Linear%20Programming:%20Modeling

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Introduction to Management Science 8th
Edition by Bernard W. Taylor III
Chapter 8 Linear Programming Modeling Examples
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Chapter Topics

• A Product Mix Example
• A Diet Example
• An Investment Example
• A Marketing Example
• A Transportation Example
• A Blend Example
• A Multi-Period Scheduling Example
• A Data Envelopment Analysis Example

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A Product Mix Example Problem Definition (1 of 7)

• Four-product T-shirt/sweatshirt manufacturing
  company.
• Must complete production within 72 hours
• Truck capacity 1,200 standard sized boxes.
• Standard size box holds12 T-shirts.
• One-dozen sweatshirts box is three times size of
  standard box.
• 25,000 available for a production run.
• 500 dozen blank T-shirts and sweatshirts in
  stock.
• How many dozens (boxes) of each type of shirt to
  produce?

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A Product Mix Example Data (2 of 7)
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A Product Mix Example Model Construction (3 of 7)
Decision Variables x1 sweatshirts, front
printing x2 sweatshirts, back and front
printing x3 T-shirts, front printing x4
T-shirts, back and front printing Objective
Function Maximize Z 90x1 125x2 45x3
65x4 Model Constraints 0.10x1 0.25x2 0.08x3
0.21x4 ? 72 hr 3x1 3x2 x3
x4 ? 1,200 boxes 36x1 48x2
25x3 35x4 ? 25,000 x1
x2 ? 500 doz.
sweatshirts
x3 x4 ? 500 doz. T-shirts
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A Product Mix Example Computer Solution with
Excel (4 of 7)
Exhibit 8.1
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A Product Mix Example Solution with Excel Solver
Window (5 of 7)
Exhibit 8.2
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A Product Mix Example Solution with QM for
Windows (6 of 7)
Exhibit 8.3
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A Product Mix Example Solution with QM for
Windows (7 of 7)
Exhibit 8.4
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A Diet Example Data and Problem Definition (1 of
5)
Breakfast to include at least 420 calories, 5
milligrams of iron, 400 milligrams of calcium, 20
grams of protein, 12 grams of fiber, and must
have no more than 20 grams of fat and 30
milligrams of cholesterol.
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A Diet Example Model Construction Decision
Variables (2 of 5)
x1 cups of bran cereal x2 cups of dry
cereal x3 cups of oatmeal x4 cups of
oat bran x5 eggs x6 slices of bacon
x7 oranges x8 cups of milk x9
cups of orange juice x10 slices of wheat
toast
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A Diet Example Model Summary (3 of 5)
Minimize Z 0.18x1 0.22x2 0.10x3 0.12x4
0.10x5 0.09x6 0.40x7 0.16x8 0.50x9
0.07x10 subject to 90x1 110x2 100x3 90x4
75x5 35x6 65x7 100x8 120x9 65x10 ?
420 2x2 2x3 2x4 5x5 3x6 4x8 x10 ?
20 270x5 8x6 12x8 ? 30 6x1 4x2 2x3
3x4 x5 x7 x10 ? 5 20x1 48x2 12x3
8x4 30x5 52x7 250x8 3x9 26x10 ?
400 3x1 4x2 5x3 6x4 7x5 2x6 x7
9x8 x9 3x10 ? 20 5x1 2x2 3x3 4x4 x7
3x10 ? 12

xi ? 0,
i1,,10.
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A Diet Example Computer Solution with Excel (4 of
5)
Exhibit 8.5
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A Diet Example Solution with Excel Solver Window
(5 of 5)
Exhibit 8.6
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An Investment Example Model Summary (1 of 4)
Maximize Z 0.085x1 0.05x2 0.065 x3
0.130x4 subject to x1
? 14,000 x1 x2 x3 x4 ?
0 x2 x3 ?
21,000 1.2x1 x2 x3 1.2 x4 ? 0
x1 x2 x3 x4 70,000
x1, x2, x3, x4 ? 0 where x1
amount invested in municipal bonds ()
x2 amount invested in certificates of deposit
() x3 amount invested in treasury
bills () x4 amount invested in
growth stock fund()
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An Investment Example Computer Solution with
Excel (2 of 4)
Exhibit 8.7
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An Investment Example Solution with Excel Solver
Window (3 of 4)
Exhibit 8.8
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An Investment Example Sensitivity Report (4 of 4)
Exhibit 8.9
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A Marketing Example Data and Problem Definition
(1 of 6)

• Budget limit 100,000
• Television time for four commercials
• Radio time for 10 commercials
• Newspaper space for 7 ads
• Resources for no more than 15 commercials
  and/or ads

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A Marketing Example Model Summary (2 of 6)
Maximize Z 20,000x1 12,000x2
9,000x3 subject to 15,000x1 6,000x 2 4,000x3
? 100,000 x1 ? 4 x2 ?
10 x3 ? 7 x1
x2 x3 ? 15 x1, x2, x3 ? 0
where x1 Exposure from Television
Commercial (people) x2 Exposure from
Radio Commercial (people) x3
Exposure from Newspaper Ad (people)
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A Marketing Example Solution with Excel (3 of 6)
Exhibit 8.10
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A Marketing Example Solution with Excel Solver
Window (4 of 6)
Exhibit 8.11
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A Marketing Example Integer Solution with Excel
(5 of 6)
Exhibit 8.12
Exhibit 8.13
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A Marketing Example Integer Solution with Excel
(6 of 6)
Exhibit 8.14
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A Transportation Example Problem Definition and
Data (1 of 3)
Warehouse supply of Retail store demand
Television Sets for television sets
1 - Cincinnati 300 A - New York 150 2 -
Atlanta 200 B - Dallas 250 3 -
Pittsburgh 200 C - Detroit 200 Total
700 Total 600
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A Transportation Example Model Summary (2 of 4)
Minimize Z 16x1A 18x1B 11x1C 14x2A
12x2B 13x2C 13x3A 15x3B 17x3C
subject to x1A x1B x1C ? 300 x2A x2B
x2C ? 200 x3A x3B x3C ? 200 x1A
x2A x3A 150 x1B x2B x3B 250 x1C
x2C x3C 200 xij ? 0, i1,2,3
and j A,B,C.
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A Transportation Example Solution with Excel (3
of 4)
Exhibit 8.15
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A Transportation Example Solution with Solver
Window (4 of 4)
Exhibit 8.16
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A Blend Example Problem Definition and Data (1 of
6)
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A Blend Example Problem Statement and Variables
(2 of 6)

• Determine the optimal mix of the three components
  in each grade of motor oil that will maximize
  profit. Company wants to produce at least 3,000
  barrels of each grade of motor oil.
• Decision variables The quantity of each of the
  three components used in each grade of motor oil
  (9 decision variables) xij barrels of
  component i used in motor oil grade j per day,
  where i 1, 2, 3 and j s (super), p (premium),
  and e (extra).

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A Blend Example Model Summary (3 of 6)
Maximize Z 11x1s 13x2s 9x3s 8x1p 10x2p
6x3p 6x1e 8x2e 4x3e subject to
x1s x1p x1e ? 4,500 x2s x2p
x2e ? 2,700 x3s x3p x3e ? 3,500
0.50(x1s x2s x3s) ? 0 0.70x2s 0.30(x1s
x3s) ? 0 0.60x1p 0.40(x2p x3p) ? 0
0.25(3x3p x1p x2p) ? 0 0.20(2x1e
3x2e- 3x3e) ? 0 0.10(9x2e x1e x3e) ?
0 x1s x2s x3s ? 3,000 x1p
x2p x3p ? 3,000 x1e x2e x3e ?
3,000 xij ? 0, i1,2,3, and js,p,e.
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A Blend Example Solution with Excel (4 of 6)
Exhibit 8.17
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A Blend Example Solution with Solver Window (5 of
6)
Exhibit 8.18
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A Blend Example Sensitivity Report (6 of 6)
Exhibit 8.19
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A Multi-Period Scheduling Example Problem
Definition and Data (1 of 5)
Production Capacity 160 computers per week
50 more computers with
overtime Assembly Costs 190 per computer
regular time 260 per computer
overtime Inventory Cost 10/comp. per week Order
schedule Week (j) Computer Orders
(dj) 1
105 2
170 3 230
4 180 5
150 6
250
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A Multi-Period Scheduling Example Decision
Variables (2 of 5)
Decision Variables rj regular production of
computers per week j (j 1,,6) -- decision
variable oj overtime production of computers
per week j (j 1,,6) -- decision variable ij
extra computers carried over as inventory in
week j (j 1,,5) computed as ij rj oj
ij1 dj, where dj is the demand (orders) as
tabulated on p.35.
Note the ij variables are being calculated, so
that they are not independent from the rj and oj
decision variables the ijs are functions of rj
and oj.
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A Multi-Period Scheduling Example Model Summary
(3 of 5)
Model summary Minimize Z 190(r1 r2 r3
r4 r5 r6) 260(o1 o2 o3 o4 o5
o6) 10(i1 i2 i3 i4 i5) subject
to rj ? 160 (j 1, , 6) oj ? 150 (j 1,
, 6) r1 o1 - i1 ? 105 r2 o2 i1 - i2 ?
170 r3 o3 i2 - i3 ? 230 r4 o4 i3 -
i4 ? 180 r5 o5 i4 - i5 ? 150 r6 o6 i5
? 250 rj, oj, ij ? 0
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A Multi-Period Scheduling Example Solution with
Excel (4 of 5)
Exhibit 8.20
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A Multi-Period Scheduling Example Solution with
Solver Window (5 of 5)
Exhibit 8.21
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A Data Envelopment Analysis (DEA) Example Problem
Definition (1 of 5)
DEA compares a number of service units of the
same type based on their inputs (resources) and
outputs. The result indicates if a particular
unit is less productive, or efficient, than other
units. Elementary school comparison Input 1
teacher to student ratio Output 1
average reading SOL score Input 2
supplementary /student Output 2
average math SOL score Input 3 parent
education level Output 3 average
history SOL score
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A Data Envelopment Analysis (DEA) Example Problem
Data Summary (2 of 5)
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A Data Envelopment Analysis (DEA)
Example Decision Variables and Model Summary (3
of 5)
Decision Variables xi a price per unit of
each output where i 1, 2, 3 yi a price per
unit of each input where i 1, 2, 3 Model
Summary Maximize Z 81x1 73x2
69x3 subject to .06y1 460y2 13.1y3
1 86x1 75x2 71x3 ?.06y1 260y2
11.3y3 82x1 72x2 67x3 ? .05y1
320y2 10.5y3 81x1 79x2 80x3 ?
.08y1 340y2 12.0y3 81x1 73x2
69x3 ? .06y1 460y2 13.1y3 xi,
yi ? 0, i1,2,3.
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A Data Envelopment Analysis (DEA)
Example Solution with Excel (4 of 5)
Exhibit 8.22
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A Data Envelopment Analysis (DEA)
Example Solution with Solver Window (5 of 5)
Exhibit 8.23
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Example Problem Solution Problem Statement and
Data (1 of 5)

• Canned cat food, Meow Chow dog food, Bow Chow.
• Ingredients/week 600 lb horse meat 800 lb fish
  1000 lb cereal.
• Recipe requirement Meow Chow at least half (1/2)
  fishBow Chow at least half (1/2) horse meat.
• 2,250 sixteen-ounce cans available each week.
• Profit /can Meow Chow 0.80 Bow Chow 0.96.
• How many cans of Bow Chow and Meow Chow should be
  produced each week in order to maximize profit?

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Example Problem Solution Model Formulation (2 of
5)
Step 1 Define the Decision Variables xij
ounces of ingredient i in pet food j per week,
where i h (horse meat), f (fish) and c
(cereal), and j m (Meow chow) and b (Bow
Chow). Step 2 Formulate the Objective
Function Maximize Z 0.05(xhm xfm xcm)
0.06(xhb xfb xcb)
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Example Problem Solution Model Formulation (3 of
5)
Step 3 Formulate the Model Constraints Amount of
each ingredient available each week xhm xhb ?
9,600 ounces of horse meat xfm xfb ? 12,800
ounces of fish xcm xcb ? 16,000 ounces of
cereal additive Recipe requirements Meow
Chow xfm/(xhm xfm xcm) ? 1/2 or - xhm
xfm- xcm ? 0 Bow Chow xhb/(xhb xfb
xcb) ? 1/2 or xhb- xfb - xcb ? 0 Can
Content Constraint xhm xfm xcm
xhb xfb xcb ? 36,000 ounces
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Example Problem Solution Model Summary (4 of 5)
Step 4 Model Summary Maximize Z 0.05xhm
0.05xfm 0.05xcm 0.06xhb 0.06xfb
0.06xcb subject to xhm xhb ? 9,600 ounces of
horse meat xfm xfb ? 12,800 ounces of
fish xcm xcb ? 16,000 ounces of cereal
additive - xhm xfm- xcm ? 0 xhb-
xfb - xcb ? 0 xhm xfm xcm xhb xfb xcb ?
36,000 ounces

xij
? 0, ih,f,m, and jm,b.
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Example Problem Solution Solution with QM for
Windows (5 of 5)
Exhibit 8.24
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