3-36
b) The linear programming spreadsheet model for this problem is shown below.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
A B C D E F G H I J K L M N O P
Button–
Wool Cashmere Silk Silk Tailored Wool Velvet Cotton Cotton Velvet Down
Slacks Sweater Blouse Camisole Skirt Blazer Pants Sweater Miniskirt Shirt Blouse
Price $300 $450 $180 $120 $270 $320 $350 $130 $75 $200 $120
L&M Cost $160 $150 $100 $60 $120 $140 $175 $60 $40 $160 $90
Material Cost $30.00 $90.00 $19.50 $6.50 $6.75 $24.75 $39.00 $3.75 $1.25 $18.00 $3.38
Net Contribution $110.00 $210.00 $60.50 $53.50 $143.25 $155.25 $136.00 $66.25 $33.75 $22.00 $26.63
Cost of Material Material
Material Material Requirements Used Available
Wool $9.00 3 2.5 25,100 <= 45,000
Acetate $1.50 2 1.5 1.5 2 28,000 <= 28,000
Cashmere $60.00 1.5 6,000 <= 9,000
Silk $13.00 1.5 0.5 18,000 <= 18,000
Rayon $2.25 2 1.5 30,000 <= 30,000
Velvet $12.00 3 1.5 9,000 <= 20,000
Cotton $2.50 1.5 0.5 30,000 <= 30,000
Button–
Wool Cashmere Silk Silk Tailored Wool Velvet Cotton Cotton Velvet Down Total
Slacks Sweater Blouse Camisole Skirt Blazer Pants Sweater Miniskirt Shirt Blouse Contribution
Items Produced 4,200 4,000 7,000 15,000 8,067 5,000 0 0 60,000 6,000 9,244 $6,862,933
<= <= <= <= <= <= <= Fixed Cost $8,960,000
Demand Forecast 7,000 4,000 12,000 15,000 5,000 5,500 6,000 Total Profit -$2,097,067
>= >= >=
Minimum Production 4,200 2,800 3,000 Also:
60% 60% Silk Camisole > = Silk Blouse
of demand of demand Cotton Miniskirt >= Cott on Sweater
6
7
B C D
Material Cost =SUMPRODUCT(CostOfMaterial,C11:C17) =SUMPRODUCT(CostOfMaterial,D11:D17)
Net Contribution =Price-LMCost–MaterialCost =Price–LMCost-MaterialCost
Rang e Name Cells
TotalProfit P24
9
10
11
12
13
14
15
16
17
N
Material
Used
= SUMPRO DUCT (C11: M11,ItemsProduced)
= SUMPRO DUCT (C12: M12,ItemsProduced)
= SUMPRO DUCT (C13: M13,ItemsProduced)
= SUMPRO DUCT (C14: M14,ItemsProduced)
= SUMPRO DUCT (C15: M15,ItemsProduced)
= SUMPRO DUCT (C16: M16,ItemsProduced)
= SUMPRO DUCT (C17: M17,ItemsProduced)
20
21
22
23
24
O P
Total
Contribution
= SUMPRO DUCT(NetContribution,ItemsProduced)
Fixed Cost 8960000
Total Profit = TotalContribution-F ixedCost
TrendLine should produce 4,200 Wool Slacks, 4,000 Cashmere Sweaters, 7,000 Silk
Blouses, 15,000 Silk Camisoles, 8,067 Tailored Skirts, 5,000 Wool Blazers, 40,000
3-37
c) If velvet cannot be sent back to the textile wholesaler, then the whole quantity will be
considered as a sunk cost and therefore added to the fixed costs. The objective function
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
A B C D E F G H I J K L M N O P
Button–
Wool Cashmere Silk Silk Tailored Wool Velvet Cotton Cotton Velvet Down
Slacks Sweater Blouse Camisole Skirt Blazer Pants Sweater Miniskirt Shirt Blouse
Price $300 $450 $180 $120 $270 $320 $350 $130 $75 $200 $120
L&M Cost $160 $150 $100 $60 $120 $140 $175 $60 $40 $160 $90
Material Cost $30.00 $90.00 $19.50 $6.50 $6.75 $24.75 $3.75 $1.25 $3.38
Net Contribut ion $110.00 $210.00 $60.50 $53.50 $143.25 $155.25 $175.00 $66.25 $33.75 $40.00 $26.63
Cost of Material Material
Material Material Requirements Used Available
Wool $9.00 3 2.5 25,100 <= 45,000
Acetate $1.50 2 1.5 1.5 2 28,000 <= 28,000
Cashmere $60.00 1.5 6,000 <= 9,000
Silk $13.00 1.5 0.5 18,000 <= 18,000
Rayon $2.25 2 1.5 30,000 <= 30,000
Velvet $12.00 3 1.5 20,000 <= 20,000
Cotton $2.50 1.5 0.5 30,000 <= 30,000
Button–
Wool Cashmere Silk Silk Tailored Wool Velvet Cotton Cotton Velvet Down Total
Slacks Sweater Blouse Camisole Skirt Blazer Pants Sweater Miniskirt Shirt Blouse Contribution
Items Produced 4,200 4,000 7,000 15,000 3,178 5,000 3,667 0 60,000 6,000 15,763 $7,085,822
<= <= <= <= <= <= <= Original Fixed Cost $8,960,000
Demand Forecast 7,000 4,000 12,000 15,000 5,000 5,500 6,000 Velvet Sunk Cost $240,000
>= >= >= Total Profit –$2,114,178
Minimum Product ion 4,200 2,800 3,000 Also:
60% 60% Silk Camisole >= Silk Blouse
of demand of demand Cotton Miniskirt > = Cotton Sweater
25
20
21
22
23
24
O P
Total
Contribution
=SUMPRODUCT(NetContribution,ItemsProduced)
Original Fixed Cost 8960000
Velvet Sunk Cost = B16*P16
The production plan changes considerably. TrendLines should produce 3,178 tailored
skirts (down from 8,067), 3,667 velvet pants (up from 0), 60,000 cotton minis (up from
40,000), and 15,763 button-down blouses (up from 9,244). The production decisions
d) When TrendLines cannot return the velvet to the wholesaler, the costs for velvet cannot
be recovered. These cost are no longer variable cost but now are sunk cost. As a
consequence the increased net contribution of the velvet items makes them more
attractive to produce. This way the revenues from selling these items can contribute to
3-38
e) The unit contribution of a wool blazer changes to $75.25.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
A B C D E F G H I J K L M N O P
Button–
Wool Cashmere Silk Silk Tailored W ool Velvet Cotton Cotton Velvet Down
Slacks Sweater Blouse Camisole Skirt Blazer Pants Sweater Miniskirt Shirt Blouse
Price $300 $450 $180 $120 $270 $320 $350 $130 $75 $200 $120
L&M Cost $160 $150 $100 $60 $120 $220 $175 $60 $40 $160 $90
Material Cost $30.00 $90.00 $19.50 $6.50 $6.75 $24.75 $39.00 $3.75 $1.25 $18.00 $3.38
Net Contribution $110.00 $210.00 $60.50 $53.50 $143.25 $75.25 $136.00 $66.25 $33.75 $22.00 $26.63
Cost of Material Material
Material Material Requirements Used Available
Wool $9.00 3 2.5 20,100 <= 45,000
Acetate $1.50 2 1.5 1.5 2 28,000 <= 28,000
Cashmere $60.00 1.5 6,000 <= 9,000
Silk $13.00 1.5 0.5 18,000 <= 18,000
Rayon $2.25 2 1.5 30,000 <= 30,000
Velvet $12.00 3 1.5 9,000 <= 20,000
Cotton $2.50 1.5 0.5 30,000 <= 30,000
Button–
Wool Cashmere Silk Silk Tailored W ool Velvet Cotton Cotton Velvet Down Total
Slacks Sweater Blouse Camisole Skirt Blazer Pants Sweater Miniskirt Shirt Blouse Contribution
Items Produced 4,200 4,000 7,000 15,000 10,067 3,000 0 0 60,000 6,000 6,578 $6,527,933
<= <= <= <= <= <= <= Fixed Cost $8,960,000
Demand Forecast 7,000 4,000 12,000 15,000 5,000 5,500 6,000 Total Profit -$2,432,067
>= >= >=
Minimum Production 4,200 2,800 3,000 Also:
60% 60% Silk Camisole >= Silk Blouse
of demand of demand Cotton Miniskirt >= Cotton Sweater
TrendLines should produce 10,067 skirts (up from 8,067), the minimum of 3,000 wool
blazers (down from 5,000), and 6,578 button-down blouses (down from 9,244). The
f) The available acetate changes from 28,000 to 38,000 square yards. The resulting
spreadsheet solution is shown below.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
A B C D E F G H I J K L M N O P
Button–
Wool Cashmere Silk Silk Tailored W ool Velvet Cotton Cotton Velvet Down
Slacks Sweater Blouse Camisole Skirt Blazer Pants Sweater Miniskirt Shirt Blouse
Price $300 $450 $180 $120 $270 $320 $350 $130 $75 $200 $120
L&M Cost $160 $150 $100 $60 $120 $140 $175 $60 $40 $160 $90
Material Cost $30.00 $90.00 $19.50 $6.50 $6.75 $24.75 $39.00 $3.75 $1.25 $18.00 $3.38
Net Contribution $110.00 $210.00 $60.50 $53.50 $143.25 $155.25 $136.00 $66.25 $33.75 $22.00 $26.63
Cost of Material Material
Material Material Requirements Used Available
Wool $9.00 3 2.5 25,100 <= 45,000
Acetate $1.50 2 1.5 1.5 2 38,000 <= 38,000
Cashmere $60.00 1.5 6,000 <= 9,000
Silk $13.00 1.5 0.5 18,000 <= 18,000
Rayon $2.25 2 1.5 30,000 <= 30,000
Velvet $12.00 3 1.5 9,000 <= 20,000
Cotton $2.50 1.5 0.5 30,000 <= 30,000
Button–
Wool Cashmere Silk Silk Tailored W ool Velvet Cotton Cotton Velvet Down Total
Slacks Sweater Blouse Camisole Skirt Blazer Pants Sweater Miniskirt Shirt Blouse Contribution
Items Produced 4,200 4,000 7,000 15,000 14,733 5,000 0 0 60,000 6,000 356 $7,581,267
<= <= <= <= <= <= <= Fixed Cost $8,960,000
Demand Forecast 7,000 4,000 12,000 15,000 5,000 5,500 6,000 Total Profit -$1,378,733
>= >= >=
Minimum Production 4,200 2,800 3,000 Also:
60% 60% Silk Camisole >= Silk Blouse
of demand of demand Cotton Miniskirt > = Cotton Sweater
TrendLines should produce 14,733 skirts (up from 8,067) and 356 button-down blouses
(down from 9,244). The production decisions for all other items are unaffected by the
3-40
3.4 a) We define 12 decision variables, one for each age group surveyed in each region. Rob’s
restrictions are easily modeled as constraints. For example, his condition that at least 20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
A B C D E F G H I J
Cost o f Survey
18 to 25 26 to 40 41 to 50 51 and over
Silicon Valley $4.75 $6.50 $6.50 $5.00
Region Big Cities $5.25 $5.75 $6.25 $6.25
Small Towns $6.50 $7.50 $7.50 $7.25
Percentage
Total Required Required
Number to Survey 18 to 25 26 to 40 41 to 50 51 and over in Region in Region in Region
Silicon Valley 600 0 0 300 900 >= 300 15%
Region Big Cities 150 550 0 0 700 >= 700 35%
Small Towns 100 0 300 0 400 >= 400 20%
Total in A.G. 850 550 300 300
>= >= >= >= Total Surveys 2000
Required in A.G. 400 550 300 300 =
Percentage Required in A.G. 20% 27.5% 15% 15% Required Surveys 2000
Total Cost $11,200
Profit Margin 15%
Bid $12,880
Age Group
Age Group
Range Name Cells
CostOfSurvey C3:F5
NumberToSurvey C9:F11
PercentageRequiredInAG C15:F15
PercentageRequiredInRegion J9:J11
RequiredInAG C14:F 14
RequiredInRegion I9:I11
RequiredSurveys J15
TotalCost J17
TotalInAG C12:F12
TotalInRegion G9:G11
TotalSurveys J13
7
8
9
10
11
G H I
Total Required
in Region in Region
= SUM(C9:F 9) >= = J9*RequiredSurveys
= SUM(C10: F10) >= = J10*RequiredSurveys
= SUM(C11: F11) >= = J11*RequiredSurveys
12
13
14
B C D
Total in A.G.=SUM(C9:C11) =SUM(D9:D11)
>= >=
Required in A.G. =C15*RequiredSurveys =D15*RequiredSurveys
13
14
15
16
17
I J
Total Surveys=SUM(NumberToSurvey)
=
Required Surveys 2000
Total Cost =SUMPRODUCT(CostOfSurvey,NumberToSurvey
3-41
b) Sophisticated Surveys will submit a bid of (1.15)($11,200) = $12,880.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A B C D E F G H I J
Cost of Su rvey
18 to 25 26 to 40 41 to 50 51 and over
Silicon Valley $4.75 $6.50 $6.50 $5.00
Region Big Cities $5.25 $5.75 $6.25 $6.25
Small Towns $6.50 $7.50 $7.50 $7.25
Percentage
Total Required Required
Number to Survey 18 to 25 26 to 40 41 to 50 51 and over in Region in Region in Region
Silicon Valley 600 50 50 200 900 >= 300 15%
Region Big Cities 50 450 150 50 700 >= 700 35%
Small Towns 200 50 100 50 400 >= 400 20%
Total in A.G. 850 550 300 300
>= >= >= >= Total Surveys 2000
Required in A.G. 400 550 300 300 =
Percentage Required in A.G. 20% 27.5% 15% 15% Required Surveys 2000
Total Cost $11,388
Minimum to Survey 18 to 25 26 to 40 41 to 50 51 and over
Silicon Valley 50 50 50 50 Profit Margin 15%
Region Big Cities 50 50 50 50 Bid $13,096
Small Towns 50 50 50 50
(Number to Survey >= Minimum to Survey)
Age G roup
Age G roup
Age G roup
The new requirement increases the bid to $13,096.
d) We include upper bounds on the total number of people surveyed in Silicon Valley and
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
A B C D E F G H I J K L
Cost of Survey
18 to 25 26 to 40 41 to 50 51 and over
Silicon Valley $4.75 $6.50 $6.50 $5.00
Region Big Cities $5.25 $5.75 $6.25 $6.25
Small Towns $6.50 $7.50 $7.50 $7.25
Percentage Max in
Total Required Required Silicon
Number to Survey 18 to 25 26 to 40 41 to 50 51 and over in Region in Region in Region Valley
Silicon Valley 100 50 50 450 650 >= 300 15% <= 650
Region Big Cities 400 450 50 50 950 >= 700 35%
Small Towns 100 50 200 50 400 >= 400 20%
Total in A.G. 600 550 300 550
>= >= >= >= T otal Surveys 2000
Required in A.G. 400 550 300 300 =
Percentage Required in A.G. 20% 27.5% 15% 15% Required Surveys 2000
<=
MaxIn18to25 600 Total Cost $11,575
Profit Margin 15%
Minimum to Survey 18 to 25 26 to 40 41 to 50 51 and over Bid $13,311
Silicon Valley 50 50 50 50
Region Big Cities 50 50 50 50
Small Towns 50 50 50 50
(Number to Survey >= Minimum to Survey)
Age Group
Age Group
Age Group
The new requirements increase the bid to $13,311.
3-42
e) The three cost factors for the age group “18 to 25” are changed.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
A B C D E F G H I J K L
Cost of Survey
18 to 25 26 to 40 41 to 50 51 and over
Silicon Valley $6.50 $6.50 $6.50 $5.00
Region Big Cities $6.75 $5.75 $6.25 $6.25
Small T owns $7.00 $7.50 $7.50 $7.25
Percentage Max in
Total Required Required Silicon
Number to Survey 18 to 25 26 to 40 41 to 50 51 and over in Region in Region in Region Valley
Silicon Valley 50 50 50 500 650 >= 300 15% <= 650
Region Big Cities 100 600 200 50 950 >= 700 35%
Small T owns 250 50 50 50 400 >= 400 20%
Total in A.G. 400 700 300 600
>= >= >= >= Total Surveys 2000
Required in A.G. 400 550 300 300 =
Percentage Required in A.G. 20% 27.5% 15% 15% Required Surveys 2000
<=
MaxIn18to25 600 Total Cost $12,025
Profit Margin 15%
Minimum to Survey 18 to 25 26 to 40 41 to 50 51 and over Bid $13,829
Silicon Valley 50 50 50 50
Region Big Cities 50 50 50 50
Small T owns 50 50 50 50
(Number to Survey > = Minimum to Survey)
Age Group
Age Group
Age Group
With the new cost factors the bid increases to $13,829.
f) We eliminate all lower and upper bounds on the age groups and regions and replace
them with Rob’s strict requirements. These requirements also ensure that exactly 2000
people are surveyed so that we can drop that constraint too.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
A B C D E F G H I J
Cost of Su rvey
18 to 25 26 to 40 41 to 50 51 and over
Silicon Valley $6.50 $6.50 $6.50 $5.00 Required Surveys 2,000
Region Big Cities $6.75 $5.75 $6.25 $6.25
Small Towns $7.00 $7.50 $7.50 $7.25
Percentage
Total Required Required
Number to Survey 18 to 25 26 to 40 41 to 50 51 and over in Region in Region in Region
Silicon Valley 50 50 50 250 400 = 400 20%
Region Big Cities 50 600 300 50 1000 = 1000 50%
Small Towns 400 50 50 100 600 = 600 30%
Total in A.G . 500 700 400 400
= = = = Total Cost $12,475
Required in A.G. 500 700 400 400
Percentage Required in A.G. 25% 35% 20% 20% Profit Margin 15%
Bid $14,346
Minimum to Survey 18 to 25 26 to 40 41 to 50 51 and over
Silicon Valley 50 50 50 50
Region Big Cities 50 50 50 50
Small Towns 50 50 50 50
(Number to Survey ³ Minimum to Survey)
Age G roup
Age G roup
Age G roup
Rob’s strict requirements increase the cost of the survey by $450. The new bid of
Sophisticated Surveys is $14,346.25.
3-43
3.5 a)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
A B C D E F G
Data: Percentage Percentage Percentage
in 6th in 7th in 8th Bussing Cost ($/Student)
Area Grade Grade Grade School 1 School 2 School 3
1 32% 38% 30% $300 $0 $700
2 37% 28% 35% – $400 $500
3 30% 32% 38% $600 $300 $200
4 28% 40% 32% $200 $500 –
5 39% 34% 27% $0 – $400
6 34% 28% 38% $500 $300 $0
Solu tion: Number of Students Assigned Total Number of
School 1 School 2 School 3 From Area Students
Area 1 0 450 0 450 = 450
Area 2 0 422.22 177.78 600 = 600
Area 3 0 227.78 322.22 550 = 550
Area 4 350 0 0 350 = 350
Area 5 366.67 0 133.33 500 = 500
Area 6 83.33 0 366.67 450 = 450
Total In School 800 1,100 1,000
<= <= <= Total
Capacity 900 1,100 1,000 Bussing
Cost
$555,556
Grade Constraints:
240 330 300 30% of total in school
<= <= <=
6th Graders 269.33 368.56 339.11
7th Graders 288.00 362.11 300.89
8th Graders 242.67 369.33 360.00
<= <= <=
288 396 360 36% of total in school
Range Name Cells
BussingCost E4:G 9
Capacity B22:D22
NumberOfStudents G14:G19
PercentageInGrade B4:D9
Solution B14:D19
TotalBussingCost G24
TotalFromArea E14:E19
TotalInSchool B20:D20
12
13
14
15
16
17
18
19
E
Total
From Area
= SUM(B14: D14)
= SUM(B15: D15)
= SUM(B16: D16)
= SUM(B17: D17)
= SUM(B18: D18)
= SUM(B19: D19)
21
22
23
24
G
Total
Bussing
Cost
=SUMPRODUCT(BussingCost,Solution)
20
A B C D
Total In School=SUM(B14:B19) =SUM(C14:C19) =SUM(D14:D19)
25
26
27
28
29
30
31
32
A B C D E
Grade Constraints:
=$E$26*TotalInSchool = $E$26*TotalInSchool =$E$26*TotalInSchool 0.3
<= <= <=
6th Graders =SUMPRODUCT(B14:B19,B4:B9)=SUMPRODUCT(C14:C19,B4:B9)=SUMPRODUCT(D14:D19,B4:B9)
7th Graders =SUMPRODUCT(B14:B19,C4:C9)= SUMPRODUCT(C14:C19,C4:C9)=SUMPRODUCT(C4:C9,D14:D19)
8th Graders =SUMPRODUCT(B14:B19,D4:D9)= SUMPRODUCT(C14:C19,D4:D9)=SUMPRODUCT(D14:D19,D4:D9)
<= <= <=
=$E$32*TotalInSchool = $E$32*TotalInSchool =$E$32*TotalInSchool 0.36
b) The recommendation to the school board is to assign students to schools as shown in
the above solution section of the spreadsheet. Quantities that are not integers must be
rounded since partial students cannot be sent.
3-44
c) The following solution decreases total bussing costs by over $135,000 but violates the
grade constraints that were imposed. Solutions will vary and those than satisfy the
grade constraints will increase the total bussing costs.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
A B C D E F G
Data: Percentage Percentage Percentage
in 6th in 7th in 8th Bussing Cost ($/Student)
Area Grade Grade Grade School 1 School 2 School 3
1 32% 38% 30% $300 $0 $700
2 37% 28% 35% – $400 $500
3 30% 32% 38% $600 $300 $200
4 28% 40% 32% $200 $500 –
5 39% 34% 27% $0 – $400
6 34% 28% 38% $500 $300 $0
Solutio n: Number of Students Assigned Total Number of
School 1 School 2 School 3 From Area Students
Area 1 0 450 0 450 = 450
Area 2 0 600 0 600 = 600
Area 3 0 0 550 550 = 550
Area 4 350 0 0 350 = 350
Area 5 500 0 0 500 = 500
Area 6 0 0 450 450 = 450
Total In School 850 1,050 1,000
<= <= <= Total
Capacity 900 1,100 1,000 Bussing
Cost
$420,000
Grade Con straints:
255 315 300 30% of total in school
<= <= <=
6th G raders 293.00 366.00 318.00
7th G raders 310.00 339.00 302.00
8th G raders 247.00 345.00 380.00
<= <= <=
306 378 360 36% of total in school
3-45
d) The number of students assigned from each area to each school changes to the solution
shown below and the total bussing cost is reduced by almost $162,000.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
A B C D E F G
Data: Percentage Percentage Percentage
in 6th in 7th in 8th Bussing Cost ($/Student)
Area Grade Grade Grade School 1 School 2 School 3
1 32% 38% 30% $300 $0 $700
2 37% 28% 35% – $400 $500
3 30% 32% 38% $600 $300 $0
4 28% 40% 32% $0 $500 –
5 39% 34% 27% $0 – $400
6 34% 28% 38% $500 $300 $0
Solutio n: Number of Students Assigned Total Number of
School 1 School 2 School 3 From Area Students
Area 1 0 450 0 450 = 450
Area 2 0 600 0 600 = 600
Area 3 0 0 550 550 = 550
Area 4 350 0 0 350 = 350
Area 5 318.18 0 181.82 500 = 500
Area 6 131.82 50 268.18 450 = 450
Total In School 800 1,100 1,000
<= <= <= Total
Capacity 900 1,100 1,000 Bussing
Cost
$393,636
Grade Con straints:
240 330 300 30% of total in school
<= <= <=
6th G raders 266.91 383.00 327.09
7th G raders 285.09 353.00 312.91
8th G raders 248.00 364.00 360.00
<= <= <=
288 396 360 36% of total in school
3-46
e) The number of students assigned from each area to each school changes to the solution
shown below and the total bussing cost is reduced by over $215,000.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
A B C D E F G
Data: Percentage Percentage Percentage
in 6th in 7th in 8th Bussing Cost ($/Student)
Area Grade Grade Grade School 1 School 2 School 3
1 32% 38% 30% $0 $0 $700
2 37% 28% 35% – $400 $500
3 30% 32% 38% $600 $0 $0
4 28% 40% 32% $0 $500 –
5 39% 34% 27% $0 – $400
6 34% 28% 38% $500 $0 $0
Solutio n: Number of Students Assigned Total Number of
School 1 School 2 School 3 From Area Students
Area 1 38.71 411.29 0 450 = 450
Area 2 0 236.56 363.44 600 = 600
Area 3 0 77.96 472.04 550 = 550
Area 4 350 0 0 350 = 350
Area 5 435.48 0 64.52 500 = 500
Area 6 75.81 374.19 0 450 = 450
Total In School 900 1,100 900
<= <= <= Total
Capacity 900 1,100 1,000 Bussing
Cost
$340,054
Grade Con straints:
270 330 270 30% of total in school
<= <= <=
6th G raders 306.00 369.75 301.25
7th G raders 324.00 352.25 274.75
8th G raders 270.00 378.00 324.00
<= <= <=
324 396 324 36% of total in school
f)
Option
Cost
# students walking
1 to 1.5 miles
# students walking
more than 1.5 miles
current
$555,556
0
0
1
$393,636
900
0
2
$340,054
900
491
g) Answers will vary.
3.6 a) Each activity corresponds to the treatment of one solid waste material prepairing it for
amalgation into one product grade. The resources are the four solid waste materials and
the limited usage of materials 1 and 3. The benefits are the collection and treatment of
3-48
3.7 a) Assign one scientist to each of the five projects to maximize the total number of bid
points.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
A B C D E F G H I
Bid Project Up Project Stable Project Choice Project Hope Project Release
Dr. Kvaal 100 400 200 200 100
Dr. Zuner 0 200 800 0 0
Dr. Tsai 100 100 100 100 600
Dr. Mickey 267 153 99 451 30
Dr. Rollins 100 33 33 34 800
Total
Assig n ment Project Up Project Stable Project Choice Project Hope Project Release Assignments Supply
Dr. Kvaal 0 1 0 0 0 1 = 1
Dr. Zuner 0 0 1 0 0 1 = 1
Dr. Tsai 1 0 0 0 0 1 = 1
Dr. Mickey 0 0 0 1 0 1 = 1
Dr. Rollins 0 0 0 0 1 1 = 1
Total Assigned 1 1 1 1 1
= = = = = T otal Bid Points
Demand 1 1 1 1 1 2551
To maximize the scientists preferences you want to assign Dr. Tsai to lead project Up,
Dr. Kvaal to lead project Stable, Dr. Zuner to lead project Choice, Dr. Mickey to lead
project Hope, and Dr. Rollins to lead project Release.
b) Dr. Rollins is not available, so his “Supply” in cell I14 is reduced to zero. Since now
Project Up would not be done.
3-49
c) Since Dr. Zooner or Dr. Mickey can lead two projects, their “Supply” in column I is
changed to 2 and the corresponding constraint changed to ≤ (in order to allow them to
do either one or two projects).
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
A B C D E F G H I
Bid Project Up Project Stable Project Choice Project Hope Project Release
Dr. Kvaal 100 400 200 200 100
Dr. Zuner 0 200 800 0 0
Dr. Tsai 100 100 100 100 600
Dr. Mickey 267 153 99 451 30
Dr. Rollins 100 33 33 34 800
Total
Assig n ment Project Up Project Stable Project Choice Project Hope Project Release Assignments Supply
Dr. Kvaal 0 1 0 0 0 1 = 1
Dr. Zuner 0 0 1 0 0 1 <= 2
Dr. Tsai 0 0 0 0 1 1 = 1
Dr. Mickey 1 0 0 1 0 2 <= 2
Dr. Rollins 0 0 0 0 0 0 = 0
Total Assigned 1 1 1 1 1
= = = = = T otal Bid Points
Demand 1 1 1 1 1 2518
Dr. Kvaal leads project Stable, Dr. Zuner leads project Choice, Dr. Tsai leads project
Release, and Dr. Mickey leads the projects Hope and Up.
d) Under the new bids of Dr. Zuner the assignment does not change:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
A B C D E F G H I
Bid Project Up Project Stable Project Choice Project Hope Project Release
Dr. Kvaal 100 400 200 200 100
Dr. Zuner 20 450 451 39 40
Dr. Tsai 100 100 100 100 600
Dr. Mickey 267 153 99 451 30
Dr. Rollins 100 33 33 34 800
Total
Assig n ment Project Up Project Stable Project Choice Project Hope Project Release Assignments Supply
Dr. Kvaal 0 1 0 0 0 1 = 1
Dr. Zuner 0 0 1 0 0 1 <= 2
Dr. Tsai 0 0 0 0 1 1 = 1
Dr. Mickey 1 0 0 1 0 2 <= 2
Dr. Rollins 0 0 0 0 0 0 = 0
Total Assigned 1 1 1 1 1
= = = = = T otal Bid Points
Demand 1 1 1 1 1 2169
e) Certainly Dr. Zuner could be disappointed that she is not assigned to project Stable,
especially when she expressed a higher preference for that project than the scientist
3-50
f) Whenever a scientist cannot lead a particular project we constrain the corresponding
changing cell (E10, F10, C13, E13, and B14) to equal 0.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
A B C D E F G H I
Bid Project Up Project Stable Project Choice Project Hope Project Release
Dr. Kvaal 86 343 171 Š Š
Dr. Zuner 0 200 800 0 0
Dr. Tsai 100 100 100 100 600
Dr. Mickey 300 Š125 Š175
Dr. Rollins Š50 50 100 600
Total
Assig n ment Project Up Project Stable Project Choice Project Hope Project Release Assignments Supply
Dr. Kvaal 0 1 0 0 0 1 = 1
Dr. Zuner 0 0 1 0 0 1 = 1
Dr. Tsai 0 0 0 0 1 1 = 1
Dr. Mickey 1 0 0 0 0 1 = 1
Dr. Rollins 0 0 0 1 0 1 = 1
Total Assigned 1 1 1 1 1
= = = = = T otal Bid Points
Demand 1 1 1 1 1 2143
Dr. Kvaal leads project Stable, Dr. Zuner leads project Choice, Dr. Tsai leads project
Release, Dr. Mickey leads project Up, and Dr. Rollins leads project Hope.
g) When we want to assign two assignees to the same task we need to duplicate that task.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
A B C D E F G H I
Bid Project Up Project Stable Project Choice Project Hope Project Release
Dr. Kvaal 86 343 171 Š Š
Dr. Zuner 0 200 800 0 0
Dr. Tsai 100 100 100 100 600
Dr. Mickey 300 Š125 Š175
Dr. Rollins Š50 50 100 600
Dr. Arriaga 250 250 0 250 250
Dr. Santos 111 1 0 333 555
Total
Assign ment Project Up Project Stable Project Choice Project Hope Project Release Assignments Supply
Dr. Kvaal 0 1 0 0 0 1 = 1
Dr. Zuner 0 0 1 0 0 1 = 1
Dr. Tsai 0 0 0 0 1 1 = 1
Dr. Mickey 1 0 0 0 0 1 = 1
Dr. Rollins 0 0 0 0 1 1 = 1
Dr. Arriaga 0 0 0 1 0 1 = 1
Dr. Santos 0 0 0 1 0 1 = 1
Total Assigned 1 1 1 2 2
= = = = = Total Bid Points
Demand 1 1 1 2 2 3226
Project Up is led by Dr. Mickey, Stable by Dr. Kvaal, Choice by Dr. Zuner, Hope by
Dr. Arriaga and Dr. Santos, and Release by Dr. Tsai and Dr. Rollins.
h) No. Maximizing overall preferences does not maximize individual preferences.
Scientists who do not get their first choice may become resentful and therefore lack the