1. 3.13.1 | 3.1a | 3.23.1a3.2 Linear Programming Formulation n I. PRODUCT MIX n II. DIET n III. FINANCE n IV. MARKETING n V. TRANSPORTATION n VII. Data Envelopment Analysis

2. PRODUCT MIX EXAMPLE

3. 3.13.1 | 3.1a | 3.23.1a3.2 4 TYPES OF SHIRTS n X1= NUMBER BOXES OF SWEATSHIRTS WITH PRINTING ON FRONT n X2 = NUMBER BOXES SWEATSHIRTS PRINTING ON BOTH SIDES n X3= NUMBER BOXES T SHIRTS FRONT n X4= NUMBER BOXES T SHIRTS BOTH

4. 3.13.1 | 3.1a | 3.23.1a3.2 SHIRT PRODUCTION n ENOUGH CAPACITY FOR 1200 “STANDARD SIZED BOXES” n BUT STANDARD SIZE HOLDS DOZEN T SHIRTS, WHILE A BOX OF A DOZEN SWEAT SHIRTS IS 3 TIMES THE SIZE OF A STANDARD SIZE BOX n 3X1+3X2+X3+X4 < 1200

5. 3.13.1 | 3.1a | 3.23.1a3.2 PROCESSING TIME TYPEHOUR PER BOX X1.1 X2.25 X3.08 X4.21

6. ALL PRODUCTION WITHIN 72 HOURS.1X1+.25X2+.08X3+.21X4 <72

7. 3.13.1 | 3.1a | 3.23.1a3.2 BUDGET=$25000 TYPECOST PER BOX X136 X248 X325 X435

8. 36X1+48X2+25X3+35X4 <25000

9. 3.13.1 | 3.1a | 3.23.1a3.2 CAPACITY=500 FOR SWEATSHIRTS AND FOR TSHIRTS n X1+X2 < 500 n X3+X4 < 500

10. 3.13.1 | 3.1a | 3.23.1a3.2 PROFIT PER BOX TYPEPROFIT X190 X2125 X345 X465

11. OBJECTIVE FUNCTION MAX 90X1+125X2+45X3+65X4

12. DIET PROBLEM

13. 3.13.1 | 3.1a | 3.23.1a3.2 DECISION VARIABLES n X1=NUMBER CUPS OATMEAL n X2=NUMBER EGGS n X3=NUMBER CUPS MILK n X4=NUMBER SLICES WHEAT TOAST

14. 3.13.1 | 3.1a | 3.23.1a3.2 MINIMIZE COST FOODCOST OATMEAL.10 EGG.10 MILK.16 TOAST.07

15. OBJECTIVE FUNCTION MIN.10X1+.10X2+.16X3+.07X4

16. 3.13.1 | 3.1a | 3.23.1a3.2 AT LEAST 420 CALORIES FOODCALORIES OATMEAL100 EGG75 MILK100 TOAST65

17. 100X1+75X2+100X3+65X4 >420

18. 3.13.1 | 3.1a | 3.23.1a3.2 NO MORE THAN 30 CHOLESTEROL FOODCHOL OATMEAL0 EGG270 MILK12 TOAST0

19. 270X2+12X3<30

20. FINANCE EXAMPLE

21. 3.13.1 | 3.1a | 3.23.1a3.2 DECISION VARIABLES n X1= $ IN MUNI BONDS n X2= $ IN CERTIF OF DEPOSIT (CD) n X3= $ IN T BILLS n X4= $ IN STOCK

22. 3.13.1 | 3.1a | 3.23.1a3.2 RETURN ON INVESTMENT INVESTMENTANNUAL RETURN MUNI.085 CD.05 T.065 STOCK.13

23. OBJECTIVE FUNCTION MAX.085X1+.05X2+.065X3+.13X4

24. 3.13.1 | 3.1a | 3.23.1a3.2 NO MORE THAN 20% OF TOTAL INVESTMENT IN MUNI n GIVEN: TOTAL INVESTMENT = $ 70,000 n YOU CALCULATE: (.20)(70000)=14000 n X1 < 14000

25. 3.13.1 | 3.1a | 3.23.1a3.2 CD CONSTRAINT n AMOUNT INVESTED IN CD SHOULD NOT EXCEED AMOUNT INVESTED IN OTHER 3 ALTERNATIVES n X2<X1+X3+X4

26. 3.13.1 | 3.1a | 3.23.1a3.2 AT LEAST 30% OF INVESTMENT IN T BILL AND CD n GIVEN: $70,000 INVESTMENT n YOU CALCULATE: (.30)(70000)=21000 n X2+X3 > 21000

27. 3.13.1 | 3.1a | 3.23.1a3.2 RATIO CONSTRAINT n MORE SHOULD BE INVESTED IN CDs AND T BILLS THAN IN MUNI AND STOCKS BY A RATIO OF AT LEAST 1.2 TO 1 n X2 +X3 > 1.2 X1+X4 n NOTE: EXCEL REQUIRES LINEAR CONSTRAINT n X2+X3 > 1.2x1+1.2x4

28. 3.13.1 | 3.1a | 3.23.1a3.2 INVEST ENTIRE $70,000 n X1+X2+X3+X4=70000

29. MARKETING EXAMPLE

30. 3.13.1 | 3.1a | 3.23.1a3.2 DECISION VARIABLES n X1 = NUMBER OF TV COMMERCIALS n X2 = NUMBER OF RADIO COMMERCIALS n X3 = NUMBER OF NEWSPAPER AD

31. 3.13.1 | 3.1a | 3.23.1a3.2 AUDIENCE MEDIUMNUMBER OF PEOPLE (EXPOSURE) TV20000 RADIO12000 PAPER9000

32. 3.13.1 | 3.1a | 3.23.1a3.2 OBJECTIVE FUNCTION n MAX 20000X1+12000X2+9000X3

33. 3.13.1 | 3.1a | 3.23.1a3.2 BUDGET CONSTRAINT: $100,000 MEDIUMCOST TV$ 15,000 RADIO$ 6,000 PAPER$ 4,000

34. 3.13.1 | 3.1a | 3.23.1a3.2 15000X1+6000X2+4000X3 < 100000

35. 3.13.1 | 3.1a | 3.23.1a3.2 TV CONSTRAINT n TV STATION HAS TIME AVAILABLE FOR 4 COMMERCIALS n X1 < 4

36. 3.13.1 | 3.1a | 3.23.1a3.2 RADIO CONSTRAINT n RADIO STATION HAS TIME AVAILABLE FOR 10 COMMERCIALS n X2 < 10

37. 3.13.1 | 3.1a | 3.23.1a3.2 AD AGENCY CONSTRAINT n AGENCY HAS STAFF AVAILABLE FOR PRODUCING NO MORE THAN A TOTAL OF 15 ADS n X1+X2+X3 < 15

38. TRANSPORTATION EXAMPLE LOGISTICS, SUPPLY CHAIN MANAGEMENT

39. 3.13.1 | 3.1a | 3.23.1a3.2 DECISION VARIABLES n X1A= NUMBER OF UNITS SHIPPED FROM WAREHOUSE 1 TO STORE A n X1B = NUMBER OF UNITS SHIPPED FROM WAREHOUSE 1 TO STORE B n X1C = ‘’’’’’’’ TO STORE C n X2A = ‘’’’’’’’’’ FROM WAREHOUSE 2 TO STORE A n … n X3C = ‘’’’’’’’ FROM W 3 TO STORE C

40. 3.13.1 | 3.1a | 3.23.1a3.2 COST TO SHIP FROM WAREHOUSE TO STORE WARE- HOUSE TO STORESUPPLY ABC 1161811300 2141213200 3131517200 DEMAND150250200

41. 3.13.1 | 3.1a | 3.23.1a3.2 OBJECTIVE FUNCTION n MIMINIZE COST n MIN 16X1A+ 14X2A+ 13X3A+ 18X1B +…+ 11X1C+…+17X3C

42. 3.13.1 | 3.1a | 3.23.1a3.2 CONSTRAINTS n (1) SUPPLY n (2) DEMAND

43. 3.13.1 | 3.1a | 3.23.1a3.2 2 CASES n TOTAL SUPPLY> TOTAL DEMAND n SHIPMENTS < SUPPLY n SHIPMENTS = DEMAND n THIS EXAMPLE n TOTAL SUPPLY < TOTAL DEMAND n SHIPMENTS = SUPPLY n SHIPMENTS < DEMAND

44. 3.13.1 | 3.1a | 3.23.1a3.2 SUPPLY CONSTRAINTS n (1) WAREHOUSE 1: X1A+X1B+X1C < 300 n (2) W. 2: X2A+X2B+X2C < 200 n (3) W. 3: X3A+X3B+X3C < 200

45. 3.13.1 | 3.1a | 3.23.1a3.2 DEMAND CONSTRAINTS n (A)STORE A: X1A+X2A+X3A = 150 n (B) STORE B: X1B+X2B+X3B = 250 n (C) STORE C: X1C+X2C+X3C = 200

46. 3.13.1 | 3.1a | 3.23.1a3.2 DEA: Data Envelopment Analysis n Compares service units for efficiency n Inputs: teacher/student ratio; funds/student n Outputs: test scores LP will compare efficiency of 1 school with other school.

47. 3.13.1 | 3.1a | 3.23.1a3.2 Inputs Teacher/student ratioFunds/student Carey school.08$340 Delancey school.06$460

48. 3.13.1 | 3.1a | 3.23.1a3.2 Outputs School Reading test scoreMath test score C8179 D8173

49. 3.13.1 | 3.1a | 3.23.1a3.2 Decision Variables n X1 = opportunity cost of output 1 (reading score) n X2 = o.c. of output 2 ( math score) n Y1 = o.c. of input 1 (teacher/student ratio) n Y2 = o.c. of input 2 (funds/student)

50. 3.13.1 | 3.1a | 3.23.1a3.2 Objective Function n MAXIMIZE Value of Delancey outputs n We will measure Delancey efficiency n Data from previous slide “output” n MAX 81x1 + 73x2

51. 3.13.1 | 3.1a | 3.23.1a3.2 Constraint #1 n Scale Delancey inputs to one, so answer is an efficiency ratio n Data from previous slide “input” n.06y1 + 460y2 = 1

52. 3.13.1 | 3.1a | 3.23.1a3.2 Constraint #2 n Efficiency = value of outputs/value of inputs n Max efficiency 100%, so value of outputs < value of inputs Carey constraint: Carey outputs < Carey inputs 81x1 + 79x2 <.08y1 + 340y2 Delancey constraint: Del outputs < Del inputs 81x1 + 73x2 <.06y1 + 460y2

53. 3.13.1 | 3.1a | 3.23.1a3.2 output n Case 1: Maximum objective function value = 1, so Delancey is efficient n Case 2: Maximum objective function value < 1, so Delancey is not efficient

54. EXCEL: USE SOLVER FOR LP MEMO asnt