
 1/9th
 of
 a
 
 milligram
 of
 vitamin
 B2.
 
 
 1x
 will
 represent
 the
 
amount
 of
 vitamin
 B2
 from
 x
 cups
 of
 rice.
 
 
A
 cup
 of
 soybeans
 contains
 1/3
 of
 a
 milligram
 of
 vitamin
 B2.
 
 (1/3)y
 will
 represent
 
the
 amount
 of
 protein
 from
 y
 cups
 of
 soybeans.
 

 
The
 amount
 of
 vitamin
 B2
 in
 our
 diet
 is
 expressed
 by:
 

1
9
!

”

#

$

%

&x+

1
3
!

”
#
$

%
&y
 

Now
 since
 this
 amount
 has
 to
 be
 greater
 than
 or
 equal
 to
 1
 milligram
 of
 
 vitamin
 B2
 
we
 define
 the
 following
 inequality
 as
 a
 requirement
 for
 the
 diet
 

1
9
!
”
#
$
%
&x+
1
3
!
”
#
$

%
&y ‘1

 

 

 

2
 Variable
 Example
 Problem
 
Part
 2:
 Analysis
 and
 set-­‐up
 

 
Non-­‐negativity
 

 
The
 last
 constraints
 are
 our
 non-­‐negativity
 constraints.
 
 
 What
 this
 means
 for
 our
 
problem
 is
 that
 we
 can’t
 have
 negative
 values
 for
 x
 and
 y
 (that
 is,
 x
 and
 y
 must
 be
 
greater
 than
 or
 equal
 to
 zero).
 
 
 This
 makes
 sense
 because
 the
 context
 of
 the
 
problem
 does
 not
 allow
 for
 us
 to
 sell
 rice
 or
 soybeans,
 nor
 does
 it
 allow
 us
 to
 
someone
 extract
 a
 cup
 of
 rice
 or
 soybeans
 OUT
 of
 someone’s
 body.
 
 

 
Thus
 we
 have
 the
 final
 set
 of
 constraints:
 

 

15x+22.5y ! 90
810x+270y !1620
1
9
x+
1
3
y !1

x ! 0,y ! 0

”
#

$
$

%
$
$

 

 

4
 Variable
 Example
 Problem
 
Part
 2:
 Analysis
 and
 set-­‐up
 

 
Analysis
 

 
There
 is
 quite
 a
 bit
 of
 information
 to
 sort
 through
 in
 these
 optimization
 problems,
 
and
 this
 example
 is
 no
 different.
 
 
 In
 this
 part
 of
 the
 problem
 we
 are
 only
 trying
 to
 
mathematically
 define
 the
 relationships
 for
 all
 the
 information
 we
 are
 given.
 
 
 
 
Do
 not
 try
 to
 “solve”
 anything
 yet
 at
 this
 point
 in
 the
 process!

 

 

 

Define
 the
 Variables
 

 
The
 most
 important
 step
 is
 to
 define
 the
 variables
 for
 the
 problem,
 which
 is
 tricky
 
because
 there
 is
 nothing
 to
 really
 point
 to
 in
 the
 presentation
 of
 the
 problem.
 
 
 The
 
main
 criteria
 you
 are
 looking
 for
 in
 a
 variable
 are:
 

 

• There
 is
 an
 unknown
 or
 unstated
 number
 or
 amount
 of
 objects/people
 
• The
 provided
 numbers
 in
 the
 problem
 relate
 directly
 to
 the
 number
 or
 

amount
 of
 the
 unknown
 objects.
 

 
The
 second
 part
 might
 be
 the
 easier
 to
 look
 at.
 
 
 Consider
 the
 following
 statements
 
from
 the
 presentation
 of
 the
 problem
 in
 part
 1:
 
“A
 Velite
 requires
 2
 gold
 to
 recruit,
 1
 gold
 to
 equip,
 and
 half
 a
 gold
 to
 train.”
 
“A
 Hopolite
 has
 a
 battle
 rating
 of:
 1
 

 
Nowhere
 is
 there
 a
 number
 provided
 for
 the
 number
 of
 Hopolites
 or
 Velites,
 and
 all
 
of
 our
 information
 (recruitment
 cost,
 equipment
 cost,
 training
 cost,
 and
 battle
 
rating)
 is
 given
 in
 the
 terms
 of
 a
 single
 Hopolite
 or
 Velite,
 and
 the
 totals
 (costs
 and
 
battle
 rating)
 will
 only
 be
 know
 if
 we
 know
 the
 numbers
 of
 each
 type
 of
 soldiers.
 

 
We
 will
 then
 define
 our
 variables
 as
 follows:
 

• x1
 :
 Number
 of
 Velites
 
• x2
 :
 Number
 of
 Hoplites
 
• x3
 :
 Number
 of
 Legionaries
 
• x4
 :
 Number
 of
 Equites
 

 

 

 
Establish
 the
 Objective
 Function
 

 
We
 are
 interested
 in
 the
 total
 battle
 rating
 of
 all
 the
 soldiers
 in
 the
 army.
 
 
 Therefore
 
we
 want
 to
 add
 the
 battle
 rating
 contributed
 by
 each
 group
 of
 soldiers:
 

 
Overall
 Battle
 Rating
 =
 Total
 Velite
 Battle
 Rating
 +
 Total
 Hopolite
 Battle
 Rating
 
+
 Total
 Legionary
 Battle
 Rating
 +
 Total
 Equite
 Battle
 Rating
 

 

4
 Variable
 Example
 Problem
 
Part
 2:
 Analysis
 and
 set-­‐up
 

 
Think:
 If
 a
 single
 Equite
 has
 a
 battle
 rating
 of
 2
 then
 two
 Equites
 will
 have
 a
 battle
 
rating
 of
 4
 (2
 x
 2
 =
 4);
 while
 10
 Equites
 will
 have
 a
 battle
 rating
 of
 20
 (2
 x
 10
 =
 20).
 
Now
 to
 find
 the
 total
 battle
 rating
 for
 each
 group
 of
 soldier,
 we
 will
 multiply
 the
 
battle
 rating
 for
 a
 single
 soldier
 by
 the
 number
 of
 soldiers.
 
 
 Now
 since
 we
 do
 not
 
have
 the
 number
 for
 the
 of
 soldiers
 of
 each
 type,
 we
 will
 use
 the
 variables.
 
 So
 we
 get
 
the
 following
 terms:
 

 

§ Total
 Velite
 Battle
 Rating
 
 =
 (0.5)×
 (x1)
 
§ Total
 Hopolite
 Battle
 Rating
 =
 
 (1)×
 (x2)
 
§ Total
 Legionary
 Battle
 Rating
 =
 (1.5)×
 (x3)
 
§ Total
 Equite
 Battle
 Rating=
 (2)×
 (x4)
 

 
And
 therefore,
 our
 objective
 function
 is:
 

BattleRating = 0.5( )x1 + 1( )x2 + 1.5( )x3 + 2( )x4
 

 

 

 
Defining
 the
 Constraint
 Inequalities
 
 

 
With
 our
 variables
 and
 objective
 function
 defined,
 we
 then
 want
 to
 look
 at
 the
 
constraint
 inequalities.
 
 
 In
 our
 problems,
 we
 are
 limited
 by
 the
 amount
 of
 gold
 
allotted
 to
 each
 category:
 recruitment,
 training,
 and
 equipment.
 
 
 We
 do
 not
 have
 to
 
use
 all
 the
 gold
 in
 a
 category,
 but
 we
 cannot
 go
 over.
 
Note:
 This
 means
 we
 will
 use
 the
 “less
 than
 or
 equal
 to”
 relationship
 between
 the
 
amount
 of
 gold
 required,
 and
 the
 total
 amount
 available
 

 
The
 mathematical
 principle
 behind
 this
 is
 much
 the
 same
 as
 the
 one
 set
 up
 in
 the
 
objective
 function:
 
 We
 will
 multiply
 the
 cost
 involved
 for
 a
 single
 soldier
 of
 each
 
type,
 and
 multiply
 by
 the
 number
 of
 soldiers.
 

 
(A
 hopolite
 costs
 2.5
 gold
 to
 recruit,
 2
 hopolites
 will
 cost
 2.5×2
 =
 5
 gold
 to
 recruit,
 20
 
hopolites
 will
 cost
 2.5×20
 =
 50
 gold
 to
 recruit
 and
 so
 on)
 

 
Thus
 we
 have
 the
 following
 constraints:
 

§ recruitment : 2×1 + 2.5×2 + 2×3 + 2×4 ! 4000
 
§ equipment :1×1 + 1.5×2 + x3 + 2×4 !1800
 
§ training : 0.5×1 + 0.5×2 + x3 + x4 ! 4000
 

 
So,
 with
 4
 variables,
 we
 will
 not
 be
 able
 to
 solve
 this
 problem
 graphically.
 
 
 This
 
means
 either
 using
 some
 linear
 algebra
 techniques
 or
 a
 solver.
 

 

 

4
 Variable
 Example
 Problem
 
Part
 2:
 Analysis
 and
 set-­‐up
 

 
To
 use
 a
 solver,
 we
 will
 have
 to
 add
 the
 values
 into
 a
 spreadsheet.
 
 
 Using
 the
 empty
 
3
 variable
 sheet
 and
 adding
 an
 extra
 column
 should
 give
 us
 a
 similar
 result
 after
 
filling
 in
 the
 values
 and
 relabeling
 some
 cells:
 

 

Army
 Battle
 Value
 
 
 
 
 
 
 
 
 
 
Variables
  X1
  X2
  X3
  X4
  Sign
  RHS
 
  LHS
  Slack
 
Objective
 Function
  0.5
  1
  1.5
  2
  =
  Max
  Battle
 Points
 
 
 
 
Recruitment
 
Constraint
  2
  2.5
  2
  2
  ≤
  4000
  Constraint
 1
 
 
 
Equipment
 
Constraint
  1
  1.5
  1
  2
  ≤
  1800
  Constraint
 2
 
 
 
Training
 Constraint
  0.5
  0.5
  1
  1
  ≤
  1000
  Constraint
 3
 
 
 

 
 
 
 
 
 
 
 
 
 

  Solutions
 
 
 
 
 
 
 
 

  Velites
  Hoplites
  Legionares
  Equites