Chapter 3:
Linear Programming
Modeling Applications
Linear Programming (LP) Can Be
Used for Many Managerial Decisions:
• Product mix • Make-buy
• Media selection
• Marketing research • Portfolio selection
For a particular application we begin with the problem scenario and data, then:
1) Define the decision variables
2) Formulate the LP model using the decision variables
• Write the objective function equation • Write each of the constraint equations
Product Mix Problem:
Fifth Avenue Industries
• Produce 4 types of men's ties
• Use 3 materials (limited resources)
Decision: How many of each type of tie to make per month?
Material Cost per yard
Yards available per month
Silk $20 1,000
Polyester $6 2,000
Cotton $9 1,250
Resource Data
Product Data
Type of Tie
Silk Polyester Blend 1 Blend 2 Selling Price
(per tie) $6.70 $3.55 $4.31 $4.81
Monthly
Minimum 6,000 10,000 13,000 6,000
Monthly
Maximum 7,000 14,000 16,000 8,500
Total material
Material Requirements
(yards per tie)
Material
Type of Tie
Silk Polyester Blend 1 (50/50)
Blend 2 (30/70)
Silk 0.125 0 0 0
Polyester 0 0.08 0.05 0.03
Cotton 0 0 0.05 0.07
Decision Variables
S = number of silk ties to make per month
P = number of polyester ties to make per month
B1 = number of poly-cotton blend 1 ties to
make per month
Profit Per Tie Calculation
Profit per tie =
(Selling price) – (material cost) –(labor cost)
Silk Tie
Objective Function (in $ of profit)
Max 3.45S + 2.32P + 2.81B1 + 3.25B2
Subject to the constraints:
Material Limitations (in yards)
0.125S < 1,000 (silk)
0.08P + 0.05B1 + 0.03B2 < 2,000 (poly)
Min and Max Number of Ties to Make 6,000 < S < 7,000
10,000 < P < 14,000 13,000 < B1 < 16,000
6,000 < B2 < 8,500
Finally nonnegativity S, P, B1, B2 > 0
Media Selection Problem:
Win Big Gambling Club
• Promote gambling trips to the Bahamas • Budget: $8,000 per week for advertising • Use 4 types of advertising
Decision: How many ads of each type?
Data
5,000 8,500 2,400 2,800
Cost
(per ad) $800 $925 $290 $380
Max Ads
Other Restrictions
• Have at least 5 radio spots per week • Spend no more than $1800 on radio
Decision Variables
T = number of TV spots per week
N = number of newspaper ads per week P = number of prime time radio spots per
week
Objective Function (in num. audience reached)
Max 5000T + 8500N + 2400P + 2800A
Subject to the constraints:
Budget is $8000
800T + 925N + 290P + 380A < 8000
No More Than $1800 per Week for Radio 290P + 380A < 1800
Max Number of Ads per Week
T < 12 P < 25 N < 5 A < 20
Finally nonnegativity T, N, P, A > 0
Portfolio Selection:
International City Trust
Has $5 million to invest among 6 investments
Decision: How much to invest in each of 6 investment options?
Data
Investment Interest Rate Risk Score
Trade credits 7% 1.7
Corp. bonds 10% 1.2
Gold stocks 19% 3.7
Platinum stocks 12% 2.4
Mortgage securities 8% 2.0
Constraints
• Invest up to $ 5 million
• No more than 25% into any one investment • At least 30% into precious metals
• At least 45% into trade credits and corporate bonds
Decision Variables
T = $ invested in trade credit
B = $ invested in corporate bonds G = $ invested gold stocks
P = $ invested in platinum stocks
Objective Function (in $ of interest earned)
Max 0.07T + 0.10B + 0.19G + 0.12P + 0.08M + 0.14C
Subject to the constraints:
Invest Up To $5 Million
No More Than 25% Into Any One Investment T < 0.25 (T + B + G + P + M + C)
B < 0.25 (T + B + G + P + M + C) G < 0.25 (T + B + G + P + M + C)
P < 0.25 (T + B + G + P + M + C) M < 0.25 (T + B + G + P + M + C)
At Least 30% Into Precious Metals
G + P > 0.30 (T + B + G + P + M + C)
At Least 45% Into
Trade Credits And Corporate Bonds
Limit Overall Risk To No More Than 2.0
Use a weighted average to calculate portfolio risk
1.7T + 1.2B + 3.7G + 2.4P + 2.0M + 2.9C < 2.0
Labor Planning:
Hong Kong Bank
Number of tellers needed varies by time of day
Decision: How many tellers should begin work at various times of the day?
Time Period Min Num. Tellers
Full Time Tellers
• Work from 9 AM – 5 PM
• Take a 1 hour lunch break, half at 11, the other half at noon
Part Time Tellers
• Work 4 consecutive hours (no lunch break) • Can begin work at 9, 10, 11, noon, or 1
• Are paid $7 per hour ($28 per day)
• Part time teller hours cannot exceed 50% of the day’s minimum
requirement
Decision Variables
F = num. of full time tellers (all work 9–5) P1 = num. of part time tellers who work 9–1
P2 = num. of part time tellers who work 10–2
P3 = num. of part time tellers who work 11–3 P4 = num. of part time tellers who work 12–4
Objective Function (in $ of personnel cost)
Min 90 F + 28 (P1 + P2 + P3 + P4 + P5)
Subject to the constraints:
Part Time Hours Cannot Exceed 56 Hours
Only 12 Full Time Tellers Available
F < 12
finally nonnegativity: F, P1, P2, P3, P4, P5 > 0
Vehicle Loading:
Goodman Shipping
How to load a truck subject to weight and volume limitations
Decision: How much of each of 6 items to load onto a truck?
Data
Item
1 2 3 4 5 6
Value $15,500 $14,400 $10,350 $14,525 $13,000 $9,625
Pounds 5000 4500 3000 3500 4000 3500
$ / lb $3.10 $3.20 $3.45 $4.15 $3.25 $2.75
Cu. ft.
Decision Variables
Wi = number of pounds of item i to load onto
truck, (where i = 1,…,6)
Objective Function (in $ of load value)
Max 3.10W1 + 3.20W2 + 3.45W3 + 4.15W4 +
3.25W5 + 2.75W6
Subject to the constraints:
Weight Limit Of 15,000 Pounds
Volume Limit Of 1300 Cubic Feet 0.125W1 + 0.064W2 + 0.144W3 +
0.448W4 + 0.048W5 + 0.018W6 < 1300
Pounds of Each Item Available W1 < 5000 W4 < 3500
W2 < 4500 W5 < 4000 W3 < 3000 W6 < 3500
Finally nonnegativity: Wi > 0, i=1,…,6
Blending Problem:
Whole Food Nutrition Center
Making a natural cereal that satisfies
minimum daily nutritional requirements
Decision: How much of each of 3 grains to include in the cereal?
Decision Variables
A = pounds of grain A to use B = pounds of grain B to use C = pounds of grain C to use
Objective Function (in $ of cost)
Min 0.33A + 0.47B + 0.38C
Subject to the constraints:
Minimum Nutritional Requirements 22A + 28B + 21C > 3 (protein) 16A + 14B + 25C > 2 (riboflavin)
8A + 7B + 9C > 1 (phosphorus) 5A + 6C > 0.425 (magnesium)
Finally nonnegativity: A, B, C > 0
Multiperiod Scheduling:
Greenberg Motors
Need to schedule production of 2 electrical motors for each of the next 4 months
Decision: How many of each type of motor to make each month?
Sales Demand Data
Month
Motor
A B
1 (January) 800 1000
2 (February) 700 1200
3 (March) 1000 1400
Production Data
Motor
(values are per motor)
A B
Production cost $10 $6
Labor hours 1.3 0.9
• Production costs will be 10% higher in months 3 and 4
Inventory Data
Motor
A B
Inventory cost
(per motor per month) $0.18 $0.13 Beginning inventory
(beginning of month 1) 0 0
Ending Inventory
(end of month 4) 450 300
Production and Inventory Balance
(inventory at end of previous period) + (production the period)
- (sales this period)
Objective Function (in $ of cost)
Min 10PA1 + 10PA2 + 11PA3 + 11PA4
+ 6PB1 + 6 PB2 + 6.6PB3 + 6.6PB4
+ 0.18(IA1 + IA2 + IA3 + IA4) + 0.13(IB1 + IB2 + IB3 + IB4)
Subject to the constraints:
Ending Inventory
IA4 = 450
IB4 = 300
Maximum Inventory level
IA1 + IB1 < 3300 (month 1)
IA2 + IB2 < 3300 (month 2)
IA3 + IB3 < 3300 (month 3)
Range of Labor Hours
2240 < 1.3PA1 + 0.9PB1 < 2560 (month 1)
2240 < 1.3PA2 + 0.9PB2 < 2560 (month 2)
2240 < 1.3PA3 + 0.9PB3 < 2560 (month 3)
2240 < 1.3PA4 + 0.9PB4 < 2560 (month 4)
finally nonnegativity: PAi, PBi, IAi, IBi > 0
