Spreadsheet Modeling & Decision Analysis
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This chapter discusses nonlinear programming and evolutionary optimization techniques. It introduces nonlinear programming problems, which have nonlinear objective functions and/or constraints. The Generalized Reduced Gradient (GRG) algorithm is commonly used to solve nonlinear programs and finds local optimal solutions. Global optimal solutions may exist but are difficult to guarantee. Examples discussed include the economic order quantity (EOQ) model, facility location problems, and nonlinear network flow problems.
Spreadsheet Modeling & Decision Analysis
- 1. Spreadsheet Modeling & Decision Analysis A Practical Introduction to Management Science 4th edition Cliff T. Ragsdale
- 2. Chapter 8 Nonlinear Programming & Evolutionary Optimization 8-2
- 3. Introduction to Nonlinear Programming (NLP) • An NLP problem has a nonlinear objective function and/or one or more nonlinear constraints. • NLP problems are formulated and implemented in virtually the same way as linear problems. • The mathematics involved in solving NLPs is quite different than for LPs. • Solver tends to mask this different but it is important to understand the difficulties that may be encountered when solving NLPs. 8-3
- 4. Possible Optimal Solutions to NLPs (not occurring at corner points) objective function objective function level curve level curve optimal solution optimal solution Feasible Feasible Region Region linear objective, nonlinear objective, nonlinear constraints linear constraints objective function level curve objective function level curves optimal solution optimal solution Feasible Region Feasible Region nonlinear objective, nonlinear objective, nonlinear constraints linear constraints 8-4
- 5. The GRG Algorithm • Solver uses the Generalized Reduced Gradient (GRG) algorithm to solve NLPs. • GRG can also be used on LPs but is slower than the Simplex method. • The following discussion gives a general (but somewhat imprecise) idea of how GRG works. 8-5
- 6. An NLP Solution Strategy X2 C D B E objective function level curves Feasible Region A (the starting point) X1 8-6
- 7. Local vs. Global Optimal Solutions X2 Local optimal solution C Local and global F optimal solution E Feasible Region G B A D X1 8-7
- 8. Comments About NLP Algorithms • It is not always best to move in the direction producing the fastest rate of improvement in the objective. • NLP algorithms can terminate at local optimal solutions. • The starting point influences the local optimal solution obtained. 8-8
- 9. Comments About Starting Points • The null starting point should be avoided. • When possible, it is best to use starting values of approximately the same magnitude as the expected optimal values. 8-9
- 10. A Note About “Optimal” Solutions • When solving a NLP problem, Solver normally stops when the first of three numerical tests is satisfied, causing one of the following three completion messages to appear: 1) “Solver found a solution. All constraints and optimality conditions are satisfied.” This means Solver found a local optimal solution, but does not guarantee that the solution is the global optimal solution. 8-10
- 11. A Note About “Optimal” Solutions • When solving a NLP problem, Solver normally stops when the first of three numerical tests is satisfied, causing one of the following three completion messages to appear: 2) “Solver has converged to the current solution. All constraints are satisfied.” This means the objective function value changed very slowly for the last few iterations. 8-11
- 12. A Note About “Optimal” Solutions • When solving a NLP problem, Solver normally stops when the first of three numerical tests is satisfied, causing one of the following three completion messages to appear: 3) “Solver cannot improve the current solution. All constraints are satisfied.” This rare message means the your model is degenerate and the Solver is cycling. Degeneracy can often be eliminated by removing redundant constraints in a model. 8-12
- 13. The Economic Order Quantity (EOQ) Problem • Involves determining the optimal quantity to purchase when orders are placed. • Small orders result in: – low inventory levels & carrying costs – frequent orders & higher ordering costs • Large orders result in: – higher inventory levels & carrying costs – infrequent orders & lower ordering costs 8-13
- 14. Sample Inventory Profiles Inventory 60 Annual Usage = 150 Number of Orders = 3 50 Order Size = 50 Avg Inventory = 25 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 Month Inventory 60 Annual Usage = 150 Number of Orders = 6 50 Order Size = 25 Avg Inventory = 12.5 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 Month 8-14
- 15. The EOQ Model D Q Total Annual Cost = DC + S + Ci Q 2 where: D = annual demand for the item C = unit purchase cost for the item S = fixed cost of placing an order i = cost of holding inventory for a year (expressed as a % of C) Q = order quantity Assumes: – Demand (or use) is constant over the year. – New orders are received in full when the inventory level drops to zero. 8-15
- 16. EOQ Cost Relationships $ 1000 800 Total Cost 600 400 Carrying Cost 200 Ordering Cost EOQ 0 0 10 20 30 40 50 Order Quantity 8-16
- 17. An EOQ Example: Ordering Paper For MetroBank • Alan Wang purchases paper for copy machines and laser printers at MetroBank. – Annual demand (D) is for 24,000 boxes – Each box costs $35 (C) – Each order costs $50 (S) – Inventory carrying costs are 18% (i) • What is the optimal order quantity (Q)? 8-17
- 18. The Model D Q MIN: DC + S + Ci Q 2 Subject to: Q ≥ 1 (Note the nonlinear objective!) 8-18
- 19. Implementing the Model See file Fig8-6.xls 8-19
- 20. Comments on the EOQ Model • Using calculus, it can be shown that the optimal value of Q is: 2DS Q = * Ci • Numerous variations on the basic EOQ model exist accounting for: – quantity discounts – storage restrictions – backlogging – etc 8-20
- 21. Location Problems • Many decision problems involve determining optimal locations for facilities or service centers. For example, – Manufacturing plants – Warehouse – Fire stations – Ambulance centers • These problems usually involve distance measures in the objective and/or constraints. • The straight line (Euclidean) distance between two points (X1, Y1) and (X2, Y2) is: Distance = (X 1 − X 2 ) + (Y1 − Y2 ) 2 2 8-21
- 22. A Location Problem: Rappaport Communications • Rappaport Communications provides cellular phone service in several mid-western states. • The want to expand to provide inter-city service between four cities in northern Ohio. • A new communications tower must be built to handle these inter-city calls. • The tower will have a 40 mile transmission radius. 8-22
- 23. Graph of the Tower Location Problem Y 50 Cleveland x=5, y=45 40 30 Youngstown Akron x=12, y=21 x=52, y=21 20 10 Canton x=17, y=5 0 40 X 8-23 0 10 20 30 50 60
- 24. Defining the Decision Variables X1 = location of the new tower with respect to the X-axis Y1 = location of the new tower with respect to the Y-axis 8-24
- 25. Defining the Objective Function • Minimize the total distance from the new tower to the existing towers ( ) ( ) ( ) ( ) 5- X 1 2 + 45 − Y 1 2 + 12- X 1 2 + 21− Y 1 2 MIN: ( ) ( ) ( ) ( ) + 17- X 1 2 + 5− Y 1 2 + 52- X 1 2 + 21− Y 1 2 8-25
- 26. Defining the Constraints • Cleveland (5- X1)2 +(45− Y1)2 ≤ 40 • Akron ( 12 - X 1 ) ( 2 + 21 − Y 1 ) 2 ≤ 40 • Canton ( 17 - X 1 ) ( ) 2 + 5− Y 1 2 ≤ 40 • Youngstown ( ) ( ) 52- X 1 2 + 21− Y 1 2 ≤ 40 8-26
- 27. Implementing the Model See file Fig8-10.xls 8-27
- 28. Analyzing the Solution • The optimal location of the “new tower” is in virtually the same location as the existing Akron tower. • Maybe they should just upgrade the Akron tower. • The maximum distance is 39.8 miles to Youngstown. • This is pressing the 40 mile transmission radius. • Where should we locate the new tower if we want the maximum distance to the existing towers to be minimized? 8-28
- 29. Implementing the Model See file Fig8-13.xls 8-29
- 30. Comments on Location Problems • The optimal solution to a location problem may not work: – The land may not be for sale. – The land may not be zoned properly. – The “land” may be a lake. • In such cases, the optimal solution is a good starting point in the search for suitable property. • Constraints may be added to location problems to eliminate infeasible areas from consideration. 8-30
- 31. A Nonlinear Network Flow Problem: The SafetyTrans Company • SafetyTrans specialized in trucking extremely valuable and extremely hazardous materials. • It is imperative for the company to avoid accidents: – It protects their reputation. – It keeps insurance premiums down. – The potential environmental consequences of an accident are disastrous. • The company maintains a database of highway accident data which it uses to determine safest routes. • They currently need to determine the safest route between Los Angeles, CA and Amarillo, TX. 8-31
- 32. Network for the SafetyTrans Problem Las 0.006 0.001 Albu- Vegas querque 0.001 +1 2 Flagstaff 8 6 Amarillo 0.003 0.010 10 0.006 0.004 Los Angeles 0.002 0.009 0.010 1 Phoenix 0.005 0.006 4 -1 0.004 0.002 0.002 Lubbock 0.003 9 Las San 0.003 Cruces Diego Tucson 7 3 5 0.010 Numbers on arcs represent the probability of an accident occurring. 8-32
- 33. Defining the Decision Variables 1, if the route from node i to node j is selected Yij = 0, otherwise 8-33
- 34. Defining the Objective Select the safest route by maximizing the probability of not having an accident, MAX: (1-P12Y12)(1-P13Y13)(1-P14Y14)(1-P24Y24)…(1-P9,10Y9,10) where: Pij = probability of having an accident while traveling between node i and node j 8-34
- 35. Defining the Constraints • Flow Constraints -Y12 -Y13 -Y14 = -1 } node 1 +Y12 -Y24 -Y26 = 0 } node 2 +Y13 -Y34 -Y35 = 0 } node 3 +Y14 +Y24 +Y34 -Y45 -Y46 -Y48 = 0 } node 4 +Y35 +Y45 -Y57 = 0 } node 5 +Y26 +Y46 -Y67 -Y68 = 0 } node 6 +Y57 +Y67 -Y78 -Y79 -Y7,10 = 0 } node 7 +Y48 +Y68 +Y78 -Y8,10 = 0 } node 8 +Y79 -Y9,10 = 0 } node 9 +Y7,10 +Y8,10 +Y9,10 = 1 } node 10 8-35
- 36. Implementing the Model See file Fig8-15.xls 8-36
- 37. Comments on Nonlinear Network Flow Problems • Small differences in probabilities can mean large differences in expected values: 0.9900 * $30,000,000 = $300,000 0.9626 * $30,000,000 = $1,122,000 • This type of problem is also useful in reliability network problems (e.g., finding the weakest “link” (or path) in a production system or telecommunications network). 8-37
- 38. A Project Selection Problem: The TMC Corporation • TMC needs to allocate $1.7 million of R&D budget and up to 25 engineers among 6 projects. • The probability of success for each project depends on the number of engineers assigned (Xi) and is defined as: Pi = Xi/(Xi + εi) Project 1 2 3 4 5 6 Startup Costs $325 $200 $490 $125 $710 $240 NPV if successful $750 $120 $900 $400 $1,110 $800 Probability Parameter εi 3.1 2.5 4.5 5.6 8.2 8.5 (all monetary values are in $1,000s) 8-38
- 39. Prob. of Success Selected Probability Functions 1.0000 0.9000 Project 2 - ε = 2.5 0.8000 Project 4 - ε = 5.6 0.7000 0.6000 0.5000 Project 6 - ε = 8.5 0.4000 0.3000 0.2000 0.1000 0.0000 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Engineers Assigned 8-39
- 40. Defining the Decision Variables 1, if project i is selected Yi = i = 1, 2, 3, ..., 6 0, otherwise Xi = the number of engineers assigned to project i, i = 1, 2, 3, …, 6 8-40
- 41. Defining the Objective Maximize the expected total NPV of selected projects 750X 1 120X 2 900X 3 800X 6 MAX: + + +L+ (X 1 + 31) (X 2 + 2.5) (X 3 + 4.5) . (X 6 + 8.5) 8-41
- 42. Defining the Constraints • Startup Funds 325Y1 + 200Y2 + 490Y3 + 125Y4 + 710Y5 + 240Y6 <=1700 • Engineers X1 + X2 + X3 + X4 + X5 + X6 <= 25 • Linking Constraints Xi - 25Yi <= 0, i= 1, 2, 3, … 6 • Note: The following constraint could be used in place of the last two constraints... X1Y1 + X2Y2+ X3Y3+ X4Y4+ X5Y5 + X6Y6 <= 25 However, this constraint is nonlinear. It is generally better to keep things linear where possible. 8-42
- 43. Implementing the Model See file Fig8-19.xls 8-43
- 44. Optimizing Existing Financial Models • It is not necessary to always write out the algebraic formulation of an optimization problem, although doing so ensures a thorough understanding of the problem. • Solver can be used to optimize a host of pre- existing spreadsheet models which are inherently nonlinear. 8-44
- 45. A Life Insurance Funding Problem • Thom Pearman owns a whole life policy with surrender value of $6,000 and death benefit of $40,000. • He’d like to cash in his whole life policy and use interest on the surrender value to pay premiums on a a term life policy with a death benefit of $350,000. • The premiums on the new policy for the next 10 years are: Year 1 2 3 4 5 6 7 8 9 10 Premium $423 $457 $489 $516 $530 $558 $595 $618 $660 $716 • Thom’s marginal tax rate is 28%. • What rate of return will be required on his $6,000 investment? 8-45
- 46. Implementing the Model See file Fig8-22.xls 8-46
- 47. The Portfolio Optimization Problem • A financial planner wants to create the least risky portfolio with at least a 12% expected return using the following stocks. Annual Return Covariance Matrix Year IBC NMC NBS IBC NMC NBS 1 11.2% 8.0% 10.9% IBC 0.00258 -0.00025 0.00440 2 10.8% 9.2% 22.0% NMC -0.00025 0.00276 -0.00542 3 11.6% 6.6% 37.9% NBS 0.00440 -0.00542 0.03677 4 -1.6% 18.5% -11.8% 5 -4.1% 7.4% 12.9% 6 8.6% 13.0% -7.5% 7 6.8% 22.0% 9.3% 8 11.9% 14.0% 48.7% 9 12.0% 20.5% -1.9% 10 8.3% 14.0% 19.1% 11 6.0% 19.0% -3.4% 12 10.2% 9.0% 43.0% Avg 7.64% 13.43% 14.93% 8-47
- 48. Defining the Decision Variables p1 = proportion of funds invested in IBC p2 = proportion of funds invested in NMC p3 = proportion of funds invested in NBS 8-48
- 49. Defining the Objective Minimize the portfolio variance (risk). n n −1 n MIN: ∑σ i =1 2 2 p i i +2 ∑∑σ i =1 j = i +1 ij pi p j σ i2 = the variance on investment i σ ij = σ ji = the covariance between investments i and j 8-49
- 50. Defining the Constraints • Expected return 0.0764 p1 + 0.1343 p2 + 0.1493 p3 >= 0.12 • Proportions p1 + p2 + p3 = 1 p1, p2, p3 >= 0 p1, p2, p3 <= 1 8-50
- 51. Implementing the Model See file Fig8-26.xls 8-51
- 52. The Efficient Frontier Portfolio Variance 0.04000 0.03500 0.03000 0.02500 0.02000 Efficient Frontier 0.01500 0.01000 0.00500 0.00000 10.00% 10.50% 11.00% 11.50% 12.00% 12.50% 13.00% 13.50% 14.00% 14.50% 15.00% Portfolio Return 8-52
- 53. Multiple Objectives in Portfolio Optimization • In portfolio problems we usually want to either: – Minimize risk (portfolio variance) – Maximize the expected return • We can deal with both objectives simultaneously as follows to generate efficient solutions: MAX: (1-r)(Expected Return) - r(Portfolio Variance) S.T.: p1 + p2 + … + pm = 1 pi >= 0 where: 0<= r <=1 is a user defined risk aversion value Note: If r = 1 we minimize the portfolio variance. If r = 0 we maximize the expected return. 8-53
- 54. Implementing the Model See file Fig8-30.xls 8-54
- 55. Sensitivity Analysis LP Term NLP Term Meaning Shadow Price Lagrange Multiplier Marginal value of resources. Reduced Cost Reduced Gradient Impact on objective of small changes in optimal values of decision variables. • Less sensitivity analysis information is available with NLPs vs. LPs. • See file Fig8-32.xls 8-55
- 56. Evolutionary Algorithms • A technique of heuristic mathematical optimization based on Darwin’s Theory of Evolution. • Can be used on any spreadsheet model, including those with “If” and/or “Lookup” functions. • Also known as Genetic Algorithms (GAs). 8-56
- 57. Evolutionary Algorithms • Solutions to a MP problem can be represented as a vector of numbers (like a chromosome) • Each chromosome has an associated “fitness” (obj) value • GAs start with a random population of chromosomes & apply – Crossover - exchange of values between solution vectors – Mutation - random replacement of values in a solution vector • The most fit chromosomes survive to the next generation, and the process is repeated 8-57
- 58. INITIAL POPULATION Chromosome X1 X2 X3 X4 Fitness 1 7.84 24.39 28.95 6.62 282.08 2 10.26 16.36 31.26 3.55 293.38 3 3.88 23.03 25.92 6.76 223.31 4 9.51 19.51 26.23 2.64 331.28 5 5.96 19.52 33.83 6.89 453.57 6 4.77 18.31 26.21 5.59 229.49 CROSSOVER & MUTATION Chromosome X1 X2 X3 X4 Fitness 1 7.84 24.39 31.26 3.55 334.28 2 10.26 16.36 28.95 6.62 227.04 3 3.88 19.75 25.92 6.76 301.44 4 9.51 19.51 32.23 2.64 495.52 Crossover 5 4.77 18.31 33.83 6.89 332.38 6 5.96 19.52 26.21 4.60 444.21 NEW POPULATION Mutation Chromosome X1 X2 X3 X4 Fitness 1 7.84 24.39 31.26 3.55 334.28 2 10.26 16.36 31.26 3.55 293.38 3 3.88 19.75 25.92 6.76 301.44 4 9.51 19.51 32.23 2.64 495.52 5 5.96 19.52 33.83 6.89 453.57 6 5.96 19.52 26.21 4.60 444.21 8-58
- 59. Example: Beating The Market • An investor would like to determine portfolio allocations that maximizes the number of times his portfolio outperforms the S&T 500. See file Fig8-37.xls 8-59
- 60. The Traveling Salesperson Problem • A salesperson wants to find the least costly route for visiting clients in n different cities, visiting each city exactly once before returning home. n (n-1)! 3 2 5 24 9 40,320 13 479,001,600 17 20,922,789,888,000 20 121,645,100,408,832,000 8-60
- 61. Example:The Traveling Salesperson Problem • Wolverine Manufacturing needs to determine the shortest tour for a drill bit to drill 9 holes in a fiberglass panel. See file Fig8-40.xls 8-61
- 62. End of Chapter 8 8-62