7-16
6. Finally, Ash wants at most one piece from Ziggy Lite displayed. We can therefore
include no more than one of “My Namesake” and “Narcissism” (R36 ≤ R38).
Constraints Imposed by Celeste
1. Celeste wants to include at least one piece from a female artist for every two pieces
included from a male artist. We have 11 pieces by female artists available: “Chaos
Reigns” by Rita Losky, “Who Has Control?” by Rita Losky, “Domestication” by Rita
2. Celeste wants at least one of the pieces “Aging Earth” and “Wasted Resources”
4. Celeste wants to include one or more of the pieces “Chaos Reigns,” “Who Has
5. Celeste knows that the museum only has enough floor space for four sculptures. We
have six sculptures available: “Perfection” by Colin Zweibell, “Burden” by Colin
Zweibell, “The Great Equalizer” by Colin Zweibell, “Aging Earth” by Norm Marson,
6. Celeste also knows that the museum only has enough wall space for 20 paintings,
collages, and drawings. We have 28 paintings, collages, and drawings available:
“Chaos Reigns,” “Who Has Control,” “Domestication,” “Innocence,” “Wasted
Resources,” “Serenity,” “Calm Before the Storm,” “Void,” “Sun,” “Storefront
7. Finally, Celeste wants “Narcissism” displayed if “Reflection” is displayed. So if the
decision variable for “Reflection” is 1, the decision variable for “Narcissism” must also
Cost Constraint
The cost of all of the pieces displayed has to be less than or equal to $4 million (C36 ≤
C38).
The problem formulation in an Excel spreadsheet follows.
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A B C D E F G H I J K L M N O P Q R S T U V W X Y Z AA
Artist Piece
Price ($thousand)
Collage?
Wire-Mesh Sculpture?
Computer-Generated Drawing?
Photo-Realistic Painting?
Cubist Painting?
Expressional Painting?
Water-Color Painting?
Oil Painting?
Painting?
Other Art Form?
Ash Briggs?
Candy Tate?
David Lyman?
Rick Rawls?
Ziggy Lite?
Female Artist?
Male Artist?
Advances Environmentalism?
Advances Native American Rights?
Advances Science?
Sits on Floor?
Hangs on Wall?
Include?
Colin Zweibell Perfection 300 1 1 1 1 0
Colin Zweibell Burden 250 1 1 1 1 0
Colin Zweibell The Great Equalizer 125 1 1 1 1 1
Rita Losky Chaos Reigns 400 1 1 1 1 1 1
Rita Losky Who Has Control? 500 1 1 1 1 1 0
Rita Losky Domestication 400 1 1 1 0
Rita Losky Innocence 550 1 1 1 0
Norm Marson Aging Earth 700 1 1 1 1 0
Norm Marson Wasted Resources 575 1 1 1 1 1 1
Candy Tate Serenity 200 1 1 1 1 1 1
Candy Tate Calm before the Storm 225 1 1 1 1 1 1
Robert Bayer Void 150 1 1 1 1 1
Robert Bayer Sun 150 1 1 1 1 0
David Lyman Storefront W indow 850 1 1 1 1 1 0
David Lyman Harley 750 1 1 1 1 1 1
Angie Oldman Consumerism 400 1 1 1 1 0
Angie Oldman Reflection 175 1 1 1 1
Angie Oldman Trojan Victory 450 1 1 1 0
Rick Rawls Rick 500 1 1 1 1 1 0
Rick Rawls Rick II 500 1 1 1 1 1 0
Rick Rawls Rick III 500 1 1 1 1 1 1
Bill Reynolds Beyond 650 1 1 1 1 1 0
Bill Reynolds Pioneers 650 1 1 1 1 1 0
Bear Canton Wisdom 250 1 1 1 1 1
Bear Canton Superior Powers 350 1 1 1 1 0
Bear Canton Living Land 450 1 1 1 1 1 0
Helen Row Study of a Violin 400 1 1 1 1 0
Helen Row Study of a Fruit Bowl 400 1 1 1 1 1
Ziggy Lite My Namesake 300 1 1 1 1 1 0
Ziggy Lite Narcissism 300 1 1 1 1 1 0
Ash Briggs All That Glitters 50 1 1 1 1 1 1
Ash Briggs The Rock 50 1 1 1 1 1 1
Ash Briggs Winding Road 50 1 1 1 1 1 1
Ash Briggs Dreams Come True 50 1 1 1 1 1 1
Total 3,950 1 1 1 1 1 1 6 1 10 5 4 2 1 1 0 5 10 1 1 1 2 13 15
7-18
Range Name Cells
Budget C38
Include? AA2:AA35
TotalPieces AA36
TotalPrice C36
In the optimal solution, 15 pieces are displayed at a cost of $3.95 million. The
following pieces are displayed:
1. “The Great Equalizer” by Colin Zweibell
2. “Chaos Reigns” by Rita Losky
b) The formulation of this problem is the same as the formulation in part (a) except that
the objective function from part (a) now becomes a constraint and the cost constraint
from part (a) now becomes the objective function. Thus, we have the new constraint
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35
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z AA
Artist Piece
Price ($thousand)
Collage?
Wire-Mesh Sculpture?
Computer-Generated Drawing?
Photo-Realistic Painting?
Cubist Painting?
Expressional Painting?
Water-Color Painting?
Oil Painting?
Painting?
Other Art Form?
Ash Briggs?
Candy Tate?
David Lyman?
Rick Rawls?
Ziggy Lite?
Female Artist?
Male Artist?
Advances Environmentalism?
Advances Native American Rights?
Advances Science?
Sits on Floor?
Hangs on Wall?
Include?
Colin Zweibell Perfection 300 1 1 1 1 1
Colin Zweibell Burden 250 1 1 1 1 1
Colin Zweibell The Great Equalizer 125 1 1 1 1 1
Rita Losky Chaos Reigns 400 1 1 1 1 1 1
Rita Losky Who Has Control? 500 1 1 1 1 1 0
Rita Losky Domestication 400 1 1 1 1
Rita Losky Innocence 550 1 1 1 0
Norm Marson Aging Earth 700 1 1 1 1 0
Norm Marson Wasted Resources 575 1 1 1 1 1 1
Candy Tate Serenity 200 1 1 1 1 1 1
Candy Tate Calm before the Storm 225 1 1 1 1 1 1
Robert Bayer Void 150 1 1 1 1 1
Robert Bayer Sun 150 1 1 1 1 1
David Lyman Storefront Window 850 1 1 1 1 1 0
David Lyman Harley 750 1 1 1 1 1 1
Angie Oldman Consumerism 400 1 1 1 1 0
Angie Oldman Reflection 175 1 1 1 1
Angie Oldman Trojan Victory 450 1 1 1 0
Rick Rawls Rick 500 1 1 1 1 1 0
Rick Rawls Rick II 500 1 1 1 1 1 0
Rick Rawls Rick III 500 1 1 1 1 1 1
Bill Reynolds Beyond 650 1 1 1 1 1 0
Bill Reynolds Pioneers 650 1 1 1 1 1 0
Bear Canton Wisdom 250 1 1 1 1 1
Bear Canton Superior Powers 350 1 1 1 1 0
Bear Canton Living Land 450 1 1 1 1 1 0
Helen Row Study of a Violin 400 1 1 1 1 1
Helen Row Study of a Fruit Bowl 400 1 1 1 1 1
Ziggy Lite My Namesake 300 1 1 1 1 1 0
Ziggy Lite Narcissism 300 1 1 1 1 1 0
Ash Briggs All That Glitters 50 1 1 1 1 1 1
Ash Briggs The Rock 50 1 1 1 1 1 1
Ash Briggs Winding Road 50 1 1 1 1 1 1
Ash Briggs Dreams Come True 50 1 1 1 1 1 1
c) This problem is also a cost minimization problem. The problem formulation is the
same as that used in part (b). A new constraint is added, however. The patron wants
The problem formulation in Excel follows.
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A B C D E F G H I J K L M N O P Q R S T U V W X Y Z AA AB
Artist Piece
Price ($thousand)
Collage?
Wire-Mesh Sculpture?
Computer-Generated Drawing?
Photo-Realistic Painting?
Cubist Painting?
Expressional Painting?
Water-Color Painting?
Oil Painting?
Painting?
Other Art Form?
Ash Briggs?
Candy Tate?
David Lyman?
Rick Rawls?
Ziggy Lite?
Rita Losky?
Female Artist?
Male Artist?
Advances Environmentalism?
Advances Native American Rights?
Advances Science?
Sits on Floor?
Hangs on Wall?
Include?
Colin Zweibell Perfection 300 1 1 1 1 0
Colin Zweibell Burden 250 1 1 1 1 1
Colin Zweibell The Great Equalizer 125 1 1 1 1 1
Rita Losky Chaos Reigns 400 1 1 1 1 1 1 1
Rita Losky Who Has Control? 500 1 1 1 1 1 1 1
Rita Losky Domestication 400 1 1 1 1 1
Rita Losky Innocence 550 1 1 1 1 1
Norm Marson Aging Earth 700 1 1 1 1 0
Norm Marson Wasted Resources 575 1 1 1 1 1 1
Candy Tate Serenity 200 1 1 1 1 1 1
Candy Tate Calm before the Storm 225 1 1 1 1 1 1
Robert Bayer Void 150 1 1 1 1 1
Robert Bayer Sun 150 1 1 1 1 1
David Lyman Storefront Window 850 1 1 1 1 1 0
David Lyman Harley 750 1 1 1 1 1 1
Angie Oldman Consumerism 400 1 1 1 1 0
Angie Oldman Reflection 175 1 1 1 1
Angie Oldman Trojan Victory 450 1 1 1 0
Rick Rawls Rick 500 1 1 1 1 1 0
Rick Rawls Rick II 500 1 1 1 1 1 0
Rick Rawls Rick III 500 1 1 1 1 1 1
Bill Reynolds Beyond 650 1 1 1 1 1 0
Bill Reynolds Pioneers 650 1 1 1 1 1 0
7-22
7.2 a) We want to maximize the total number of kitchen sets, so each of the 20 kitchen sets
becomes a decision variable. But the kitchen sets are not our only decision variables.
Because we assume that any particular item composing a kitchen set is replenished
immediately, we only need to stock one of each item. A particular item may compose
A handful of constraints exist in this problem.
1. We cannot indicate that a kitchen set is in stock unless all the items composing that
kitchen set are also in stock. Thus, a kitchen set decision variable is 1 only if all the
decision variables for the items composing that kitchen set are also 1. For example, for
set 1 this constraint equals (Set 1) <= (T2+W2+L4+C2+O4+S2+D2+R2) / 8.
4. A maximum of two different styles of light fixtures can be in stock.
5. A maximum of two different styles of cabinets can be in stock.
6. A maximum of three different styles of countertops can be in stock.
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A B C D E F G H I J K L M N O P Q R S T U V W X Y Z AA AB AC AD AE AF AG AH AI
Set In Fraction of
T1 T2 T3 T4 W1 W2 W3 W4 L1 L2 L3 L4 C1 C2 C3 C4 O1 O2 O3 O4 D1 D2 S1 S2 S3 S4 R1 R2 R3 R4 Stock? Items in Stock
Set 1 1 1 1 1 1 1 1 1 0 <= 0.625
Set 2 1 1 1 1 1 1 1 1 0 <= 0.75
Set 3 1 1 1 1 1 1 1 1 0 <= 0.625
Set 4 1 1 1 1 1 1 1 1 0 <= 0.5
Set 5 1 1 1 1 1 1 1 1 0 <= 0.375
Set 6 1 1 1 1 1 1 1 1 0 <= 0.625
Set 7 1 1 1 1 1 1 1 1 0 <= 0.375
Set 8 1 1 1 1 1 1 1 1 1 <= 1
Set 9 1 1 1 1 1 1 1 1 0 <= 0.625
Set 10 1 1 1 1 1 1 1 1 0 <= 0.5
Set 11 1 1 1 1 1 1 1 1 0 <= 0.75
Set 12 1 1 1 1 1 1 1 1 0 <= 0.75
Set 13 1 1 1 1 1 1 1 1 0 <= 0.375
Set 14 1 1 1 1 1 1 1 0 <= 0.286
Set 15 1 1 1 1 1 1 1 1 <= 1
Set 16 1 1 1 1 1 1 1 0 <= 0.429
Set 17 1 1 1 1 1 1 1 0 <= 0.143
Set 18 1 1 1 1 1 1 1 1 <= 1
Set 19 1 1 1 1 1 1 1 0 <= 0.429
Set 20 1 1 1 1 1 1 1 1 <= 1
Item In Stock? 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 1 0 1 1 0 1 0 0 1 1 1
Floor Tile
Wallpaper
Light Fixtures
Cabinets
Countertops
Sinks
Dwashers & Ranges
Total Sets in Stock
Total Items 2 2 2 2 3 2 4 4
<= <= <= <= <= <= <=
Sinks
Ranges
Light Fixtures
Cabinets
Countertops
Dwash
(>= 20 sq. ft.)
(>= 5 rolls)
Floor Tile
Wallpaper
Range Name Cells
Capacit y I 29:O 29
FractionInStock AI3:AI22
ItemInStock? B24:AE24
SetInStock? AG3:AG 22
T ot alItems I 27:O 27
TotalSetsInStock AG27
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AI
Fraction of
Items in Stock
=SUMPRODUCT(B3:AE3,ItemInStock?)/SUM(B3:AE3)
=SUMPRODUCT(B4:AE4,ItemInStock?)/SUM(B4:AE4)
=SUMPRODUCT(B5:AE5,ItemInStock?)/SUM(B5:AE5)
=SUMPRODUCT(B6:AE6,ItemInStock?)/SUM(B6:AE6)
=SUMPRODUCT(B7:AE7,ItemInStock?)/SUM(B7:AE7)
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H I J K L M N O
Floor Tile
Wallpaper
Light Fixtures
Cabinets
Countertops
Sinks
Dwashers & Ranges
Total Items =SUM(B24:E24) =SUM(F24:I24) =SUM(J24:M24) =SUM(N24:Q24) =SUM(R24:U24) =SUM(X24:AA24) =SUM(V24:W24,AB24:AE24)
b) We should stock the following items: T2, T3; W1, W3; L1, L3; C1, C2; O1, O2, O4;
D2; S1, S3; and R2, R3, and R4. This combination allows four different kitchen sets to
c) We model this new problem by changing the capacity constraint for the dishwashers
and ranges. Now, instead of being able to stock a combination of only four different
styles of dishwashers and ranges, we can stock a maximum of two different styles of
The formulation of the problem in Excel follows:
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A B C D E F G H I J K L M N O P Q R S T U V W X Y Z AA AB AC AD AE AF AG AH AI
Set In Fraction of
T1 T2 T3 T4 W1 W2 W3 W4 L1 L2 L3 L4 C1 C2 C3 C4 O1 O2 O3 O4 D1 D2 S1 S2 S3 S4 R1 R2 R3 R4 Stock? Items in Stock
Set 1 1 1 1 1 1 1 1 1 0 <= 0.25
Set 2 1 1 1 1 1 1 1 1 0 <= 0.5
Set 3 1 1 1 1 1 1 1 1 0 <= 0.75
Set 4 1 1 1 1 1 1 1 1 1 <= 1
Set 5 1 1 1 1 1 1 1 1 0 <= 0.625
Set 6 1 1 1 1 1 1 1 1 0 <= 0.5
Set 7 1 1 1 1 1 1 1 1 0 <= 0.75
Set 8 1 1 1 1 1 1 1 1 1 <= 1
Set 9 1 1 1 1 1 1 1 1 0 <= 0.625
Set 10 1 1 1 1 1 1 1 1 0 <= 0.75
Set 11 1 1 1 1 1 1 1 1 1 <= 1
Set 12 1 1 1 1 1 1 1 1 0 <= 0.5
Set 13 1 1 1 1 1 1 1 1 0 <= 0.625
Set 14 1 1 1 1 1 1 1 0 <= 0.571
Set 15 1 1 1 1 1 1 1 1 <= 1
Set 16 1 1 1 1 1 1 1 0 <= 0.714
Set 17 1 1 1 1 1 1 1 0 <= 0.429
Set 18 1 1 1 1 1 1 1 0 <= 0.571
Set 19 1 1 1 1 1 1 1 0 <= 0.286
Set 20 1 1 1 1 1 1 1 1 <= 1
Item In Stock? 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 0 1 0 1 1
Floor Tile
Wallpaper
Light Fixtures
Cabinets
Countertops
Sinks
Ranges
Total Sets in Stock
Total Items 2 2 2 2 3 2 3 5
<= <= <= <= <= <= <=
(>= 20 sq. ft.)
(>= 5 rolls)
Floor Tile
Wallpaper
Sinks
Ranges
Light Fixtures
Cabinets
Countertops
Dwash
With the extra space, the number of kitchen sets we can stock increases from four to
five—now sets 4, 8, 11, 15, and 20. To keep these five sets in stock, we stock the
7-26
7.3 a) Since a residential area cannot be split across multiple schools, the decision becomes
which school to send each area’s students to. This is formulated with a binary variable
for each area/school combination, representing the yes-or-no decision of whether that
The number of students in each school is then calculated in StudentAssignments
(B24:D29) based upon the results in AreaAssignments (B14:D19).
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41
42
A B C D E F G H
Data: Percentage Percentage Percentage
Number in 6th in 7th in 8th Bussing Cost ($/Student)
Area of Students Grade Grade Grade School 1 School 2 School 3
1 450 32% 38% 30% $300 $0 $700
2 600 37% 28% 35% Ğ$400 $500
3 550 30% 32% 38% $600 $300 $200
4 350 28% 40% 32% $200 $500 Ğ
5 500 39% 34% 27% $0 Ğ$400
6 450 34% 28% 38% $500 $300 $0
Area Assig n ments Total
School 1 School 2 School 3 Assignments Supply
Area 1 1 0 0 1 = 1
Area 2 0 1 0 1 = 1
Area 3 0 0 1 1 = 1
Area 4 0 1 0 1 = 1
Area 5 0 0 1 1 = 1
Area 6 1 0 0 1 = 1
Stud ent Assignments
School 1 School 2 School 3
Area 1 450 0 0
Area 2 0 600 0
Area 3 0 0 550 Total
Area 4 0 350 0 Bussing
Area 5 0 0 500 Cost
Area 6 450 0 0 $1,085,000
Total In School 900 950 1,050
<= <= <=
Capacity 900 1,100 1,000
Grade Con straints:
270 285 315 30% of total in school
7-27
The formulas for the spreadsheet formulation are shown below.
Range Name Cells
AreaAssignments B14:D19
BussingCost F4:H9
Capacity B32:D32
NumberOfStudents B4:B9
PercentageInGrade C4:E9
StudentAssignments B24:D29
Supply G14:G 19
TotalAssignments E14:E19
TotalBussingCost G29
TotalInSchool B30:D30
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14
15
16
17
18
19
E
Total
Assignments
= SUM(B14:D14)
= SUM(B15:D15)
= SUM(B16:D16)
= SUM(B17:D17)
= SUM(B18:D18)
= SUM(B19:D19)
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25
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27
28
29
30
A B C D
Student Assignm
School 1 School 2 School 3
Area 1 = $B4*B14 = $B4*C14 =$B4*D14
Area 2 = $B5*B15 = $B5*C15 =$B5*D15
Area 3 = $B6*B16 = $B6*C16 =$B6*D16
Area 4 = $B7*B17 = $B7*C17 =$B7*D17
Area 5 = $B8*B18 = $B8*C18 =$B8*D18
Area 6 = $B9*B19 = $B9*C19 =$B9*D19
Total In School = SUM(B24:B29) = SUM(C24:C29) =SUM(D24:D29)
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G
Total
Bussing
Cost
=SUMPRODUCT(BussingCost,StudentAssignments)
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37
38
39
40
41
42
A B C D E
Grad e Constraints:
=$E$36*TotalInSchool = $E$36*TotalInSchool = $E$36*TotalInSchool 0.3
<= <= <=
6th G raders =SUMPRODUCT(B24:B29,$C$4:$C$9) = SUMPRODUCT(C24:C29,$C$4:$C$9) = SUMPRODUCT(D24:D29,$C$4:$C$9)
7th G raders =SUMPRODUCT(B24:B29,$D$4:$D$9) = SUMPRODUCT(C24:C29,$D$4:$D$9) = SUMPRODUCT(D24:D29,$D$4:$D$9)
8th G raders =SUMPRODUCT(B24:B29,$E$4:$E$9) = SUMPRODUCT(C24:C29,$E$4:$E$9) = SUMPRODUCT(D24:D29,$E$4:$E$9)
<= <= <=
=$E$42*TotalInSchool = $E$42*TotalInSchool = $E$42*TotalInSchool 0.36
b) Without prohibiting the splitting of residential areas, the total cost was $555,556. Thus,
adding this restriction increases the cost by $1,085,000 – $555,556 = $529,443.
7-28
c) As shown in the spreadsheet, the solution remains the same, but the bussing costs are
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A B C D E F G H
Data: Percentage Percentage Percentage
Number in 6th in 7th in 8th Bussing Cost ($/Student)
Area of Students Grade Grade Grade School 1 School 2 School 3
1 450 32% 38% 30% $300 $0 $700
2 600 37% 28% 35% Š$400 $500
3 550 30% 32% 38% $600 $300 $0
4 350 28% 40% 32% $0 $500 Š
5 500 39% 34% 27% $0 Š$400
6 450 34% 28% 38% $500 $300 $0
Area Assig n ments Total
School 1 School 2 School 3 Assignments Supply
Area 1 1 0 0 1 = 1
Area 2 0 1 0 1 = 1
Area 3 0 0 1 1 = 1
Area 4 0 1 0 1 = 1
Area 5 0 0 1 1 = 1
Area 6 1 0 0 1 = 1
Stud ent Assignments
School 1 School 2 School 3
Area 1 450 0 0
Area 2 0 600 0
Area 3 0 0 550 Total
Area 4 0 350 0 Bussing
Area 5 0 0 500 Cost
Area 6 450 0 0 $975,000
Total In School 900 950 1,050
<= <= <=
Capacity 900 1,100 1,000
Grade Con straints:
270 285 315 30% of total in school
<= <= <=
6th G raders 297 320 360
7th G raders 297 308 346
8th G raders 306 322 344
<= <= <=
324 342 378 36% of total in school
7-29
d) Again, the solution remains the same, but the bussing costs are reduced to $840,000.
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5
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25
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A B C D E F G H
Data: Percentage Percentage Percentage
Number in 6th in 7th in 8th Bussing Cost ($/Student)
Area of Students Grade Grade Grade School 1 School 2 School 3
1 450 32% 38% 30% $0 $0 $700
2 600 37% 28% 35% Š$400 $500
3 550 30% 32% 38% $600 $0 $0
4 350 28% 40% 32% $0 $500 Š
5 500 39% 34% 27% $0 Š$400
6 450 34% 28% 38% $500 $0 $0
Area Assig n ments Total
School 1 School 2 School 3 Assignments Supply
Area 1 1 0 0 1 = 1
Area 2 0 1 0 1 = 1
Area 3 0 0 1 1 = 1
Area 4 0 1 0 1 = 1
Area 5 0 0 1 1 = 1
Area 6 1 0 0 1 = 1
Stud ent Assignments
School 1 School 2 School 3
Area 1 450 0 0
Area 2 0 600 0
Area 3 0 0 550 Total
Area 4 0 350 0 Bussing
Area 5 0 0 500 Cost
Area 6 450 0 0 $840,000
Total In School 900 950 1,050
<= <= <=
Capacity 900 1,100 1,000
Grade Con straints:
270 285 315 30% of total in school
<= <= <=
6th G raders 297 320 360
7th G raders 297 308 346
8th G raders 306 322 344
<= <= <=
324 342 378 36% of total in school
f) Arguments can be made for all three alternatives. Answers will vary.
