Product Mix Problem: Fifth Avenue Industries. Linear Programming (LP) Can Be Used for Many Managerial Decisions:

Transcription

1 Linear Programming (LP) Can Be Used for Many Managerial Decisions: Product mix Make-buy Media selection Marketing research Portfolio selection Shipping & transportation Multiperiod scheduling 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 1

2 Resource Data Material Silk Polyester Cotton Cost per yard $2 $6 $9 Yards available per month 1, 2, 1,25 Labor cost is $.75 per tie Product Data Type of Tie Silk Polyester Blend 1 Blend 2 Selling Price (per tie) $6.7 $3.55 $4.31 $4.81 Monthly Minimum 6, 1, 13, 6, Monthly Maximum 7, 14, 16, 8,5 Total material (yards per tie) Material Requirements (yards per tie) Type of Tie Material Silk Polyester Blend 1 (5/5) Blend 2 (3/7) Silk.125 Polyester Cotton.5.7 Total yards

3 Decision Variables S = number of silk ties to make per month P = number of polyester ties to make per month B 1 = number of poly-cotton blend 1 ties to make per month B 2 = 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 = $6.7 (.125 yds)($2/yd) - $.75 = $3.45 per tie Objective Function (in $ of profit) Max 3.45S P B B 2 Subject to the constraints: Material Limitations (in yards).125s < 1, (silk).8p +.5B 1 +.3B 2 < 2, (poly).5b 1 +.7B 2 < 1,25 (cotton) 3

4 Min and Max Number of Ties to Make 6, < S < 7, 1, < P < 14, 13, < B1 < 16, 6, < B2 < 8,5 Finally nonnegativity S, P, B1, B2 > Go to file 3-1.xls Media Selection Problem: Win Big Gambling Club Promote gambling trips to the Bahamas Budget: $8, per week for advertising Use 4 types of advertising Decision: How many ads of each type? Objective: Maximize audience reached Data Audience Reached (per ad) Cost (per ad) Max Ads Per week TV Spot 5, $8 12 Advertising Options Radio Newspaper (prime time) 8,5 $ ,4 $29 25 Radio (afternoon) 2,8 $38 2 4

5 Other Restrictions Have at least 5 radio spots per week Spend no more than $18 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 5T + 85N + 24P + 28A Subject to the constraints: Budget is $8 8T + 925N + 29P + 38A < 8 At Least 5 Radio Spots per Week P + A > 5 No More Than $18 per Week for Radio 29P + 38A < 18 Max Number of Ads per Week T < 12 P < 25 N < 5 A < 2 Finally nonnegativity T, N, P, A > Go to file 3-3.xls 5

6 Portfolio Selection: International City Trust Has $5 million to invest among 6 investments Decision: How much to invest in each of 6 investment options? Objective: Maximize interest earned Data Investment Trade credits Corp. bonds Gold stocks Platinum stocks Mortgage securities Construction loans Interest Rate 7% 1% 19% 12% 8% 14% Risk Score Constraints Invest up to $ 5 million No more than 25% into any one investment At least 3% into precious metals At least 45% into trade credits and corporate bonds Limit overall risk to no more than 2. 6

7 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) Max.7T +.1B +.19G +.12P +.8M +.14C Subject to the constraints: Invest Up To $5 Million T + B + G + P + M + C < 5,, No More Than 25% Into Any One Investment T <.25 (T + B + G + P + M + C) B <.25 (T + B + G + P + M + C) G <.25 (T + B + G + P + M + C) P <.25 (T + B + G + P + M + C) M <.25 (T + B + G + P + M + C) C <.25 (T + B + G + P + M + C) 7

8 At Least 3% Into Precious Metals G + P >.3 (T + B + G + P + M + C) At Least 45% Into Trade Credits And Corporate Bonds T + B >.45 (T + B + G + P + M + C) Limit Overall Risk To No More Than 2. Use a weighted average to calculate portfolio risk 1.7T + 1.2B + 3.7G + 2.4P + 2.M + 2.9C < 2. T + B + G + P + M + C OR 1.7T + 1.2B + 3.7G + 2.4P + 2.M + 2.9C < 2. (T + B + G + P + M + C) finally nonnegativity: T, B, G, P, M, C > 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 8

9 Time Period Min Num. Tellers Total minimum daily requirement is 112 hours Full Time Tellers Work from 9 AM 5 PM Take a 1 hour lunch break, half at 11, the other half at noon Cost $9 per day (salary & benefits) Currently only 12 are available Part Time Tellers Work 4 consecutive hours (no lunch break) Can begin work at 9, 1, 11, noon, or 1 Are paid $7 per hour ($28 per day) Part time teller hours cannot exceed 5% of the day s minimum requirement (5% of 112 hours = 56 hours) 9

10 Decision Variables F = num. of full time tellers (all work 9 5) P 1 = num. of part time tellers who work 9 1 P 2 = num. of part time tellers who work 1 2 P 3 = num. of part time tellers who work 11 3 P 4 = num. of part time tellers who work 12 4 P 5 = num. of part time tellers who work 1 5 Objective Function (in $ of personnel cost) Min 9 F + 28 (P 1 + P 2 + P 3 + P 4 + P 5 ) Subject to the constraints: Part Time Hours Cannot Exceed 56 Hours 4 (P 1 + P 2 + P 3 + P 4 + P 5 ) < 56 Minimum Num. Tellers Needed By Hour Time of Day F + P 1 > 1 (9-1) F + P 1 + P 2 > 12 (1-11).5 F + P 1 + P 2 + P 3 > 14 (11-12).5 F + P 1 + P 2 + P 3 + P 4 > 16 (12-1) F + P 2 + P 3 + P 4 + P 5 > 18 (1-2) F + P 3 + P 4 + P 5 > 17 (2-3) F + P 4 + P 5 > 15 (3-4) F + P 5 > 1 (4-5) 1

11 Only 12 Full Time Tellers Available F < 12 finally nonnegativity: F, P 1, P 2, P 3, P 4, P 5 > 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 Data Item Value $15,5 $14,4 $1,35 $14,525 $13, $9,625 Pounds $ / lb $3.1 $3.2 $3.45 $4.15 $3.25 $2.75 Cu. ft. per lb

12 Decision Variables W i = number of pounds of item i to load onto truck, (where i = 1,,6) Truck Capacity 15, pounds 1,3 cubic feet Objective Function (in $ of load value) Max 3.1W W W W W W 6 Subject to the constraints: Weight Limit Of 15, Pounds W 1 + W 2 + W 3 + W 4 + W 5 + W 6 < 15, Volume Limit Of 13 Cubic Feet.125W W W W W W 6 < 13 Pounds of Each Item Available W 1 < 5 W 4 < 35 W 2 < 45 W 5 < 4 W 3 < 3 W 6 < 35 Finally nonnegativity: W i >, i=1,,6 Go to file 3-7.xls 12

13 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 Grain $ per pound A $.33 B $.47 C $.38 Minimum Daily Requirement Protein per pound Riboflavin per pound Phosphorus per pound Magnesium per pound 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 13

14 Objective Function (in $ of cost) Min.33A +.47B +.38C Subject to the constraints: Total Blend is 2 Ounces, or.125 Pounds A + B + C =.125 (lbs) Minimum Nutritional Requirements 22A + 28B + 21C > 3 (protein) 16A + 14B + 25C > 2 (riboflavin) 8A + 7B + 9C > 1 (phosphorus) 5A + 6C >.425 (magnesium) Finally nonnegativity: A, B, C > 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 14

15 Decision Variables P At = number of motor A to produce in month t (t=1,,4) P Bt = number of motor B to produce in month t (t=1,,4) I At = inventory of motor A at end of month t (t=1,,4) I Bt = inventory of motor B at end of month t (t=1,,4) Sales Demand Data Month 1 (January) 2 (February) 3 (March) 4 (April) Motor A B Production cost Production Data Motor (values are per motor) A B $1 $6 Labor hours Production costs will be 1% higher in months 3 and 4 Monthly labor hours most be between 224 and

16 Inventory Data Inventory cost (per motor per month) Beginning inventory (beginning of month 1) Ending Inventory (end of month 4) Motor A B $ $.13 3 Max inventory is 33 motors 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 1P A1 + 1P A2 + 11P A3 + 11P A4 + 6P B1 + 6 P B P B P B4 +.18(I A1 + I A2 + I A3 + I A4 ) +.13(I B1 + I B2 + I B3 + I B4 ) Subject to the constraints: (see next slide) 16

17 Production & Inventory Balance + P A1 8 = I A1 (month 1) + P B1 1 = I B1 I A1 + P A2 7 = I A2 (month 2) I B1 + P B2 12 = I B2 I A2 + P A3 1 = I A3 (month 3) I B2 + P B3 14 = I B3 I A3 + P A4 11 = I A4 (month 4) I B3 + P B4 14 = I B4 Ending Inventory I A4 = 45 I B4 = 3 Maximum Inventory level I A1 + I B1 < 33 (month 1) I A2 + I B2 < 33 (month 2) I A3 + I B3 < 33 (month 3) I A4 + I B4 < 33 (month 4) Range of Labor Hours 224 < 1.3P A1 +.9P B1 < 256 (month 1) 224 < 1.3P A2 +.9P B2 < 256 (month 2) 224 < 1.3P A3 +.9P B3 < 256 (month 3) 224 < 1.3P A4 +.9P B4 < 256 (month 4) finally nonnegativity: P Ai, P Bi, I Ai, I Bi > Go to file 3-11.xls 17