Presented By:Calvin Mendes 71 Nestor Mascarenhas - 91 Rohit Gupta 101 Sonia Durairaj - 111 Yvette Ferrao -121

Linear Programmingy Linear programming is a form of optimization where

you optimize an objective function with a system of linear inequalities called constraints.

Steps in Researchy Identify the Problem or Opportunity y Understand the system y Formulate a mathematical Model y Verify the Model y Select the Best Alternative y Present the result of the Analysis y Implement And Evaluate

Product Mix Problem: Fifth Avenue Industriesy Produce 4 types of men's ties y Use 3 materials (limited resources)

Decision: How many of each type of tie to make per month? Objective: Maximize profit

Resource DataMaterial Silk Polyester Cotton Yards available Cost per yard per month $20 1,000 $6 $9 2,000 1,250

Labor cost is $0.75 per tie

Product DataType of Tie Silk Selling rice(per tie)

olyester Blend 1 Blend 2 $3.55 10,000 14,000 0.08 $4.31 13,000 1 ,000 0.10 $4.81 ,000 8,500 0.10

$ .70 ,000 7,000 0.125

ont ly ini u ont ly xi u Tot l teri l(y rds per tie)

Material Requirements (yards per tie)Type of Tie Silk Material Silk olyester Cotton Total yards 0.125 0 0 0.125 Blend 1 olyester (50/50) 0 0.08 0 0.08 0 0.05 0.05 0.10 Blend 2 (30/70) 0 0.03 0.07 0.10

Decision VariablesS = number of silk ties to make per month P = number of polyester ties to make per B1 = number of poly-cotton blend 1 ties to month B2 = number of poly-cotton blend 2 ties to month month make per make per

Profit Per Tie CalculationProfit per tie = (Selling price) (material cost) (labor cost) Silk Tie Profit = $6.70 (0.125 yds)($20/yd) - $0.75 = $3.45 per tie Polyester= $2.32 per tie Blend 1 = $2.81 per tie Blend 2 = $3.25 per tie

Blending Problem: Whole Food Nutrition CenterMaking a natural cereal that satisfies minimum daily nutritional requirements Decision: How much of each of the 3 grains (A,B and C) to include in the cereal? Objective: Minimize cost of a 2 ounce/0.125 pounds serving of cereal

$ per po nd rotein per po nd Riboflavin per po nd hosphor s per po nd Magnesi m per po nd

Grain Minim m A B C Daily $0.33 $0.47 $0.38 Req irement 22 1 8 5 28 14 7 0 21 25 9 3 2 1 0.425

Decision VariablesA = 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 + 0.38CSubject 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 5A (phosphorus) (magnesium) + 6C > 0.425

Finally non-negativity: A, B, C > 0

Vehicle Loading: Goodman ShippingHow 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

DataItem 1Val eo nds

24500 $3.20 0.0 4

33000 $3.45 0.144

43500 $4.15 0.448

54000 $3.25 0.048 3500 $2.75 0.018

$15,500 $14,400 $10,350 $14,525 $13,000 $9, 25

5000 $3.10 0.125

$ / lb C . ft. per lb

Decision VariablesWi = number of pounds of item i to load onto truck , (where i = 1, ,6) Truck Capacity y 15,000 pounds y 1,300 cubic feet

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 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

Portfolio Selection: International City TrustHas $5 million to invest among 6 investments Decision: How much to invest in each of 6 investment options? Objective: Maximize interest earned

DataInvestment Tra e cre its Corp. bon s Gol stocks latin m stocks ort a e sec rities Constr ction loans Interest Rate % 0% 9% 2% % 4% Risk Score . .2 3. 2.4 2.0 2.9

Constraintsy Invest up to $ 5 million y No more than 25% into any one investment y At least 30% into precious metals y At least 45% into trade credits and corporate bonds y Limit overall risk to no more than 2.0

Decision VariablesT = $ 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) Max 0.07T + 0.10B + 0.19G + 0.12P + 0.08M + 0.14CSubject to the constraints:

Invest Up To $5 Million T + B + G + P + M + C < 5,000,000

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) C < 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 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

Application in Advertising Strategy

Midtown Motors Inc.Midtown Motors Inc. hired a marketing firm to develop an advertising strategy for promoting its used-car sales They recommended Midtown use spot announcements on both television and radio as the advertising media for the proposed promotional campaign

GUIDELINES:1. Use at least 30 announcements for combined television and radio coverage 2. Do not use more than 25 radio announcements 3. The no. of radio announcements cannot be less than the no. of television announcements

y The television station has quoted a cost of $1200 per

announcement, and the radio station has quoted a cost of $300 per spot announcement.y Midtown s advertising budget has been set at

$25,500y The marketing services firm has rated the various

advertising media alternatives in terms of audience coverage and recall power of the advertisement

For Midtown s media alternatives, the television announcement is rated at 600, and the radio announcement is rated at 200

Midtown s president would like to know how many television and how many radio spot announcements should be used to maximize the overall rating of the advertising campaign.

SolutionDecision variables: y X be the no. of television spot announcements y Y be the no. of radio spot announcements Objective : Maximize 600X + 200Y Constraints, X + Y > 30 Y < 25 Y>X 1200X + 300Y < 25,500

(Coverage & recall)

(Advertising strategy guidelines) (Cost of announcements)

Thank you!