1. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Linear Programming

2. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Module Objectives  Explain the importance of optimization to service operations management.  Give a real example of an operations management decision modeled with linear programming.  Demonstrate how to develop linear programming models.  Show how waiting line models can be solved using EXCEL SOLVER.  Discuss required assumptions and LP modeling complications.  Demonstrate 0-1, transportation, and assignment models.

3. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Concepts OPTIMIZATION - find best possible answer OBJECTIVE - what it is you want to accomplish OBJECTIVE FUNCTION measure of attainment of objective

4. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. History 1930s - transportation method WWII - Dantzig: simplex method degeneracy (computer could cycle) size limit 1979 - Khachian: ellipsoid algorithm 1984 - Karmarkar: Bell Labs algorithm

5. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. LP Model optimize {max or min} objective function (if min, maximize the negative) subject to limits (constraints) ( ) for i=1,m

6. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Assumptions of LP Linearity - all functions linear no diminishing returns, no economies of scale Certainty - all coefficients ( ) are known constants no distributions Continuity - solution liable to contain fractional values for

7. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Applications Resource allocation blending, product mix, inventory budget allocation, cash flow Planning & scheduling Diet Transportation Assignment

8. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Production Mix Example Decision - how many cans to produce each day Variables – H&B, JHB, LB, JLB, JP Objective - maximize profit Objective function - how it is measured 0.21H&B+0.20JHB+0.10LB+0.15JLB+0.10JP Limits – cans, ham, beans, max, min

9. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Model Max0.21 H&B + 0.20 JHB + 0.10 LB + 0.15 JLB + 0.10 JP s.t. 1 H&B + 1 JHB + 1 LB + 1 JLB + 1 JP  24000 4 H&B + 3 JHB  30000 9 H&B + 9 JHB + 14 LB + 12 JLB  100000 H&B  5000H&B  10000 JHB  1000JHB  4000 LB  1000LB  6000 JLB  2000JLB  4000 JP  0JP  1000

10. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Constraints Can function  10,000 cans/day Ham function  30,000 oz/day Bean function  100,000 oz/day highly predictable, or would be uncertain

11. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Diet Management Decide what food to give hospitals army CAMP - IBM system library Finnish school menu - lowered fat, reduced cost Newer models - incorporate preference

12. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. LP Can optimize many business problems Must consider the assumptions that are made –LP gives extreme solutions –if assumptions not appropriate, answer can be widely off

13. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Sensitivity Analysis Impact of changing any parameter economic interpretation

14. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Issues Reduced cost - effect of change in contribution coefficients –will the optimal solution change? Dual price (shadow price; marginal value) –economic interpretation of resource (constraint) value

15. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Reduced Cost Amount a nonbasic decision variable needs to be improved before it would be attractive enough to include in the optimal solution Relates to Nonbasic - not included in the set of m variables in simplex solution –therefore, solution value = 0

16. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Example VariableValueReduced cost Objective coefficient Allowable increase Allowable decrease H&B588800.21Infinity0.01 JHB1000-0.010.200.01Infinity LB1000-0.230.100.23Infinity JLB2000-0.130.150.13Infinity JP10000.10.10infinity0.1

17. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Reduced Cost Example H&B current profit 0.21, basic (in solution) –If price increases, no change (in solution) –If price decreases 0.01, another solution better HJB current profit 0.20, in solution but at minimum –INSUFFICIENT –Reduced cost –0.01 –Would need 0.20+0.01 >= 0.21 to make worthwhile producing more

18. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Reduced Cost Example Cj analysis APPLIES IF ONLY ONE CHANGE IN MODEL COEFFICIENTS If more than one Cj changes, can apply 100% RULE NO CHANGE IN OPTIMAL SOLUTION IF: sum of ratios D/[allowable D] over all decision variables is <= 1.0 –note D is change

19. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. 100% Rule Example If all profit rates dropped by.005 DallowableD/allowable H&B:-.005-.01.5 JHB:-.005 -infinity0 LB:-.005-infinity0 JLB:-.005-infinity0 JP:-.005-0.1.05 SO NO CHANGE.55 < 1 PROFIT would drop (still better than any other)

20. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. 100% rule A ONE-WAY test: if sum > 1.0, doesn’t prove anythingif all drop.01 DallowableD/allowable H&B:-.01-.011.0 JHB:-.01 -infinity0 LB:-.01-infinity0 JLB:-.01-infinity0 JP:-.01-0.1.1 SO DON’T KNOW1.1 > 1

21. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Dual Price Analysis DUAL PRICE: rate of change in objective function per one unit increase in right-hand- side Range analysis: NOT FOR OPTIMALITY, but for APPLICABILITY OF DUAL PRICE –how much right-hand-side could vary before DUAL PRICE would change

22. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Dual Prices VariableValueDual priceLimitAllowable increase Allowable decrease Cans10888024000Infinity13112 Ham26556030000Infinity3444 Beans1000000.02310000077508000

23. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Dual Price Example ECONOMIC INTERPRETION IF Beans increased 1 unit, NEW SOLUTION: objective function increase of $0.023 per ounce/day IF Ham increased 1 unit, NO CHANGE IN OPTIMAL SOLUTION have 3444 spare units per day IF Cans increased 1 unit, NO CHANGE IN SOLUTION: have 13,111 spare cans per day

24. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Dual Range Analysis How much RIGHT-HAND-SIDE can change before DUAL PRICE changes Cans: if drop 13112 (to 10888), starts to make a difference Ham: if drops 3444 (to 26556), starts to make a difference Beans: same 0.023 dual price unless increases 7750 (to 107,750) or drops 8000 (to 92,000) assumes ONE CHANGED parameter

25. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Dual Range Analysis Beans: if could acquire more beans, would be willing to pay up to 0.073 per ounce –Currently paying $0.05/ounce –Value is 0.05 + 0.023 more Not suggesting paying more IT WOULD BE ECONOMICALLY WORTHWHILE

26. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Sensitivity Analysis Can gauge the stability of the solution ONLY CAN CONSIDER ONE CHANGED PARAMETER AT A TIME –except for 100% rule DUAL PRICE can show impact on objective of changing right-hand-side

27. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Integer & 0-1 Programming the assumption of continuity not appropriate in all cases models, solution methods

28. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Types of Models All integer –require round numbers in solution –how many trucks to buy, buildings to build Zero-one –binary (incur cost or don’t; fund project or don’t) Mixed –some variables continuous; others integer or 0-1

29. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Solution Methods Round off Graphical Branch & bound Various enumeration methods

30. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Example Max 9X + 10YMax 10X + 22Y st 3X + 4Y <=30st 2X + 5Y <=20 2X + Y <=12 6X + 4Y <=24 solution: X = 3.6X = 1 9/11 Y = 4.8Y = 3 3/11 obj = 90.18obj = 86

31. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Rounding Max 9X + 10YMax 10X + 22Y st 3X + 4Y <=30st 2X + 5Y <=20 2X + Y <=12 6X + 4Y <=24 X = 3.64X = 1 9/112 Y = 4.85Y = 3 3/113 obj = 90.18 86obj = 90.18 86 but not feasible feasible, 3X + 4Y = 32not optimal 2X + Y = 13(X=0,Y=4,obj 88)

32. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Graphical Requires there be only two variables Plot all integer values (dots) Plot iso-objective function lines Balance on dot with greatest objective

33. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Branch & Bound Systematic search Use LP to generate optimal solutions Impose constraints to force integrality Need to branch whenever constrained

34. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. B&b Problem A Max 9X + 10YX=3.6, Y=4.8, Z=90.18 st 3X + 4Y <= 30 2X + Y <= 12 BRANCH: X =4 X=3, Y=5.25, Z=79.5X=4, Y=4, Z=76 not integerfeasible (LB) BRANCH: Y =6 X=3, Y=5, Z=77X=2, Y=6, Z=78 *OPT

35. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Problem A Sequence Original:two fractional variables BRANCH X 4 fractional answer integral:fathomed BRANCH Y 5 integral infeasible fathomed OPTIMAL must fathom each branch

36. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Enumeration Need to be able to identify all possible answers if zero-one, this is possible (a problem if n>30 [1,073,741,824]) simply take the greatest objective (a search problem)

37. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Capital Budgeting Decision: which investments to adopt Have a budget May have other constraints –scarce skills –required features Each project is a 0-1 variable 0 - don’t fund;1 - fund

38. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Inventory Order costs are do or don’t (0-1) –set-up costs the same Holding costs are continuous Variables for: –order for each time period (0-1) –quantity on hand (continuous) –quantity added (received), deleted (sold) Mixed integer model

39. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Project Selection Just like capital budgeting decision: which projects to fund (1) or not (2)

40. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Black Fly Control Solomon, et al., Interfaces [1992] Western Africa - black flies carry disease 1974 - program to control eggs 11 day reproductive cycle sprayed from helicopter LP: min cost st sufficient dosage variables: treatments by point by time (0-1) saved $400,000 per year

41. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Bellcore Project Selection Hoadley, et al., Interfaces [1993] research product: work needed to deliver technology 1991 - 222 products, 1046 projects projects: infrastructure benefit all usage sensitive - shared by few elective - user specific nonlinear integer program max utility index 0-1 variables for participation greater overall satisfaction

42. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Telecommunications Net Cox, et al., Interfaces [1993] US West - 14 western states fiber-optic new, copper old,cable links need redundency least-cost expansion plans reduce cost, ensure service variables: flows between links links were 0-15 minute to hours to solve saved 10% over existing, $100 million

43. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Summary Many useful models require variables be integer or 0-1 For integer, LP based best (branch & bound using LP to generate) For 0-1, enumeration often better

44. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Transportation Models problem structure solution method applications

45. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Transportation Model Special class of problem –set of demands with known requirements –set of sources with known supply –find the set of assignment quantities minimizing cost Variables: quantity from each source to each demand Constraints for each demand, supply –a ij coefficients always 1

46. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Why A commonly encountered managerial decision –firms have multiple warehouses to serve many retailers There is an efficient solution method –Hitchcock [1941]

47. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Example Refining SOURCES:DESTINATIONS: HoustonSan Antonio Corpus ChristiTexarkana Fort WorthEl Paso

48. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Example VARIABLES & COSTS Hou-SA 20CC-SA 6FW-SA 2 Hou-Tx 9CC-Tx 10FW-Tx 15 Hou-EP 5CC-EP 18FW-EP 12 Objective: minimize sum of variable quantities times costs

49. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Example CONSTRAINTS for each SUPPLY: Hou-SA + Hou-Tx + Hou-EP 150 CC-SA + CC-Tx + CC-EP 100 FW-SA + FW-Tx + FW-EP 250 for each DEMAND: Hou-SA + CC-SA + FW-SA 200 Hou-Tx + CC-Tx + FW-Tx 120 Hou-EP + CC-EP + FW-EP 180

50. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Balance Transportation model is BALANCED if: SUPPLY = DEMAND If NOT Make equal by creating DUMMY source or demand (whichever is smaller)

51. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Example Optimal Solution HouCCFW 200120180 SA150 20 -9 -5 150 Tx100 6 - 10 100 18 - EP250 2 200 15 20 12 30

52. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Electrical Pricing Aarvik & Randolph, Interfaces [1975] Norwegian national electrical cooperative 95% of Norway’s generating capacity 75 companies in grid system –transmission, exchange, connection fees Used transportation to get transmission fees for each subscriber

53. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Nuclear Waste Disposal Rautman, Reid & Ryder, Operations Research [1993] DOE - Yucca Mountain repository Congressional mandates, time behavior of materials, geologic capacity Identify optimal schedule for permanent storage of reactor waste –min total disposal area required –considered type of waste by time

54. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Assignment Problem problem structure solution method applications

55. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Problem Assign tasks to work units A restricted form of transportation model –each supply, demand equals exactly 1 Fewer places where it is useful A simplified transportation model –Each RHS = 1

56. McGraw-Hill/Irwin © 2003 The McGraw-Hill Companies, Inc., All Rights Reserved. Application Franz & Miller, Operations Research [1993] Assign medical residents to clinics Max resident preferences subject to training goals & staffing requirements Priorities used as weights in LP 12 3rd year, 12 2nd year residents 3180 0-1 variables, 856 constraints Much faster, better solutions