Managerial decision modeling with spreadsheets by stair render chapter 03

For a particular application we begin withthe problem scenario and data, then: 1 Define the decision variables 2 Formulate the LP model using the decision variables • Write the objective

Chapter 3:

Linear Programming Modeling Applications

Linear Programming (LP) Can Be Used

for Many Managerial Decisions:

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

3) Implement the model in Excel

4) Solve with Excel’s Solver

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?

Objective: Maximize profit

Material Cost per yard

Yards available per month

Product Data

Type of Tie Silk Polyester Blend 1 Blend 2 Selling Price

(per tie) $6.70 $3.55 $4.31 $4.81 Monthly

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

B2 = number of poly-cotton blend 2 ties to make per month

Profit Per Tie Calculation

Profit per tie =

(Selling price) – (material cost) –(labor cost)

Silk Tie

Profit = $6.70 – (0.125 yds)($20/yd) - $0.75

= $3.45 per tie

Objective Function (in $ of profit)

Max 3.45S + 2.32P + 2.81B 1 + 3.25B 2

Subject to the constraints:

Material Limitations (in yards) 0.125S < 1,000 (silk)

0.08P + 0.05B 1 + 0.03B 2 < 2,000 (poly)

0.05B 1 + 0.07B 2 < 1,250 (cotton)

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

Go to file 3-1.xls

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?

Objective: Maximize audience reached

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

A = number of afternoon 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

At Least 5 Radio Spots per Week

P + A > 5

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

Go to file 3-3.xls

Portfolio Selection:

International City Trust

Has $5 million to invest among 6 investments

Objective: Maximize interest earned

• 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

• Limit overall risk to no more than 2.0

Decision Variables

T = $ invested in trade credit

B = $ invested in corporate bonds

G = $ invested gold stocks

P = $ invested in platinum stocks

M = $ invested in mortgage securities

C = $ invested in construction loans

Objective Function (in $ of interest earned)

No More Than 25% Into Any One Investment

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

T + B > 0.45 (T + B + G + P + M + C)

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

T + B + G + P + M + C

OR 1.7T + 1.2B + 3.7G + 2.4P + 2.0M + 2.9C <

2.0 (T + B + G + P + M + C) finally nonnegativity: T, B, G, P, M, C > 0

Go to file 3-5.xls

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?

Objective: Minimize personnel cost

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

• Cost $90 per day (salary & benefits)

• Currently only 12 are available

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

(50% of 112 hours = 56 hours)

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

P5 = num of part time tellers who work 1–5

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

4 (P1 + P2 + P3 + P4 + P5) < 56

Minimum Num Tellers Needed By Hour

Only 12 Full Time Tellers Available

F < 12

finally nonnegativity: F, P1, P2, P3, P4, P5 > 0

Go to file 3-6.xls

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?

Objective: Maximize the value shipped

Objective Function (in $ of load value)

Subject to the constraints:

Weight Limit Of 15,000 Pounds

W1 + W2 + W3 + W4 + W5 + W6 < 15,000

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

Go to file 3-7.xls

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?

Objective: Minimize cost of a 2 ounce serving of cereal

Minimum Daily Requirement

Decision Variables

A = pounds of grain A to use

B = pounds of grain B to use

C = pounds of grain C to use

Note: grains will be blended to form a 2 ounce serving of cereal

Objective Function (in $ of cost)

Min 0.33A + 0.47B + 0.38C

Subject to the constraints:

Total Blend is 2 Ounces, or 0.125 Pounds

A + B + C = 0.125 (lbs)

Minimum Nutritional Requirements 22A + 28B + 21C > 3 (protein)

16A + 14B + 25C > 2 (riboflavin) 8A + 7B + 9C > 1 (phosphorus)

Finally nonnegativity: A, B, C > 0

Go to file 3-9.xls

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?

Objective: Minimize total production and inventory cost

Decision Variables

PAt = number of motor A to produce in

month t (t=1,…,4) PBt = number of motor B to produce in

month t (t=1,…,4)

IAt = inventory of motor A at end of

month t (t=1,…,4) IBt = inventory of motor B at end of

month t (t=1,…,4)

Sales Demand Data

Production Data

Motor (values are per motor)

Production and Inventory Balance

(inventory at end of previous period)

+ (production the period)

- (sales this period)

= (inventory at end of 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:

(see next slide)

Production & Inventory Balance

0 + PA1 – 800 = IA1 (month 1)

Ending Inventory IA4 = 450 IB4 = 300

Maximum Inventory level IA1 + IB1 < 3300 (month 1) IA2 + IB2 < 3300 (month 2) IA3 + IB3 < 3300 (month 3) IA4 + IB4 < 3300 (month 4)

Range of Labor Hours