Examples of operational research

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Simple introduction to operations research

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Examples of operational research

  1. 1. HOW A FRENCH GUY CAN TEACH IN A MANDARIN-SPEAKING COUNTRYProblem: I speak French, you speak Mandarin.Solutions:+ English+ Images+ Stupid questions (I do not ask stupid questions because of stupidity, its just for checking that theres no misunderstanding)+ Interrupt me + send emails+ Feel free of discussing in Mandarin
  2. 2. I dont want so speak Chinese to youIn France, we say “this is Chinese to me”for “I cant catch a word”.==> Please tell me if I “speak Chinese to you”.
  3. 3. DISCLAIMERFact 1: Many applications in these lessons are about military applications.Reason: Easy to understand + many precursor operation research works are military.Fact 2: My main application fields are ecological and energetical.Implication: We will see such applications later, with much more details.
  4. 4. WHAT IS THE POINT IN THESE 3 HOURS ?This is about Operations Research (OR).Key points: - OR can help a company/country to save up plenty of $. - new OR works usually do not originate at the head of companies / countries; originate in young people who want to do new things. - you can be very useful if you understand the goal of OR.
  5. 5. Operations Research everywhere. I hope that from now on, you will see plentyof OR opportunities everywhere.
  6. 6. POLLThis teaching is adaptive.Tell me what you want, and Ill try to adapt.Email: olivier.teytaud@inria.fr (emails will be considered as private, no forward, no archive; only an anonymous agregation is public)More mathematics ?More illustrative examples ?More C / C++ / Java / Matlab examples ?More examples in domain X or Y ?More algorithmic elements ?Today, illustrative examples mainly ==> simple.
  7. 7. Overview1) Basic definitions2) Illustrative example: UK in WWII3) Categories of Op. Research (OR) a) based on optimization b) based on statistics c) based on time-decomposition d) others e) all-in-one
  8. 8. OP. RESEARCH: DefinitionsGoal = near optimal decisions tocomplex decision-making problems.Similar to: - Decision sciences - Management sciences (part of)Close to artificial intelligence==> we will see plenty of vocabulary, because vocabulary = crucial for finding relevant literature.
  9. 9. OP. RESEARCH: History- Some precursors- “Real” birth in world war II- Then widely spread everywhere - industry - business
  10. 10. Overview1) Basic definitions2) Illustrative example: UK in WWII3) Categories of Op. Research (OR) a) based on optimization b) based on statistics c) based on time-decomposition d) others e) all-in-one
  11. 11. WORLD WAR II viewed from Europe- France + others quickly defeated ==> Hitler close to complete victory- But UK resists, so Germany must fight both on East and West- Also UK territory can be used by US for bombing germany- The surprising resistance of UK is a key element of World War II- Op. Research = key element of WW II.
  12. 12. WORLD WAR II- Modern techniques everywhere- Supply chain = critical- Both tactical and strategical elements.
  13. 13. UK is an island- attacks = bombing- defense = anti-aircraft artillery + radar + planes
  14. 14. VS
  15. 15. An optimization problem: defending UK We know that Q German bombers might take off at locations A, B, C and D. We know that their flights have maximal length L. We can build N radar antennas. We can set up M anti-aircraft artillery. We can distribute P air-fighters on airports E, F, G and H.Choose: - the positionning of the N radar antenna - the anti-aircraft artillery - the positionning of the air-fighters
  16. 16. An optimization problem: defending UKStep 1: defining the variables & constraints.- Two “Float” variables for each positionning (antenna, artillery). ==> 2(N+M) variables- One “int” variables for each airport (number of airplanes), ==> e=nb of planes in E, ==> f=nb of planes in F, ==> g=nb of planes in G, ==> h=nb of planes in H.Constraints:- Number of planes: e+f+g+h=P- Antennas and artillery not in the sea
  17. 17. An optimization problem: defending UKKeeping 2(N+M)+4 variables take too much room.Lets say x = vector ( e,f,g,h,x1,x2,x3,...,x(2(N+M)+4) ).Step 2: defining the objective function.= something which quantifies to which extent a proposed solution is good or not.f(x) = very big number if x is a bad strategy. = very small number if x is a very good strategy.Then, we will look for x* such that f(x*) = minimum.Definition: x* is the optimum of f.Can you define f for our radars / antenna / planes ?
  18. 18. An optimization problem: defending UKReminder:We know that German bombers mighttake off at locations A, B, C and D.We know that their flights have maximal length L.We dont know which starting point they will choose.f(x) = maximum expected damage of the possible German attacksNew variables: a = nb of planes which take off at A, b,c,d = nb of planes which take off at B,C,D. z = vector defining German trajectories==> y=(a,b,c,d,z)
  19. 19. An optimization problem: defending UKComputing g(x,y) on a computer: - simulate the time steps of a airplane flight / bombing - evaluate the damage - return the damage Not so easy!So we can write g(x,y) = damage if strategy y.But what is f(x) ? f(x) = max g(x,y) (maximum on y)Is it clear for you ?
  20. 20. Defending UK: the complete pictureMain function: Find x* such f(x*) is minimum. // x* = optimal // UK strategyFunction f(x): Find y such that g(x,y) is maximum. Return g(x,y)Function g(x,y) Return damage if - UK has strategy x - Germany has strategy y
  21. 21. Two optimization levelsMain function: Find x* such f(x*) is minimum. // x* = optimal // UK strategyFunction f(x): Main optimization Find y such that g(x,y) is maximum. problem Return g(x,y)Function g(x,y) Return damage if - UK has strategy x - Germany has strategy y
  22. 22. The second one is antagonistMain function: Find x* such f(x*) is minimum. // x* = optimal // UK strategyFunction f(x): Find y such that g(x,y) is maximum. Return g(x,y) Secondary optimizationFunction g(x,y) problem: the German army will optimize its behavior Return damage if - UK has strategy x - Germany has strategy y
  23. 23. Overview1) Basic definitions2) Illustrative example: UK in WWII3) Categories of Op. Research (OR) Sometimes OR = solved by just modelization. submarine
  24. 24. Overview1) Basic definitions2) Illustrative example: UK in WWII3) Categories of Op. Research (OR) Example: submarine UK trying to destroy U-boats (which destroyed UK convoys).
  25. 25. Overview1) Basic definitions2) Illustrative example: UK in WWII3) Categories of Op. Research (OR) Sometimes OR = solved by just modelization. Approaches U-boat: Escapes! submarine
  26. 26. Overview1) Basic definitions2) Illustrative example: UK in WWII3) Categories of Op. Research (OR) Sometimes OR = solved by just modelization. Approaches Bomb! Escapes! submarine
  27. 27. OverviewSimulation: at which depth should the bombexplode in order to maximize the probability ofdestroying the U-boatA relevant simulator helped a lot UK. Approaches Escapes! Bomb! submarine
  28. 28. OverviewSimulation: based the plane speed, the detectiontime (by the U-boat), the U-boat behavior, theU-boat speed, the bombs damage radius...Draw a figure with this and you have the solution. Approaches Escapes! Bomb! submarineJust a relevant pen&paper model ==> big improv.by changing the depth of explosion.
  29. 29. Overview1) Basic definitions2) Illustrative example: UK in WWII3) Categories of Op. Research (OR)Modelization is crucial – nothing workswithout a good model.But often we need maths&algorithms:optimization, statistics, Zermelo-like algos,... a) based on optimization b) based on statistics c) based on time-decomposition d) others e) all-in-one
  30. 30. Optimizing plane is important; turbulence matters Real-worldSimulations
  31. 31. Optimizing aerodynamics is importantBetteraerodynamics==> less fuel.Turbulencesgenerated byplanesat takeoff/landingcan be dangerous.(or even en-route) Feedback from experience: often crucial
  32. 32. Optimization-based ORPlenty of parameters in a plane:length of wings, width of wings,position of gas turbines, etc.Lets say x1, x2, … , xk.If you have a flight simulator, you can define a functionf(x1,x2,.;.,xk) = efficiency (aerodynamics)Or for short: f(x) = efficiencyThen, it is important to find x* such that f(x*) is minimum.Also important for- trains- hard drives- automobiles
  33. 33. Optimization: definitionsIt is important to find x* such thatf(x*) is minimum.Definition: such a x* is termed a “minimum” of f. It is an “optimum” (the best solution).Minimization: optimum = minimumMaximization: optimum = maximum
  34. 34. QuestionsMinimum of: f(x) = x2 2 f(x) = (x-1) 2 2 f(x)= ( (x-1) -1) f(x)= -x2 f(x)= -x2 for x >0Final question: how to find the maximum of f if you just have a program for finding minima ?
  35. 35. Minimization: the simplestalgorithm everdouble * RandomSearch( double (*f)(double* x) ){ bestValue=-MAXDOUBLE; for ( 10000 times ) { x=randomVector() value = f(x) If (value<bestValue) { bestValue=value; xstar=x;} } return xstar;}
  36. 36. Minimization: the simplestalgorithm ever, random searchI sample plenty of points. Here!I evaluate the value for each point.I keep the best one.
  37. 37. Minimization: the simplestalgorithm ever, random searchRandomSearch( f ){ bestValue=-MAXDOUBLE; for ( 10000 times ) { x=randomVector() value = f(x) If (value<bestValue) { bestValue=value; xstar=x;} } return xstar;}
  38. 38. Sorry, we must stop, we need a tool: Gaussian random variables.Just a brief overview:standard Gaussianvectors (normalized)= random vectors - distributed around 0, - with density decreasing with the distance, - equally in all directions.
  39. 39. Sorry, we must stop, we need a tool: Gaussian random variables.Just a brief overview:standard Gaussianvectors (normalized)= random vectors - distributed around 0, - with density decreasing with the distance, - equally in all directions.
  40. 40. Minimization: almost as simple,the (1+1)-evolution strategyOnePlusOne( f ){ x=randomVector;s=1; for ( 10000 times ) { xp = x+s*randomGaussianVector() value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x;}
  41. 41. Minimization: almost as simple,the (1+1)-evolution strategyOnePlusOne( f ){ x=randomVector;s=1; for ( 10000 times ) { xp = x+s*randomGaussianVector() value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x;}
  42. 42. The (1+1)-evolution strategy:should I stay or should I go ?OnePlusOne( f ){I start x=randomVector;s=1; here.( 10000 times ) for { Xp = x+s*randomGaussianVector() value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x;}
  43. 43. I should go!OnePlusOne( f ){ x=randomVector;s=1; for (I10000 times ) test { here: better! Xp = x+s*randomGaussianVector() value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x;}
  44. 44. ...OnePlusOne( f ){ x=randomVector;s=1; forSo here times ) ( 10000 { I go. Xp = x+s*randomGaussianVector() value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x;}
  45. 45. … we can not always succeed .. ...andOnePlusOne( f ) here{ x=randomVector;s=1; fortest. I ( 10000 times ) { Xp = x+s*randomGaussianVector() value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x;}
  46. 46. Should I stay or should I go now ?OnePlusOne( f ){ x=randomVector;s=1; for ( 10000 times ) { Xp = x+s*randomGaussianVector() ...I stay and I test f(x) value = if (value<bestValue) again { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x;}
  47. 47. Should I stay or should I go now ? OnePlusOne( f ) { x=randomVector;s=1; ... for ( 10000 times ) {and I try again Xp = x+s*randomGaussianVector() ... value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x; }
  48. 48. Should I stay or should I go now ?OnePlusOne( f ){ x=randomVector;s=1; for ( 10000 times ) { Xp = x+s*randomGaussianVector() value = f(x) ... and(value<bestValue) if again { bestValue=value; x=xp;s=2*s} else s=0.84*s; ... } return x;}
  49. 49. Should I stay or should I go now ?OnePlusOne( f ) ...{ x=randomVector;s=1; and for ( 10000 times ) { again ... Xp = x+s*randomGaussianVector() value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x;}
  50. 50. Should I stay or should I go now ?OnePlusOne( f ){ x=randomVector;s=1; for ( 10000 times ) { ... Xp = x+s*randomGaussianVector() value = f(x) Found! if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x;}
  51. 51. Should I stay or should I go now ?OnePlusOne( f ){ x=randomVector;s=1; for ( 10000 times ) { Xp = x+s*randomGaussianVector() value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x;}
  52. 52. Minimization: almost as simple,the (1+1)-evolution strategyOnePlusOne( f ) There is a{ x=randomVector;s=1; for the size s parameter of for ( 10000 times ) modifications { Xp = x+s*randomGaussianVector() value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x;}
  53. 53. Minimization: almost as simple,the (1+1)-evolution strategyOnePlusOne( f ){ x=randomVector;s=1; for ( 10000 times ) { Xp = x+s*randomGaussianVector() value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x; If success, then I increase the size of} modifications!
  54. 54. Minimization: almost as simple,the (1+1)-evolution strategyOnePlusOne( f ){ x=randomVector;s=1; for ( 10000 times ) { Xp = x+s*randomGaussianVector() value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } Otherwise, I return x; decrease the size of} modifications!
  55. 55. Minimization: almost as simple,the (1+1)-evolution strategyOnePlusOne( f ){ x=randomVector;s=1; for ( 10000 times ) { Xp = x+s*randomGaussianVector() value = f(x) if (value<bestValue) { bestValue=value; x=xp;s=2*s} else s=0.84*s; } return x; There are plenty of variants of this algorithm.} These variants are evolution strategies.
  56. 56. Other applicationI have many shops.They sell products from my factory.Bringing products factory → shops is hard.I want to install 7 warehouses:- reduced costs for transportation- less breakdownsLets simplify: theres no breakdown.Where should be the 7 warehouses ?
  57. 57. Other applicationI have many shops.They sell products from my factory.Bringing products factory → shops is hard.I want to install 7 warehouses:- reduced costs for transportation- less breakdownsLets simplify: theres no breakdown.Where should be the 7 warehouses ?
  58. 58. Without warehouses● Either many small long-distance transportations (huge cost)● or one big transportation (big delay, big cost)
  59. 59. Warehouses● Less travels with big transportation● Total price decreased● Maximum delay reduced (==> no breakdown)● Less pollution
  60. 60. Formalizationx=position of the warehouses.f(x) = benefit when using the 7 warehouses = cost of transportation with no warehouse - cost of transportation with the warehouses==> with more sophisticated models, a very important case of operation research
  61. 61. FormalizationFor computing f(x), I need:(1) cost of transportation with no warehouse(2) cost of transportation with the warehouses==> it is a secondary optimization problem: optimizing the tactical use. Cost = minimum cost_if_policy(y)==> very usual: - one strategical level: positionning the infrastructure - one tactical level: optimal use of the Infrastructure (policy)
  62. 62. Sometimes many levelsLevel 1: size of the infrastructure- number/size of factories- number/size of trucks- number/size of warehouses- number/size of trains, boats (…)Level 2: positionning- where is factory A- where is factory B- where is warehouse C (…)Level 3: scheduling of supply chain- when/where does truck X start- when/where does train Y start (…)
  63. 63. Logistics: combining all this for fast supply chain. Not only positionning, for a fixed number of trucks, men, boats. You can also see which number of boats, trucks, warehouses, is best.
  64. 64. Overview1) Basic definitions2) Illustrative example: UK in WWII3) Categories of Op. Research (OR) a) based on optimization b) based on statistics c) based on time-decomposition d) others e) all-in-one
  65. 65. Statistics in a (small) nutshellSample = some examples= e.g. x1, x2, x3,...., xn.- You are a sample of Taiwanese students;- I am a (small) sample of French researchers.Sample average m = (x1+...+xn)/nStandard deviation:
  66. 66. Statistics crucial notion 1: confidence intervalWith probability 95%, the true mean isbetween m - 2/sqrt(n) and m + 2/sqrt(n).Under some conditions: - no bias in the sample - no correlation in the sample Sorry for statisticians, I know I simplify so much...
  67. 67. Statistics crucial notion 1: 2: confidence interval biasAmerican election in 1936(during great depression).Who will be elected, Landon or Roosevelt ?Literary Digests poll: sample = 2.3 millions ==> predicts Landon. Sample=people motivated for answering the poll = minority which hates Roosevelt. ==> they all vote for Landon. Big bias.Gallup poll: sample = 50 000 persons (more properly sampled). Predicted the right answer.
  68. 68. Example of application: industrial owens
  69. 69. StatisticsSteel quality(for my factory) Carbon quantity
  70. 70. Statistics crucial notion 3: conditional probabilitiesProbability for a dice ?P(3) = 1/6P(odd) = P(even) = 1/2P(1 or 2) = 1/3P(1 | 1,2,3,or 4 ) = ¼ <== this is conditioning = probability of 1, given that we get 1, 2, 3 or 4.
  71. 71. Statistics crucial notion 4: Bayes theoremBayes published a theological book: Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures (1731)But he also developped a special case ofBayes theorem:P(A | B) = P(A and B) / P(B)P(3 | 1,2 or 3) = (1/6) / (½) = 1/3
  72. 72. Lets see a nice, counter-intuitiveexample of application ofstatistics:==> protecting bombers
  73. 73. Lets come back to anti-aircraft artillery.UK bombers also attacked Germany. Bombing with (HamburgBut, many anti-bombers artillery. incendiary bombs; 1000°C, ~190km/hMany planes destroyed. Wind – death toll 42600)Problem: how to protect planes ?
  74. 74. Using statistics for protecting bombers 1) For each part X of the plane, estimate q(X)= P( part X damaged | plane came back)E.g.q(part1) = 0.1q(part2) = 0.0q(part3) = 0.02 2) Assume that P(part X damaged) = constant (does not depend on X) 3) Then, which part should we reinforce ?
  75. 75. Using statistics for protecting bombers q(X)= P( part X damaged | plane came back) = P ( PXD | PCB)E.g. q(part1) = 0.1 q(part2) = 0.0 q(part3) = 0.02 Define c = P(PCB)/P(PXD) = constant.P(PXD | PCB) = P(PXD and PCB) / P( PCB)P(PCB | PXD) = P(PCD and PXD) / P( PXD) = P(PXD | PCB) * P(PCB)/P(PXD) = c * q(X)
  76. 76. Using statistics for protecting bombers q(X)= P( part X damaged | plane came back) = P ( PXD | PCB)E.g. q(part1) = 0.1 q(part2) = 0.0 q(part3) = 0.02 Define c = P(PCB)/P(PXD) = constant. P(PCB | PXD) = c * q(X)==> Probability that plane survives to “part X is damaged” increases if q(x) increase.==> Conclusion: reinforce part2 ! ! !
  77. 77. Using statistics for sending mailsDivide customers in 5 categories- Women ( < 25 y.o.)- Men (<25 y.o.)- Women (25-45)- Men (25-45)- >45You have money for sending Q advertisements.You have P1, P2, P3, P4, P5 customersin categories above.
  78. 78. Using statistics for sending mailsYou have P1, P2, P3, P4, P5 customersin categories above.Data: Sample of past advertisements: m1=frequency of positive answers in category 1. m2=frequency of positive answers in category 2. …==> Who should receive advertisements ?
  79. 79. Using statistics for sending mails m1=frequency of positive answers in category 1 = number of positive answers / number of adse.g. if 17 ads sent, m1=mean(0,1,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0)=5/17
  80. 80. Using statistics for sending mailsm1=frequency of positive answers in category 1.m2=frequency of positive answers in category 2. …1 = std deviation for category 12 = std deviation for category 2 …b=benefit per customer with positive answer. If ( (m1+21/sqrt(n1)) x b < price of one mailing), then discard category 1. Send ads to non-discarded categories.
  81. 81. Overview1) Basic definitions2) Illustrative example: UK in WWII3) Categories of Op. Research (OR) a) based on optimization b) based on statistics c) based on time-decomposition: Zermelo (remember, irrigation, water, crops, etc) d) others e) all-in-one
  82. 82. Our river in Paris is the “Seine”. A French politician said he would soon swim across it.After all, he never did it. For your health, dont do it. Nevertheless, we try to keep it as clean as possible.
  83. 83. Another beautiful applicationThis is Paris.Beautiful town.With plenty of people(10 millions in IDF).
  84. 84. Another beautiful applicationThis is Paris.Beautiful town.With plenty of people(10 millions in IDF).Producing plenty of fecalmatter ==> dirty water.
  85. 85. Dirty water should be separated from the Seine.And usually it is.Something like this: Seine Dirty water
  86. 86. Problem: if big rainfalls reach dirty water, then dirty water might pollute the Seine SeineDirtywater
  87. 87. No typhoon in France.But we can have heavy rains/winds in Paris:- 0.96 dm in 24 hours happened in 1987.- gusts at 169 km/h in 1999 (very unusual in France) Problem: if big rainfalls reach dirty water, then dirty water might pollute the Seine Seine Dirty water (yes, in Taiwan it is more impressive, sometimes it is 16.7 dm in 24 hours and gusts can reach 250 km/h...)
  88. 88. No typhoon in France.But we can have heavy rains/winds in Paris:- 0.96 dm in 24 hours happened in 1987.- gusts at 169 km/h in 1999 (very unusual in France) Problem: if big rainfalls reach dirty water, then dirty water might pollute the Seine Seine Dirty water (yes, in Taiwan it is more impressive, sometimes it is 16.7 dm in 24 hours and gusts can reach 250 km/h...)
  89. 89. No typhoon in France.But we can have heavy rains/winds in Paris:- 0.96 dm in 24 hours happened in 1987.- gusts at 169 km/h in 1999 (very unusual in France) Problem: if big rainfalls reach dirty water, then dirty water might pollute the Seine Seine Dirty water → Seine! (yes, in Taiwan it is more impressive, sometimes it is 16.7 dm in 24 hours and gusts can reach 250 km/h...)
  90. 90. Another beautiful applicationThree water networks:- dirty water: should go to cleaning stations- clean water: can go to the Seine, but cant be drunk- drinkable water (France: tap water = drinkable)
  91. 91. Big water networkDirty Dirty Dirty Dirtywater water water waterClean Clean Clean Cleanwater water water water
  92. 92. Water vs dirty waterChallenge:Summer storms.Not comparable to a Taiwanese typhoon.But a lot of water.Can make dirty water become very big.Can invade clean water.Your mission:- Get read of dirty water- Protect clean water
  93. 93. Water vs dirty waterState: level in each stock, valves status (open or closed)At each time step, i(x) liters of water reach stock x. you can open or close valves ==> get a new state.Your mission: - Get read of dirty water - Protect clean water ===> Zermelo.
  94. 94. Water vs dirty water Typically:(0, 1, 0, 0, 0, 1, 0, 1, 0.42, 0.2, 0.0, 0.8, 0.3) (valves) (stock levels)Plenty of rules:- if (valve 4 opens, then water from stock 1 3 goes to stock 2 at rate 0.02m /s)- if (stock[2]>0.3) then dirty water ==> Seine, 3 0.1m /s==> Maximize your number of points at the end of the storm
  95. 95. Shrinking horizonToo many time steps!At each time step, make a decisionusing only 30 time steps.Example of heuristic function:zermeloValue(state)=the margin before any stock can go to the Seine.
  96. 96. Shrinking horizonToo many time steps!At each time step, make a decisionusing only 30 time steps.Example of heuristic function:zermeloValue(state)=the margin before any stock can go to the Seine.
  97. 97. Shrinking horizonToo many time steps!At each time step, make a decisionusing only 30 time steps.Example of heuristic function:zermeloValue(state)=the margin before any stock can go to the Seine.
  98. 98. Shrinking horizonToo many time steps!At each time step, make a decisionusing only 30 time steps.Example of heuristic function:zermeloValue(state)=the margin before any stock can go to the Seine.
  99. 99. Shrinking horizonmoving window of 30 time steps
  100. 100. Overview1) Basic definitions2) Illustrative example: UK in WWII3) Categories of Op. Research (OR) a) based on optimization b) based on statistics c) based on time-decomposition d) others e) all-in-one
  101. 101. Overview1) Basic definitions2) Illustrative example: UK in WWII3) Categories of Op. Research (OR) a) based on optimization b) based on statistics c) based on time-decomposition d) others e) all-in-one

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