LP Formulation
Set 2
Agricultural planning : narrative
Three farming communities are developing a joint agricultural
production plan for the coming year.
Production capacity of each community is limited by their land
and water.
Community
1
2
3
Land (Acres)
400
600
300
Water (Acres Feet)
600
800
375
The crops suited for this region include sugar beets, cotton, and
sorghum. These are the three being considered for the next year.
Information regarding the maximum desired production of each
product, water consumption , and net profit are given below
LP-Formulation
Ardavan Asef-Vaziri
June-2013
2
Agricultural planning : narrative
Crop
1
2
3
Max desired
(Acres)
600
500
325
Water consumption
(Acre feet / Acre)
3
2
1
Net return
($/Acre)
1000
750
250
Because of the limited available water, it has been agreed that
every community will plant the same proportion of its available
irritable land. For example, if community 1 plants 200 of its
available 400 acres, then communities 2 and 3 should plant 300
out of 600, and 150 out of 300 acres respectively.
However, any combination of crops may be grown at any
community.
Goal : find the optimal combination of crops in each community,
in order to maximize total return of all communities
LP-Formulation
Ardavan Asef-Vaziri
June-2013
3
Agricultural planning : decision variables
x11 = Acres allocated to Crop 1 in Community 1
x21 = Acres allocated to Crop 2 in Community 1
x31 = Acres allocated to Crop 3 in Community 1
x12 = Acres allocated to Crop 1 in Community 2
x22 = Acres allocated to Crop 2 in Community 2
x32 = Acres allocated to Crop 3 in Community 2
……………..
xij = Acres allocated to Crop i in Community j
i for crop j for community, we could have switched them
Note that x is volume not portion, we could have had it as
portion
LP-Formulation
Ardavan Asef-Vaziri
June-2013
4
Agricultural planning : Formulation
Land
x11+x21+x31  400
x12+x22+x32  600
x13+x23+x33  300
Water
3x11+2x21+1x31  600
3x12+2x22+1x32  800
3x13+2x23+1x33  375
LP-Formulation
Ardavan Asef-Vaziri
June-2013
5
Agricultural planning : Formulation
Crops
x11+ x12 + x13  600
x21 +x22 +x23  500
x31 +x32 +x33  320
Proportionality of land use
x11+x21+x31
x12+x22+x32
400
600
x11+x21+x31
x13+x23+x33
400
LP-Formulation
300
Ardavan Asef-Vaziri
June-2013
6
Agricultural planning : Formulation
Crops
x11+ x12 + x13  600
x21 +x22 +x23  500
x31 +x32 +x33  320
Proportionality of land use
x11+x21+x31
x12+x22+x32
400
600
x11+x21+x31
x13+x23+x33
400
LP-Formulation
300
Ardavan Asef-Vaziri
June-2013
7
Agricultural planning : all variables on LHS
Proportionality of land use
600(x11+x21+x31 ) - 400(x12+x22+x32 ) = 0
300(x11+x21+x31 ) - 400(x13+x23+x33 ) = 0
600x11+ 600 x21+ 600 x31 - 400x12- 400 x22- 400 x32 = 0
300x11+ 300 x21+ 300 x31 - 400x13- 400 x23- 400 x33 = 0
x11, x21,x31, x12, x22, x32, x13, x23, x33  0
LP-Formulation
Ardavan Asef-Vaziri
June-2013
8
SAVE-IT Company : Narrative
A reclamation center collects 4 types of solid waste material,
treat them, then amalgamate them to produce 3 grades of
product. Techno-economical specifications are given below
Grade Specifications
A
M1 :  30% of total
M2 :  40% of total
Processing cost/pound Sales price/ pound
3
8.5
M3 :  50% of total
M4 : exactly 20%
M1 :  50% of total
B
M2 :  10% of total
M4 : exactly 10%
2.5
7
C
M1 :  70% of total
2
5.5
LP-Formulation
Ardavan Asef-Vaziri
June-2013
9
SAVE-IT Company : Narrative
Availability and cost of the solid waste materials M1, M2, M3,
and M4 per week are given below
Material
Pounds available / week Treatment cost / pound
M1
3000
3
M2
2000
6
M3
4000
4
M4
1000
5
Due to environmental considerations, a budget of
$30000 / week should be used to treat these material.
Furthermore, for each material, at least half of the pounds
per week available should be collected and treated.
LP-Formulation
Ardavan Asef-Vaziri
June-2013
10
SAVE-IT Co. Mixture Specification
A1: weight of solid waste 1 in grade A
A1, A2, A1, A4, B1, B2, B3, B4, C1, C2, C3, C4
Mixture Specifications:
Grade A:
A1  0.3 (A1+A2+A3+A4)
A2  0.4 (A1+A2+A3+A4)
A3  0.5 (A1+A2+A3+A4)
A3 = 0.2 (A1+A2+A3+A4)
Grade B:
B1  0.5(B1+B2+B3+B4)
B2  0.1(B1+B2+B3+B4)
B4 = 0.1(B1+B2+B3+B4)
Grade C:
LP-Formulation
C1  0.3 (C1+C2+C3+C4)
Ardavan Asef-Vaziri
June-2013
11
SAVE-IT Co. Material Availability and usage
Availability of material
A1+B1+C1  3000
A2+B2+C2  2000
A3+B3+C3  4000
A4+B4+C4  1000
At least half of the material treated
A1+B1+C1  1500
A2+B2+C2  1000
A3+B3+C3  2000
A4+B4+C4  500
LP-Formulation
Ardavan Asef-Vaziri
June-2013
12
SAVE-IT Co. Treatment and Processing Costs, and Profit
Spend all the treatment budget
3(A1+B1+C1)+6(A2+B2+C2)+4(A3+B3+C3)+5(A4+B4+C4) = 30000
Maximize profit Z
(8.5-3)(A1+A2+A3+A4)+(7-2.5) (B1+B2+B3+B4)+(5.5-2)
(C1+C2+C3+C4)
– 3(A1+B1+C1)-6(A2+B2+C2)-4(A3+B3+C3)-5(A4+B4+C4))
A1, A2, A1, A4, B1, B2, B3, B4, C1, C2, C3, C4 0
LP-Formulation
Ardavan Asef-Vaziri
June-2013
13
SAVE-IT Co. Treatment and Processing Costs, and Profit
Pounds of Material j in Product i
A
B
C
1
750
2250
0
2 1000
0
0
3
250
2250
0
4
500
500
0
SUM
3000
1000
2500
1000
Processing $/Unit Available
3
3000
6
2000
4
4000
5
1000
Sum
2500
5000
0
Cost
5.5
4.5
3.5
A1
A3
B1
B2
C1
A4
0.3
0.5
0.5
0.1
0.7
0.2
0
-1000
-250
-500
0
0
<=
<=
<=
<=
<=
=
0
0
0
0
0
0
B4
A2
0.1
0.4
0
0
=
>=
0
0
LP-Formulation
6250
Ardavan Asef-Vaziri
At Least Half
1500
1000
2000
500
Budget
30000
June-2013
30000
14
Capital budgeting : Narrative representation
There are 3 investment projects offered to the public.
We may invest in any portion of one or more projects.
Investment requirements of each project in each year ( in millions
of dollars) is given below. The Net Present Value (NPV) of total
cash flow is also given.
Year
Project 1
Project 2
Project 3
0
40
80
90
1
60
80
60
2
90
80
20
3
10
70
60
NPV
45
70
50
LP-Formulation
Ardavan Asef-Vaziri
June-2013
15
Capital budgeting : Narrative representation
If we invest in 5% of project 1, then we need to invest 2, 3, 4.5, and
0.5 million dollars in years 0, 1, 2, 3 respectively. The NPV of our
investment would be also equal to 5% of the NPV of this project,
i.e. 2.25 million dollars.
Year
0
1
2
3
NPV
LP-Formulation
Project 1
40
60
90
10
45
5% of Project 1
2
3
4.5
0.5
2.25
Ardavan Asef-Vaziri
June-2013
16
Capital budgeting : Narrative representation
Based on our budget forecasts,
Our total available money to invest in year 0 is 25M.
Our total available money to invest in years 0 and 1 is 45M
Our total available money to invest in years 0, 1, 2 is 65M
Our total available money to invest in years 0, 1, 2, 3 is 80M
To clarify, in year 0 we can not invest more than 25M.
In year 1 we can invest 45M minus what we have invested in year
0.
The same is true for years 2 and 3.
The objective is to maximize the NPV of our investments
LP-Formulation
Ardavan Asef-Vaziri
June-2013
17
Capital budgeting : Formulation
x1 = proportion of project 1 invested by us.
x2 = proportion of project 2 invested by us.
x3 = proportion of project 3 invested by us.
Maximize NPV Z = 45x1 + 70 x2 + 50 x3
subject to
Year 0 : 40 x1 + 80 x2 + 90 x3  25
Year 1 : Investment in year 0 + Investment in year 1  45
LP-Formulation
Ardavan Asef-Vaziri
June-2013
18
Capital budgeting : Formulation
Investment in year 0 = 40 x1 + 80 x2 + 90 x3
Investment in year 1 = 60 x1 + 80 x2 + 60 x3
Year 1 : 60 x1 + 80 x2 + 60 x3 + 40 x1 + 80 x2 + 90 x3  45
Year 1 : 100x1 + 160 x2 + 150 x3  45
Year 2 : 90x1 + 80x2 + 20 x3 + 100x1 + 160 x2 + 150 x3  65
Year 2 : 190x1 + 240x2 + 170 x3  65
Year 3 : 10x1 + 70x2 + 60 x3 + 190x1 + 240x2 + 170 x3  80
Year 3 : 200x1 + 310x2 + 230 x3  80
x1 , x2, x3  0.
LP-Formulation
Ardavan Asef-Vaziri
June-2013
19
Refresh
We need to lease warehouse space. The estimated required
space ( in 1000 sq ft) is given below.
Month
1
2
3
4
5
Space required
30
20
40
10
50
If the leasing cost was fixed the best strategy was to lease as
needed. But this is not the case
Leasing period (months) 1
2
3
4
5
Cost per sq-feet leased
65
100
135 160
190
Now it may be more economical to lease for more than one
month and take advantage of the lower rates for longer periods.
Find the optimal leasing strategy to minimize leasing costs.
LP-Formulation
Ardavan Asef-Vaziri
June-2013
20
Decision Variables
Xij spaced leased in month i and kept until month j months.
i = 1, 2, 3, 4, 5. j= i, i+1, …, 5
Min z = 65X11 +100 X12 +135 X13 +160 X14+190 X15
+ 65X22+100 X23 +135 X24 +160 X25
+ 65X33+100 X34 +135 X35
+ 65X44+100 X45
+ 65X55
LP-Formulation
Ardavan Asef-Vaziri
June-2013
21
Constraints
X11 + X12 + X13 + X14+ X15 
30,000
X12 + X13 + X14+ X15 + X22+ X23 + X24 + X25  20,000
X13 + X14+ X15 + X23 + X24 + X25 + X33+ X34 + X35  40,000
X14+ X15 + X24 + X25 + X34 + X35 + X44+ X45 10,000
X15 + X25 + X35 + X45 + X55  50,000
X11 , X12 , X13 , X14 , X15 , X22 , X23 , X24 , X25 , X33 , X34 , X35
X44 , X45 , X55  0
LP-Formulation
Ardavan Asef-Vaziri
June-2013
22
excel; Format 1
x11 x12 x13 x14 x15 x22 x23 x24 x25 x33 x34 x35 x44 x45 x55
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
0
1
1
1
1
1
1
0
0
0
0
0
0
1
1
0
0
1
1
0
1
1
1
1
0
0
0
0
0
1
0
0
0
1
0
0
1
0
1
1
65 100 135 160 190
LP-Formulation
65 100 135 160
65 100 135
65 100
Ardavan Asef-Vaziri
30
30
40
30
50
>=
>=
>=
>=
>=
30
20
40
10
50
65
June-2013
23
excel; Format 2
x11
x12
x22
x13
x23
x33
x14
x24
x34
x44
x15
x25
x35
x45
x55
0
0
0
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
30
0
0
0
20
65
100000
100000
100000
100000
100
65
100000
100000
100000
135
100
65
100000
100000
160
135
100
65
100000
190
160
135
100
65
30
30
40
30
50
>=
>=
>=
>=
>=
30
20
40
10
50
7650
LP-Formulation
Ardavan Asef-Vaziri
June-2013
24
excel; Best Format (Ctrl)
x11
x12
x22
x13
x23
x33
x14
x24
x34
x44
x15
x25
x35
x45
x55
0
0
0
0
0
10
0
0
0
0
30
0
0
0
20
65
100
65
135
100
65
160
135
100
65
190
160
135
100
65
30
30
40
30
50
>=
>=
>=
>=
>=
30
20
40
10
50
7650
LP-Formulation
Ardavan Asef-Vaziri
June-2013
25