What is Monte Carlo Simulation? Monte Carlo Simulation

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1 Wht is Monte Crlo Simultion? Monte Crlo methods re widely used clss of computtionl lgorithms for simulting the ehvior of vrious physicl nd mthemticl systems, nd for other computtions. Monte Crlo lgorithm is often used to find solutions to mthemticl numericl Monte Crlo method l prolems which my hve mny vriles tht cnnot esily e solved, e.g. integrl clculus, or other numericl methods Monte Crlo Simultion A scheme employing rndom numers which is used to solve certin stochstic or deterministic prolems where the pssge of time plys no sustntive role. Common prolem is estimtion of where f is function, is vector nd Ω is domin of integrtion. Specil cse: Estimte nd limits of integrtion, f d Ω f d, for sclr 2 1

2 2 3 Monte Crlo Simultion Let X e uniform rndom vrile on the intervl [, ] with density nd let 1,, n e rndom smple from X. Then = = = = n i i f n X f E d p f d p p f d f 1. ] [ p =, 1 4 Monte Crlo Simultion Emple: Estimte We pproimte this y where 1,, n re smple from uniform [0, ] rndom vrile. d 0. sin, sin 1 = n i i n

3 Monte Crlo Simultion Emple: Estimte 0 sin d. n = 10 n = 100 n = 1000 n = 2000 = nswer = = 2 nswer = There is considerle vriility in the qulity of solution; ccurcy of numericl integrtion sensitive to integrnd nd domin of integrtion 5 Cse Study Cke s shop prolem An owner of kery shop would like to determine how mny 10- inch irthdy ckes he should produce ech dy in order to mimize his profit. His present method of determining the quntity to ke is sed on his est guess. 3

4 Cke s shop prolem The production costs re $2.00 per cke. And the profit for ech cke is $2.5. However, If over estimtes the dily demnd, some ckes will e left over t the end of the dy. The policy is to sell ll leftover ckes to locl store tht specilizes in dy-old items. He is currently receiving $1.50 per cke for the surplus ckes, thus incurring loss of $0.50 per cke. Cke s shop prolem Cse 1: The production quntity is less thn or equl to demnd If <= d, z = 2.5 Cse 1: The production quntity is greter thn the demnd If > d z = 2.5d + -d -0.5 Z = 3.00 d

5 Cke s shop prolem Generlizeing: p = selling price for ech cke c = cost of ech unit s = dy-old price If <= d, z = p-c If > d z = p-c d + -d s-c Z = p-s d s-c Historicl dy demnd for the irthdy ckes Dily demn Frequency oreltive freq Totl 20 1 frequency _ of _ oservtion reltive _ frequency = totl _ numer _ of _ oservtions 5

6 Hnd Simultion Tke sheet of pper nd cut it into twenty equl pieces. Follow the historicl dily demnd frequency in the tle, write the numer zero on one piece. On two of the remining pieces write the numer one, which stnds for the demnd of one unit. Check the numers you hve written crefully, ecuse this deck of twenty Hnd Simultion The first step is the selection of the production quntity, Assume =3. use the deck of twenty slips of pper to generte demnd y selecting one slip of pper t rndom. Suppose the first slip drwn hs 5 written on it. We shll then use demnd of 5 ckes for the first simulted dy of kery shop opertion. i.e. underproduction of 2 ckes. 6

7 Hnd Simultion Since <d, we cn computer our first dy s profit using the epression 2.5=253=$7.5. i.e. Totl profit of $7.5. Generte the demnd for second dy reshuffle nd drw piece suppose d=1 Since >d use the second cse z= = $1.5 So the totl profit is = 9 10-dy simultion results for production quntity =3 Dy Generted demnd Dily profit Totl profit

8 Hnd Simultion Now we perform the sme ten dy simultion for nother quntity production =1,2,3 8 Compre the totl profit for ech one Pick the est profit to e the suggested production quntity Of course if we run the simultion for more dys we get more ccurte estimte. 10-dys Simultion Results for vrious production quntities Production Size Ten Dy Simulted profit $ From the tle it is cler tht the est production quntity tht mimizes the profit is t =6. The results re sed on only 10-dy simultion. 8

9 The role of rndom numers in simultion Suppose we select rndom numers in sets of two digits. This will provide us with 100 two-digit rndom numers from 00 to 99 with ech two-digit rndom numer hving 1/100 chnce of eing selected 0 unites, the reltive frequency of 0 is 5% Thus we wnt 5% of the 100 possile two-digit rndom numers to correspond to demnd of 0 units. While choosing ny five numers of the 100 numers will do we my ssign demnd of 0 to the first 5 numers i.e. 00, 01,02,03, nd 04 Rndom numer Intervls nd the dily demnd Dily Dem reltive FrequencIntervl of Rndom num to to to to to to to to to 99 9

10 Results of simulting ten dily demnds Rndom numer Simulted dily demnd simulted dily demnd The role of rndom numers For ny simultion prolem in which reltive frequency distriution of vrile cn e developed, It is esy to pply the ove rndom numer sed procedure to simulte vlues of the vrile. First, develop tle of intervls y ssociting n intervl of rndom numers with ech possile vlue of the vrile Then s ech rndom numer is selected, you cn simply check the corresponding intervl nd find the ssocited vlue of the vrile. 10

11 The role of rndom numers Oviously, for long nd comple simultions tht require numerous clcultions, high speed computer simultion process is desirle. In computer simultion pseudo- rndom numers re used in ectly the sme wy s the rndom numers selected from rndom numer tles ove. It would e very risky to mke decision sed on the results of such short period of simultion. The role of rndom numers When we think of performing the simultion clcultions for simulted period s long s 500 dys, the prolems of crrying out the simultion for even cse s smll s the kery shop prolem re significnt. For emple let us consider the 500 dys. The mthemticl model does not chnge ut the work we hve to go through to evlute the results does chnge ut epnds. Now we cn crete tle similr to the ten-dy tle to evlute ech order size for 500 dys of opertion. 11