LP Extra Issues and Examples

2

Special Cases in LP

• Infeasibility

• Unbounded Solutions

• Redundancy

• Degeneracy

• More Than One Optimal Solution

3

A Problem with No Feasible Solution

X2

X1

8

6

4

2

02 4 6 8

Region Satisfying3rd Constraint

Region Satisfying First 2 Constraints

4

A Solution Region That is Unbounded to the Right

X2

X1

15

10

5

05 10 15

Feasible Region

X1 > 5 X2 < 10

X1 + 2X2 > 10

5

A Problem with a Redundant Constraint

X2

X1

30

25

20

15

10

5

05 10 15 20 25 30

Feasible Region

2X1 + X2 < 30

X1 < 25

X1 + X2 < 20

RedundantConstraint

6

Sensitivity Analysis

• Changes in the Objective Function Coefficient

• Changes in Resources (RHS)

• Changes in Technological (LHS) Coefficients

Minimization Example

X1 = number of tons of black-and-white picture chemical produced

X2 = number of tons of color picture chemical produced

Minimize total cost = 2,500X1 + 3,000X2

Subject to:X1 ≥ 30 tons of black-and-white chemicalX2 ≥ 20 tons of color chemicalX1 + X2 ≥ 60 tons totalX1, X2 ≥ $0 nonnegativity requirements

Minimization ExampleTable B.9

60 –

50 –

40 –

30 –

20 –

10 –

–| | | | | | |0 10 20 30 40 50 60

X1

X2

Feasible region

X1 = 30X2 = 20

X1 + X2 = 60

b

a

Minimization Example

Total cost at a = 2,500X1 + 3,000X2

= 2,500 (40) + 3,000(20)= $160,000

Total cost at b = 2,500X1 + 3,000X2

= 2,500 (30) + 3,000(30)= $165,000

Lowest total cost is at point a

LP ApplicationsProduction-Mix Example

Department

Product Wiring Drilling Assembly Inspection Unit Profit

XJ201 .5 3 2 .5 $ 9XM897 1.5 1 4 1.0 $12TR29 1.5 2 1 .5 $15BR788 1.0 3 2 .5 $11

Capacity MinimumDepartment (in hours) Product Production Level

Wiring 1,500 XJ201 150Drilling 2,350 XM897 100Assembly 2,600 TR29 300Inspection 1,200 BR788 400

LP ApplicationsX1 = number of units of XJ201 producedX2 = number of units of XM897 producedX3 = number of units of TR29 producedX4 = number of units of BR788 produced

Maximize profit = 9X1 + 12X2 + 15X3 + 11X4

subject to .5X1 + 1.5X2 + 1.5X3 + 1X4 ≤ 1,500 hours of wiring3X1 + 1X2 + 2X3 + 3X4 ≤ 2,350 hours of drilling2X1 + 4X2 + 1X3 + 2X4 ≤ 2,600 hours of assembly

.5X1 + 1X2 + .5X3 + .5X4 ≤ 1,200 hours of inspection X1 ≥ 150 units of XJ201

X2 ≥ 100 units of XM897X3 ≥ 300 units of TR29X4 ≥ 400 units of BR788

LP ApplicationsDiet Problem Example

A 3 oz 2 oz 4 ozB 2 oz 3 oz 1 ozC 1 oz 0 oz 2 ozD 6 oz 8 oz 4 oz

Feed

Product Stock X Stock Y Stock Z

LP ApplicationsX1 = number of pounds of stock X purchased per cow each monthX2 = number of pounds of stock Y purchased per cow each monthX3 = number of pounds of stock Z purchased per cow each month

Minimize cost = .02X1 + .04X2 + .025X3

Ingredient A requirement: 3X1 + 2X2 + 4X3 ≥ 64Ingredient B requirement: 2X1 + 3X2 + 1X3 ≥ 80Ingredient C requirement: 1X1 + 0X2 + 2X3 ≥ 16Ingredient D requirement: 6X1 + 8X2 + 4X3 ≥ 128

Stock Z limitation: X3 ≤ 80X1, X2, X3 ≥ 0

Cheapest solution is to purchase 40 pounds of grain X at a cost of $0.80 per cow

LP ApplicationsLabor Scheduling Example

Time Number of Time Number ofPeriod Tellers Required Period Tellers Required

9 AM - 10 AM 10 1 PM - 2 PM 1810 AM - 11 AM 12 2 PM - 3 PM 1711 AM - Noon 14 3 PM - 4 PM 15Noon - 1 PM 16 4 PM - 5 PM 10

F = Full-time tellersP1 = Part-time tellers starting at 9 AM (leaving at 1 PM)P2 = Part-time tellers starting at 10 AM (leaving at 2 PM)P3 = Part-time tellers starting at 11 AM (leaving at 3 PM)P4 = Part-time tellers starting at noon (leaving at 4 PM)P5 = Part-time tellers starting at 1 PM (leaving at 5 PM)

LP Applications

= $75F + $24(P1 + P2 + P3 + P4 + P5) Minimize total daily

manpower cost

F + P1 ≥ 10 (9 AM - 10 AM needs)F + P1 + P2 ≥ 12 (10 AM - 11 AM needs)

1/2 F + P1 + P2 + P3 ≥ 14 (11 AM - 11 AM needs)1/2 F + P1 + P2 + P3 + P4 ≥ 16 (noon - 1 PM needs)

F + P2 + P3 + P4 + P5 ≥ 18 (1 PM - 2 PM needs)F + P3 + P4 + P5 ≥ 17 (2 PM - 3 PM needs)F + P4 + P5 ≥ 15 (3 PM - 7 PM needs)F + P5 ≥ 10 (4 PM - 5 PM needs)F ≤ 12

4(P1 + P2 + P3 + P4 + P5) ≤ .50(10 + 12 + 14 + 16 + 18 + 17 + 15 + 10)

LP Applications

= $75F + $24(P1 + P2 + P3 + P4 + P5) Minimize total daily

manpower cost

F + P1 ≥ 10 (9 AM - 10 AM needs)F + P1 + P2 ≥ 12 (10 AM - 11 AM needs)

1/2 F + P1 + P2 + P3 ≥ 14 (11 AM - 11 AM needs)1/2 F + P1 + P2 + P3 + P4 ≥ 16 (noon - 1 PM needs)

F + P2 + P3 + P4 + P5 ≥ 18 (1 PM - 2 PM needs)F + P3 + P4 + P5 ≥ 17 (2 PM - 3 PM needs)F + P4 + P5 ≥ 15 (3 PM - 7 PM needs)F + P5 ≥ 10 (4 PM - 5 PM needs)F ≤ 124(P1 + P2 + P3 + P4 + P5) ≤ .50(112)

F, P1, P2 , P3, P4, P5 ≥ 0

LP Applications

= $75F + $24(P1 + P2 + P3 + P4 + P5) Minimize total daily

manpower cost

F + P1 ≥ 10 (9 AM - 10 AM needs)F + P1 + P2 ≥ 12 (10 AM - 11 AM needs)

1/2 F + P1 + P2 + P3 ≥ 14 (11 AM - 11 AM needs)1/2 F + P1 + P2 + P3 + P4 ≥ 16 (noon - 1 PM needs)

F + P2 + P3 + P4 + P5 ≥ 18 (1 PM - 2 PM needs)F + P3 + P4 + P5 ≥ 17 (2 PM - 3 PM needs)F + P4 + P5 ≥ 15 (3 PM - 7 PM needs)F + P5 ≥ 10 (4 PM - 5 PM needs)F ≤ 124(P1 + P2 + P3 + P4 + P5) ≤ .50(112)

F, P1, P2 , P3, P4, P5 ≥ 0

There are two alternate optimal solutions to this problem but both will cost $1,086 per day

F = 10 F = 10P1 = 0 P1 = 6P2 = 7 P2 = 1 P3 = 2 P3 = 2P4 = 2 P4 = 2P5 = 3 P5 = 3

First SecondSolution Solution