Stochastic Programming Modeling

Transcription

1 Stochastic Programming Modeling IMA New Directions Short Course on Mathematical Optimization Jeff Linderoth Department of Industrial and Systems Engineering University of Wisconsin-Madison August 8, 2016 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 1 / 77

2 Week #2 The first week focused on theory and algorithms for continuous optimization problems where problem parameters are known with certainty. This week we will focus on two different topics: 1 Stochastic Programming: Used for Optimization under data uncertainty 2 Integer Programming: Used for modeling discrete decisions Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 2 / 77

3 Today s Outline About This Week About Us About You Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 3 / 77

4 Today s Outline About This Week About Us About You Stochastic Programming What is it?/why Should we Do it? A Newsvendor Recourse Models and Extensive Form How to implement in a modeling language Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 3 / 77

5 This Week Resources Our exercises will be done with AMPL: A Mathematical Programming Language We added you all to a Dropbox: There you can get AMPL, templates for the exercises, and the lecture slides. Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 4 / 77

6 This Week The Dream Team Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 5 / 77

7 This Week The Dream Team Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 5 / 77

8 This Week Optimization Dream Team Monday: Dave Morton, Northwestern, Sample Average Approximation Tuesday: Shabbir Ahmed, Georgia Tech, Multistage Stochastic Programming Wednesday: Robert Hildebrand, IBM, Lenstra s Algorithm Thursday: Santanu Dey, Georgia Tech, Cutting Plane Theory Friday: Dan Bienstock, Columbia, Mixed Integer Nonlinear Programming Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 6 / 77

9 This Week Week Overview Social Events! Monday: Stub and Herb s Wednesday: Twins Game Thursday: Surly Brewing Company Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 7 / 77

10 This Week Recommended Texts Stochastic Programming?: Very good. Requires strong math background?: A more gentle introduction, but still covers the whole field quite well.?: FREE!. It s in the Dropbox Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 8 / 77

11 This Week Recommended Texts Stochastic Programming?: Very good. Requires strong math background?: A more gentle introduction, but still covers the whole field quite well.?: FREE!. It s in the Dropbox Integer Programming?: Classic reference.?: A more gentle treatment?: Very nice geometric intuition?: My (new) favorite book Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 8 / 77

12 This Week Course Level/Expectations We will use AMPL ( to solve problems and prototype algorithms: If nothing else, you can get to learn a new language for modeling and solving mathematical optimization problems We will do a few proofs, but we will not require significant mathematical sophistication beyond a reasonable understanding of LP duality We assume some basic background in probability theory (no measure theory required) what is a random variable, expected value, law of large numbers, some basic statistics (CLT) We will expect some basic linear algebra knowledge Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 9 / 77

13 This Week About us... B.S. (G.E.), UIUC, M.S., OR, GA Tech, Ph.D., GA Tech, : MCS, ANL : Axioma, Inc : Lehigh University Research Areas: Large Scale Optimization, High Performance Computing. Married. One child, Jacob. Now 13. He is awesome. Hobbies: Golf, Integer Programming, Human Pyramids. Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 10 / 77

14 This Week About Jim... B.S. (I.E.), UW-Madison, 2001 M.S., OR, GA Tech, Ph.D., GA Tech, : IBM : UW-Madison Research Areas: Discrete Optimization, Stochastic Optimization, Applications Married. Three children: Rowan, Camerson, Remy. They are awesome Hobbies: Boxing, Integer Programming, Human Pyramids. Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 11 / 77

15 This Week Picture Time Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 12 / 77

16 This Week About You Quiz #1! 1 Name 2 Nationality 3 Education Background. 4 Research Interests/Thesis Topic? 5 (Optimization) Modeling Languages you know: (AMPL, GAMS, Mosel, CVX,... 6 Programming Languages you know: (C, Python, Matlab, Julia, FORTRAN, Java,...) 7 Anything specific you hope to accomplish/learn this week? 8 One interesting fact about yourself you think we should know. 9 Do you like human pyramids? :-) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 13 / 77

17 Introduction to SP Background Stochastic Programming $64 Question What does Programming mean in Mathematical Programming, Linear Programming, etc...? Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 14 / 77

18 Introduction to SP Background Stochastic Programming $64 Question What does Programming mean in Mathematical Programming, Linear Programming, etc...? A. Planning. Mathematical Programming (Optimization) is about decision making, or planning. Stochastic Programming is about decision making under uncertainty. View it as Mathematical Programming with random parameters Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 14 / 77

19 Introduction to SP Background Dealing With Randomness In most applications of optimization, randomness is ignored Otherwise, it is dealt with via: Sensitivity analysis For large-scale problems, sensitivity analysis is useless Careful determination of instance parameters No matter how careful you are, you can t get rid of inherent randomness. Stochastic Programming is the way! 1 1 This is not necessarily true, but we will assume it to be so for the next two days Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 15 / 77

20 Introduction to SP Newsvendor Hot Off the Presses A paperboy (newsvendor) needs to decide how many papers to buy in order to maximize his profit. He doesn t know at the beginning of the day how many papers he can sell (his demand). Each newspaper costs c. He can sell each newspaper for a price of s. He can return each unsold newspaper at the end of the day for r. (Note that s > c > r). The demand (unknown when we purchase papers) is D Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 16 / 77

21 Introduction to SP Newsvendor Hot Off the Presses A paperboy (newsvendor) needs to decide how many papers to buy in order to maximize his profit. He doesn t know at the beginning of the day how many papers he can sell (his demand). Each newspaper costs c. He can sell each newspaper for a price of s. He can return each unsold newspaper at the end of the day for r. (Note that s > c > r). The demand (unknown when we purchase papers) is D Newsvendor Profit F (x, D) = { (s c)x if x D sd + r(x D) cx if x > D Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 16 / 77

22 Introduction to SP Newsvendor Pictures of Function Marginal profit: (s c) if can sell all: x D Marginal loss: (c r) if have to salvage F (x, D) x = D x Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 17 / 77

23 Introduction to SP Newsvendor What Should We Do? Optimize, silly: max F (x, D). x 0 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 18 / 77

24 Introduction to SP Newsvendor What Should We Do? Optimize, silly: max F (x, D). x 0 This problem does not make sense! You can t optimize something random! Chewbacca_defense Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 18 / 77

25 Introduction to SP Newsvendor The Function is Random F (x, D 1 ) F (x, D 2 ) x = D 1 x x = D 2 x One x can t simultaneously optimize both functions Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 19 / 77

26 Introduction to SP Newsvendor (Silly) Idea #1 Suppose D is a random variable with cdf H(t) def = P(D t) Silly Idea: Plan for Average Case Let µ def = E[D] be the mean value of demand In this case: (proof by picture) max x 0 F (x, µ) x = µ. In this case, the optimal policy is to purchase µ We will see that this can be far from optimal when your problem takes more uncertainty into account Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 20 / 77

27 Introduction to SP Newsvendor Idea #2 Robust Plan for Worst Case Suppose D [l, u], and we wish to do the best we can given that the worst outcome for our objective will occur: max x 0 min D [l,u] F (x, D) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 21 / 77

28 Introduction to SP Newsvendor Idea #2 Robust Plan for Worst Case Suppose D [l, u], and we wish to do the best we can given that the worst outcome for our objective will occur: Note that we can write: max x 0 min D [l,u] F (x, D) F (x, D) = min{(s c)x, D(s r) + (r c)x} max x 0 min D [l,u] F (x, D) = max x 0 min D [l,u] min{(s c)x, D(s r) + (r c)x} = max min{(s c)x, l(s r) + (r c)x} x 0 = max x 0 F (x, l) x = l Robust optimization say some nice things. But we will not cover in Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 21 / 77

29 Introduction to SP Newsvendor Idea #3: Maximize Long-Run Profit The best idea Treat F (x, D) as a proper random variable, and maximize long-run profit. i.e. solve the optimization problem: max E[F (x, D)]. x 0 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 22 / 77

30 Introduction to SP Newsvendor Idea #3: Maximize Long-Run Profit The best idea Treat F (x, D) as a proper random variable, and maximize long-run profit. i.e. solve the optimization problem: max E[F (x, D)]. x 0 In this case, the objective may make sense. The newsvendor will make a purchase every day Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 22 / 77

31 Introduction to SP Newsvendor Optimizing for the Newsvendor Given only knowledge of the random variable D, given as the cdf H D (t), how many newspapers should the newsvendor buy? Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 23 / 77

32 Introduction to SP Newsvendor Optimizing for the Newsvendor Given only knowledge of the random variable D, given as the cdf H D (t), how many newspapers should the newsvendor buy? With some old-school calculus (Chain rule, Fundmental theorem of calculus), one can show that the optimal closed form solution to the Newsvendor problem is ( ) s c x = H 1 s r the (s c)/(s r) quantile of the distribution H Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 23 / 77

33 Introduction to SP Newsvendor Optimizing for the Newsvendor Given only knowledge of the random variable D, given as the cdf H D (t), how many newspapers should the newsvendor buy? With some old-school calculus (Chain rule, Fundmental theorem of calculus), one can show that the optimal closed form solution to the Newsvendor problem is ( ) s c x = H 1 s r the (s c)/(s r) quantile of the distribution H It Ain t Always That Easy The newsvendor is about the only stochastic program that admits such a simple closed form solution. In general, we must solve instances numerically (and also approximately) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 23 / 77

34 Introduction to SP Newsvendor Simulating (with Scenarios) newsboy.xls s = 2, c = 0.3, r = 0.05 Demand: Normally distributed. µ = 100, σ = 20 Mean Value Solution Buy 100. TRUE long run profit 154 Stochastic Solution Buy 123 TRUE long run profit 162 The difference between the two solutions ( ) is called the value of the stochastic solution. (Duh!) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 24 / 77

35 Introduction to SP Newsvendor Do You Feel Lucky, Punk? Should we always optimize the random variable F (x, D) in expectation? We may be risk-averse min ρ[f (x, D)] x 0 If ρ(a) = E[a]: Standard stochastic program If ρ(a) = E[a] + λv(a) for λ R, we have a mean-variance stochastic program Risk measures are discussed in the second lecture Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 25 / 77

36 Introduction to SP Newsvendor Another Possible Newsvendor Problem Suppose the newsvendor is lazy. He just wants to usually make enough money to go to Stub and Herb s, but he doesn t want to hurt his back carrying too may papers Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 26 / 77

37 Introduction to SP Newsvendor Another Possible Newsvendor Problem Suppose the newsvendor is lazy. He just wants to usually make enough money to go to Stub and Herb s, but he doesn t want to hurt his back carrying too may papers Chance Constraints min{x P {F (x, D) b} 1 α} x 0 Minimize the number of papers to purchase to ensure that the probability that you make at least b in profit is at least 1 α Note that F (x, D) is a random variable Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 26 / 77

38 Introduction to SP Newsvendor Another Possible Newsvendor Problem Suppose the newsvendor is lazy. He just wants to usually make enough money to go to Stub and Herb s, but he doesn t want to hurt his back carrying too may papers Chance Constraints min{x P {F (x, D) b} 1 α} x 0 Minimize the number of papers to purchase to ensure that the probability that you make at least b in profit is at least 1 α Note that F (x, D) is a random variable Jim will discuss this a bit as well Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 26 / 77

39 Introduction to SP Newsvendor Take Away Message The Flaw of Averages The flaw of averages occurs when uncertainties are replaced by single average numbers planning. Joke: Did you hear the one about the statistician who drowned fording a river with an average depth of three feet. Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 27 / 77

40 Introduction to SP Newsvendor Take-away Message: Point Estimates If you are planning using point estimates, then you are planning sub-optimally It doesn t matter how carefully you choose the point estimate it is impossible to hedge against future uncertainty by considering one realization of the uncertainty in your planning process Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 28 / 77

41 Stages Stages and Decisions The newsvendor problem is a classical recourse problem : Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 29 / 77

42 Stages Stages and Decisions The newsvendor problem is a classical recourse problem : 1 We make a decision now (first-period decision) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 29 / 77

43 Stages Stages and Decisions The newsvendor problem is a classical recourse problem : 1 We make a decision now (first-period decision) 2 Nature makes a random decision ( stuff happens) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 29 / 77

44 Stages Stages and Decisions The newsvendor problem is a classical recourse problem : 1 We make a decision now (first-period decision) 2 Nature makes a random decision ( stuff happens) 3 We make a second period decision that attempts to repair the havoc wrought by nature in (2). (recourse) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 29 / 77

45 Stages Stages and Decisions The newsvendor problem is a classical recourse problem : 1 We make a decision now (first-period decision) 2 Nature makes a random decision ( stuff happens) 3 We make a second period decision that attempts to repair the havoc wrought by nature in (2). (recourse) Key Idea The evolution of information is of paramount importance Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 29 / 77

46 Stages Newsvendor Again Newsvendor Profit F (x, D) = min{(s c)x, (s + r)d + (r c)x} D a random variable with cdf H D (t) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 30 / 77

47 Stages Newsvendor Again Newsvendor Profit F (x, D) = min{(s c)x, (s + r)d + (r c)x} D a random variable with cdf H D (t) We showed that ( ) s c x = H 1. s r Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 30 / 77

48 Stages Newsvendor Again Newsvendor Profit F (x, D) = min{(s c)x, (s + r)d + (r c)x} D a random variable with cdf H D (t) We showed that ( ) s c x = H 1. s r Suppose that Ω = {d 1, d 2,... d S } So there are a finite set of scenarios S, each with associated probability p j. ( j S p j = 1) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 30 / 77

49 Stages Newsvendor SP Parameters d s : Demand for newspapers in scenario s p s : Probability of scenario s Writing an optimization model for the newsvendor Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 31 / 77

50 Stages Newsvendor SP Parameters d s : Demand for newspapers in scenario s p s : Probability of scenario s Writing an optimization model for the newsvendor Variables x: Number to purchase y s : Number to sell in scenario s z s : Number to salvage in scenario s Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 31 / 77

51 Stages Newsvendor Stochastic LP max cx + p s (qy s + rz s ) s S s.t. y s d s s S x y s z s = 0 s S x 0 y s, z s 0 s S Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 32 / 77

52 Stages Put Another Way We could write the objective for the newsvendor problem in the form: where Q(x, D) = F (x, D) = cx + EQ(x, D), max {qy + rz y D, y + z = x}. y 0,z 0 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 33 / 77

53 Stages Put Another Way We could write the objective for the newsvendor problem in the form: where Q(x, D) = F (x, D) = cx + EQ(x, D), max {qy + rz y D, y + z = x}. y 0,z 0 Q(x, D) is the optimal recourse function: Given that we have chosen x and observed demand D, what should I do to maximize profit? Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 33 / 77

54 Stages It s Not Always So Easy For the newsvendor the recourse function: Q(x, D) has a simple closed form: Q(x, D) = min{sx, sd + r(x D)} In general the recourse function may not be simple Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 34 / 77

55 Stages It s Not Always So Easy For the newsvendor the recourse function: Q(x, D) has a simple closed form: Q(x, D) = min{sx, sd + r(x D)} In general the recourse function may not be simple In fact, for two-stage stochastic linear programs, the recourse function will be the optimal value of a linear program Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 34 / 77

56 Stages Scenario Modeling The most common representation of uncertainty (in stochastic programming) is via a list of scenarios, which are specific representations of how the future will unfold. Think of these as random variables ξ 1, ξ 2,... ξ S, with ξ j Ξ Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 35 / 77

57 Stages Scenario Modeling The most common representation of uncertainty (in stochastic programming) is via a list of scenarios, which are specific representations of how the future will unfold. Think of these as random variables ξ 1, ξ 2,... ξ S, with ξ j Ξ What we CAN T do Planners often generate a solution for each scenario generated What-if analysis. Each solution yields a prescription of what should be done if the scenario occurs, but there is no theoretical guidance about the compromise between those prescriptions Can we combine these prescriptions in a natural way? Stochastic Programming does this! Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 35 / 77

58 Farmer Ted Background Farmer Ted In this example, the farmer has recourse that is, he can do something at step (3). Not just sell his newspapers. Farmer Ted can grow Wheat, Corn, or Beans on his 500 acres. Farmer Ted requires 200 tons of wheat and 240 tons of corn to feed his cattle These can be grown on his land or bought from a wholesaler. Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 36 / 77

59 Farmer Ted Background More Constraints Any excess production can be sold for $170/ton (wheat) and $150/ton (corn) Any shortfall must be bought from the wholesaler at a cost of $238/ton (wheat) and $210/ton (corn). Farmer Ted can also grow beans Beans sell at $36/ton for the first 6000 tons Due to economic quotas on bean production, beans in excess of 6000 tons can only be sold at $10/ton Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 37 / 77

60 Farmer Ted Background The Data 500 acres available for planting Wheat Corn Beans Yield (T/acre) Planting Cost ($/acre) Selling Price ( 6000T) 10 (>6000T) Purchase Price N/A Minimum Requirement N/A Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 38 / 77

61 Farmer Ted Background Formulate the LP Decision Variables x W,C,B Acres of Wheat, Corn, Beans Planted w W,C,B Tons of Wheat, Corn, Beans sold (at favorable price). e B Tons of beans sold at lower price y W,C Tons of Wheat, Corn purchased. Note that Farmer Ted has recourse. After he observes the weather event, he can decide how much of each crop to sell or purchase! Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 39 / 77

62 Farmer Ted Background Formulation max 150x W 230x C 260x B 238y W + 170w W 210y C + 150w C + 36w B + 10e B subject to x W + x C + x B x W + y W w W = 200 3x C + y C w C = x B w B e B = 0 w B 6000 x W, x C, x B, y W, y C, e B, w W, w C, w B 0 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 40 / 77

63 Farmer Ted Background Solution with (expected) yields Wheat Corn Beans Plant (acres) Production Sales Purchase Profit: $118,600 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 41 / 77

64 Farmer Ted Background It s the Weather, Stupid! Farmer Ted knows well enough to know that his yields aren t always precisely Y = (2.5, 3, 20). He decides to run two more scenarios Good weather: 1.2Y Bad weather: 0.8Y. Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 42 / 77

65 Farmer Ted Making the SuperModel Creating a Stochastic Model Here is a general procedure for making a (scenario-based) 2-stage stochastic optimization problem For a nominal state of nature (scenario), formulate an appropriate LP model Decide which decisions are made before uncertainty is revealed, and which are decided after All second stage variables get scenario index Constraints with scenario indices must hold for all scenarios Second stage variables in the objective function should be weighted by the probability of the scenario occurring Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 43 / 77

66 Farmer Ted Making the SuperModel What does this mean in our case? First stage variables are the x (or planting variables) Second stage variables are the y, w, e (purchase and sale variables) We have one copy of the y, w, e for each scenario! Attach a scenario subscript s = 1, 2, 3 to each of the purchase and sale variables. 1: Good, 2: Average, 3: Bad w C2 : Tons of corn sold at favorable price in scenario 2 e B3 : Tons of beans sold at unfavorable price in scenario 3. Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 44 / 77

67 Farmer Ted Making the SuperModel Expected Profit The second stage cost for each submodel appears in the overall objective function weighted by the probability that nature will choose that scenario 150x W 230x C 260x B +1/3( 238y W w W 1 210y C w C1 + 36w B1 + 10e B1 ) +1/3( 238y W w W 2 210y C w C2 + 36w B2 + 10e B2 ) +1/3( 238y W w W 3 210y C w C3 + 36w B3 + 10e B3 ) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 45 / 77

68 Farmer Ted Making the SuperModel Constraints x W + x C + x B 500 3x W + y W 1 w W 1 = x W + y W 2 w W 2 = 200 2x W + y W 3 w W 3 = 200 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 46 / 77

69 Farmer Ted Making the SuperModel Constraints (cont.) 3.6x C + y C1 w C1 = 240 3x C + y C2 w C2 = x C + y C3 w C3 = x B w B1 e B1 = 0 20x B w B2 e B2 = 0 16x B w B3 e B3 = 0 w B1, w B2, w B All vars 0 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 47 / 77

70 Farmer Ted Making the SuperModel Optimal Solution Wheat Corn Beans s Plant (acres) Production Sales Purchase Production Sales Purchase Production Sales Purchase Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 48 / 77

71 Farmer Ted Statistics:VSS The Value of the Stochastic Solution (VSS) Suppose we just replaced the random quantities (the yields) by their mean values and solved that problem. Would we get the same expected value for the Farmer s profit? How can we check? Solve the mean-value problem to get a first stage solution x. Fix the first stage solution at that value x, and solve all the scenarios to see Farmer Ted s profit in each. Take the weighted (by probability) average of the optimal objective value for each scenario Alternatively (and probably faster), we can fix the x variables and solve the stochastic programming problem we created. Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 49 / 77

72 Farmer Ted Statistics:VSS Computing FT s VSS Mean yields Y = (2.5, 3, 20) (We already solved this problem). x W = 120, x C = 80, x B = 300 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 50 / 77

73 Farmer Ted Statistics:VSS Fixed Policy Average Yield Scenario maximize 150x W 230x C 260x B 238y W + 170w W 210y C + 150y C + 36w B + 10e B subject to x W = 120 x C = 80 x B = 300 x W + x C + x B x W + y W w W = 200 3x C + y C w C = x B w B e B = 0 w B 6000 x W, x C, x B, y W, y C, e B, w W, w C, w B 0 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 51 / 77

74 Farmer Ted Statistics:VSS Fixed Policy Average Yield Scenario Solution Wheat Corn Beans Plant (acres) Production Sales Purchase Profit: $118,600 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 52 / 77

75 Farmer Ted Statistics:VSS Fixed Policy Bad Yield Scenario maximize 150x W 230x C 260x B 238y W + 170w W 210y C + 150y C + 36w B + 10e B subject to x W = 120 x C = 80 x B = 300 x W + x C + x B 500 2x W + y W w W = x C + y C w C = x B w B e B = 0 w B 6000 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 53 / 77

76 Farmer Ted Statistics:VSS Fixed Policy Bad Yield Scenario maximize 150x W 230x C 260x B 238y W + 170w W 210y C + 150y C + 36w B + 10e B subject to x W = 120 x C = 80 x B = 300 x W + x C + x B 500 2x W + y W w W = x C + y C w C = x B w B e B = 0 w B 6000 Objective Value: $55,120 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 53 / 77

77 Farmer Ted Statistics:VSS Fixed Policy Good Yield Scenario maximize 150x W 230x C 260x B 238y W + 170w W 210y C + 150y C + 36w B + 10e B subject to x W = 120 x C = 80 x B = 300 x W + x C + x B 500 3x W + y W w W = x C + y C w C = x B w B e B = 0 w B 6000 Objective Value: $148,000 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 54 / 77

78 Farmer Ted Statistics:VSS What s it Worth to Model Randomness? If Farmer Ted implemented the policy based on using only average yields, he would plant x W = 120, x C = 80, x B = 300 He would expect in the long run to make an average profit of... 1/3(118600) + 1/3(55120) + 1/3(148000) = If Farmer Ted implemented the policy based on the solution to the stochastic programming problem, he would plant x W = 170, x C = 80, x B = 250. From this he would expect to make Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 55 / 77

79 Farmer Ted Statistics:VSS VSS The difference of the values is the Value of the Stochastic Solution : $1150. Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 56 / 77

80 Farmer Ted Statistics:VSS VSS The difference of the values is the Value of the Stochastic Solution : $1150. It would pay off $1150 per growing season for Farmer Ted to use the stochastic solution rather than the mean value solution. $1150 is precisely the value of implementing a planting policy based on the stochastic solution, rather than the mean-value solution. Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 56 / 77

81 Farmer Ted Statistics:VSS (General) Stochastic Programming A Stochastic Program def f(x) = E ω [F (x, ξ(ω))] min x X 2 Stage Stochastic LP w/recourse F (x, ω) def = c T x + Q(x, ω) c T x: Pay me now Q(x, ω): Pay me later The Recourse Problem Q(x, ω) def = min q(ω) T y W (ω)y = h(ω) T (ω)x y 0 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 57 / 77

82 Extensive Form Two Stage Stochastic Linear Program Assume Ω = {ω 1, ω 2,... ω S } R r, P(ω = ω s ) = p s, s = 1, 2,..., S T s def = T (ω s ), h s def = h(ω s ), q s def = q(ω s ), W s = W (ω s ) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 58 / 77

83 Extensive Form Two Stage Stochastic Linear Program Assume Ω = {ω 1, ω 2,... ω S } R r, P(ω = ω s ) = p s, s = 1, 2,..., S T s def = T (ω s ), h s def = h(ω s ), q s def = q(ω s ), W s = W (ω s ) where for s = 1,..., S min c x + S s=1 p sq s (x) s.t. Ax b x R n 1 + Q s (x) def = Q(x, ω s ) = min q s y s.t. W s y = h s T s x y R n 2 + Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 58 / 77

84 Extensive Form Extensive Form When we have a finite number of scenarios, or if we approximate the problem with a finite number of scenarios 2, we can write an equivalent extensive form linear program: c T x + p 1 q1 T y 1 + p 2 q2 T y p s qs T y s s.t. Ax = b T 1 x + W 1 y 1 = h 1 T 2 x + W 2 y 2 = h T S x + W S y s = h s x X y 1 Y y 2 Y y s Y 2 Stay Tuned for Dave Morton s Lecture Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 59 / 77

85 Extensive Form The Upshot This is just a larger linear program It is a larger linear program that also has special structure Jim explains how to exploit this structure tomorrow Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 60 / 77

86 Extensive Form The Upshot This is just a larger linear program It is a larger linear program that also has special structure Jim explains how to exploit this structure tomorrow c T x + p 1 q1 T y 1 + p 2 q2 T y p s qs T y s s.t. Ax = b T 1 x + W 1 y 1 = h 1 T 2 x + W 2 y 2 = h T S x + W S y s = h s x X y 1 Y y 2 Y y s Y Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 60 / 77

87 Extensive Form Building the Supermodel Weird Science A general technique for creating two-stage resource problems. 1 Write a nominal (one scenario) model 2 Decide which variables are first stage, and second stage 3 Give s scenario index to all second stage variables and random parameters 4 Give context to all scenarios Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 61 / 77

88 Facility Location Example Facility Location and Distribution Facilities: I Customers: J Fixed cost f i, capacity u i for facility i I Demand d j : for j J Per unit Delivery cost: c ij i J, j J Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 62 / 77

89 Facility Location Example Facility Location and Distribution Facilities: I Customers: J Fixed cost f i, capacity u i for facility i I Demand d j : for j J Per unit Delivery cost: c ij i J, j J min i I f i x i + i I y ij d j i I y ij u i x i 0 j J x i {0, 1}, y ij 0 c ij y ij j J j J i I i I, j J Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 62 / 77

90 Facility Location Example AMPL for FL 1 var x{i} binary; 2 var y{i,j} >= 0; 3 4 minimize Cost: AMPL Code 5 sum{i in I} f[i]*x[i] + sum{i in I, j in J} c[i,j]*y[i,j] ; 6 7 subject to MeetDemand{j in J}: 8 sum{i in I} y[i,j] >= d[j] ; 9 10 subject to FacCapacity{i in I}: 11 sum{j in J} y[i,j] - u[i]*x[i] <= 0 ; Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 63 / 77

91 Facility Location Example Evolution of Information 1 Build facilities now 2 Demand becomes known. One of the scenarios S = {d 1, d 2,... d S } happens 3 Meet demand from open facilities Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 64 / 77

92 Facility Location Example Evolution of Information 1 Build facilities now 2 Demand becomes known. One of the scenarios S = {d 1, d 2,... d S } happens 3 Meet demand from open facilities First stage variables: x i Second stage variables: y ijs Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 64 / 77

93 Facility Location Example The SuperModel min f i x i + i I s S p s c ij y ijs i I j J y ijs d js j J s S i I y ijs u i x i 0 j J x i {0, 1}, y ijs 0 i I, s S i I, j J, s S Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 65 / 77

94 Facility Location Example Modeling Discussion Do we always want to meet demand? Regardless of the outcome d s? What happens on the off chance that our product is so popular that we can t possibly meet demand, even if we opened all of the facilities? Does the world end? Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 66 / 77

95 Facility Location Example Modeling Discussion Do we always want to meet demand? Regardless of the outcome d s? What happens on the off chance that our product is so popular that we can t possibly meet demand, even if we opened all of the facilities? Does the world end? Two Ideas 1 We could penalize not meeting demand of customers. 2 We only want to meet demand most of the time. (Chance constraint) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 66 / 77

96 Facility Location Example SP Definitions A 2-stage stochastic optimization problem has complete recourse if for every scenario, there always exists a feasible second solution: Q s (x) < + x R n, s = 1,..., S A 2-stage stochastic optimization problem has relatively complete recourse if for every scenario, and for every feasible first stage solution, there always exists a feasible second solution: Q s (x) < + x X, s = 1,..., S Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 67 / 77

97 Facility Location Example Penalize Shortfall: A Recourse Formulation min f i x i + p s c ij y ijs + λe js i I s S i I j J y ijs + e js d js j J s S i I y ijs u i x i 0 j J x i {0, 1}, y ij 0 i I, s S i I, j J Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 68 / 77

98 AMPL Hints Stop. AMPL Time. Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 69 / 77

99 AMPL Hints AMPL Hints 1 All chapters of AMPL book are available for download: http: //ampl.com/resources/the-ampl-book/chapter-downloads/ 2 You can change the solver with the command option solver cplex; (or replace cplex with baron, conopt, gurobi, knitro, loqo, minos, snopt, xpress.) 3 Use var to declare variables; You may also put >= 0 on the same line if the variables are constrained to be non-negative. 4 One your AMPL model is complete, you can type model <filename>; at the ampl: prompt. This will tell you if you have syntax errors. 5 If you have syntax errors. Fix them. Save the file, and type reset; Then go to 4. 6 If no errors, type solve; Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 70 / 77

100 AMPL Hints AMPL Entities Data Sets: lists of products, materials, etc. Parameters: numerical inputs such as costs, etc. Model Variables: The values to be decided upon. Objective Function. Constraints. Data and Model typically stored in different files! Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 71 / 77

101 AMPL Hints Template of Typical AMPL File Define Sets Define Parameters Define Variables Also can define variable bound constraints in this section Define Objective Define Constraints Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 72 / 77

102 AMPL Hints Important AMPL Keywords/Syntax model file.mod; data file.mod; reset; quit; set param var maximize (minimize) subject to Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 73 / 77

103 AMPL Hints Important AMPL Notes The # character starts a comment All statements must end in a semi-colon; Names must be unique! A variable and a constraint cannot have the same name AMPL is case sensitive. Keywords must be in lower case. Even if the AMPL error message is cryptic, look at the location where it shows an error this will often help you deduce what is wrong. Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 74 / 77

104 AMPL Hints Important AMPL Notes The # character starts a comment All statements must end in a semi-colon; Names must be unique! A variable and a constraint cannot have the same name AMPL is case sensitive. Keywords must be in lower case. Even if the AMPL error message is cryptic, look at the location where it shows an error this will often help you deduce what is wrong. Learning Data Input Look at examples Look at Chapter 9 of AMPL Book: http: //ampl.com/resources/the-ampl-book/chapter-downloads/ Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 74 / 77

105 AMPL Hints Some AMPL Tips option show stats 1; shows the problem size Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 75 / 77

106 AMPL Hints Conclusions Replacing uncertain parameters with point estimates may lead to sub-optimal planning: the flaw of averages Two-stage recourse problems: Decision Event Decision The Value of the Stochastic Solution Creating the extensive form/supermodel Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 76 / 77

107 AMPL Hints VSS: Value of the Stochastic Solution Let zs be the optimal solution value to zs def = min E[F (x, ξ(ω))] x X Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 77 / 77

108 AMPL Hints VSS: Value of the Stochastic Solution Let zs be the optimal solution value to zs def = min E[F (x, ξ(ω))] x X Let xmv be an optimal solution to the mean-value problem: xmv arg min F (x, E[ξ(ω)]) x X Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 77 / 77

109 AMPL Hints VSS: Value of the Stochastic Solution Let zs be the optimal solution value to zs def = min E[F (x, ξ(ω))] x X Let xmv be an optimal solution to the mean-value problem: xmv arg min F (x, E[ξ(ω)]) x X Let zmv be the long run cost if you plan based on the policy obtained from the average scenario: zmv def = EF (xmv, ξ(ω)) Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 77 / 77

110 AMPL Hints Simple HW: Prove vss 0 Jeff Linderoth (UW-Madison) Stochastic Programming Modeling Lecture Notes 77 / 77 VSS: Value of the Stochastic Solution Let zs be the optimal solution value to zs def = min E[F (x, ξ(ω))] x X Let xmv be an optimal solution to the mean-value problem: xmv arg min F (x, E[ξ(ω)]) x X Let zmv be the long run cost if you plan based on the policy obtained from the average scenario: zmv def = EF (xmv, ξ(ω)) Value of Stochastic Solution vss def = zmv zs