DSC3703-2013-10-E-1.pdf - UNIVERSITY EXAMINATIONS U N I SA...

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Unformatted text preview: UNIVERSITY EXAMINATIONS UNIVERSITEITSEKSAMENS U N I SA 3mm DSC3703 OctoberfNovember 2013 SIMULATION Duration 2 Hours 80 Marks EXAMINERS ' FIRST DR S MUKERU SECOND MS J LE ROUX EXTERNAL DR A DE VILLIERS Programmable pocket calculator is permissible. Closed book examination. This examination question paper remains the property of the University of South Africa and may not be removed from the examination venue. This paper consists of 5 pages plus a list of formulae (pp. i — ii). ANSWER ALL THE QUESTIONS Question 1 The forty (40) numbers U1, U2, , U40 (on Appendix A) are aSSumed to follow a U[O; 1) distribution But as an operations research analyst, it is important that you check whether they pass some statistical tests before usmg them Assume that you want to perform a chi-square goodness-of—fit test to check whether these numbers are really from a U[0, 1) distribution (a) Usmg ten (10) sub—intervals, calculate the corresponding chi-Square test value X2 [7] (b) Can you reject the null hypothesrs that the numbers are indeed from a U [0, 1) distribution at the 5% Significance level? Justify your answer [3] [TURN OVER] DSC3703 2 Oct/ Nov 2013 Questions 2, 3, 4 and 5 are based on the following problem (Problem 1). PROBLEM 1. One of the oldest and most successful practlces of revenue management 1n the airline industry has been to do overbookmg (prov1d1ng more reservatlons than the number of seats avallable on a flight, to allow for the cons1derable number of no—shows that usually occur) Therefore, a large amount of additional revenue can be obtalned by domg a Sigmficant amount of overbooklng. However, the penaltles have become substant1al for denying admlssmn to a flight for someone Wlth a reservation, so careful analys1s must be done to achieve an approprlate trade- ofi' between the addltlonal revenue from overbooking and the risk of incurrlng these penalties. Transco Alrhnes 15 an airline company It has a daily fl1ght from City A to Clty B that 13 mainly used by busmess travellers There are 150 seats available 1n the smgle cabln The average fare per seat 18 R6 000. This is a nonrefundable fare, so no—shows (absentees) forfeit the entire fare When a passenger IS bumped from the fllght (that IS, he 18 demed admissmn to the flight because the flight IS full due to overbookmg), Transco Alrhnw arranges to put the passenger on the next available filght on another airline and gives the passenger a voucher for use on a future fhght and an addrtlonal amount for the intangible cost of a loss of goodwill on the part of the bumped passenger The total cost of bumping a passenger lS denoted by TC Based on previous experience, the operations research analyst of the company has estimated that the relative frequency of the number of no~shows (independent of the exact number of reservations) IS as shown below Number of no—shows Relative frequency 0 1 2 3 4 5 6 7 8 9 Question 2 [E] (This question is based on Problem 1 ) The company wants to determme how much overbookmg should be made when such an oppor- tumty 15 available. For a partlcular flight, denote by Q the number of overbooklngs made, by NS the number of no—shows and by T0, the total cost of bumping a passenger Explain why the nett profit of the company (Profit) IS given by Profzt = 6000(150 + Q) — TCmax{0, Q — NS} [TURN OVER] DSC3703 3 Oct / Nov 2013 Question 3 (This question. as based on. Problem 1 ) Usmg the random numbers U1, U2, U3 (from Appendlx A) generate three values of NS (number of no—shows) and find the corresponding company profit for both values Q m 4 and Q = 6 when the total cost of bumpmg a passenger IS R20 000 (that rs T0 = 20 000) Accordmg to the average profit for the two values of Q, Wthh number of over—bookings should you advise the company to make? Present your answers 1n the form of the followmg table: Corresponding NS Profit (for Q = 4) Profit (for Q = 6) Question 4 (This question. 25 based on Problem 1 ) After carefully consrdermg the cost of loss of goodw111 on the part of a bumped passenger and all the arrangements that the company has to make (w1th other airlines) to accommodate the bumped passenger, 1t IS now estlmated that the total cost (T0) of bumplng a passenger is a random variable of which the cumulative dlStl‘lbutIOD functlon (cdf) F(cc) IS glven by 56% 1f 05x<10000 my): 563% 1f 10000£x<20000 1 1f $220000 (a) Assuming that you only have moms to U[0, 1) random numbers, glve an algorithm that can be used to generate values for the varlable TC. [6] (b) Us1ng random numbers U4, U5, U5 (from Appendix A), generate three (3) values for TC Show all your calculations. Fmd the new correspondmg values of the profit when NS = 2 and Q = 6 [6] Question 5 [E (Thts questwn 15 based on Pmblem 1 ) A srmulat1on model for the system 18 run for 900000 rephcatrons for the value Q = 6. The computer output shows that the average profit is R919 103,00 wrth a standard devration of 27 307,00 Find the corresponding 95% confidence interval for the profit [TURN OVER] DSC3703 4 Oct / Nov 2013 Questions 6, 7, 8 and 9 are based on the following problem (Problem 2) PROBLEM 2. Mr Jerry IS the productlon manager of a manufacturlng company that pro— cesses metal sheets (as raw materlal) to produce wmg sectlons of planes Metal sheets arrive randomly at the productlon statlon of the company and the mterarrlval times are estlmated to be exponentlally dlstrlbuted Wlth rate two (2) sheets per hour (that Is, w1th a mean of half an hour (0,5 hours)) The production statlon has two (2) 1dent1cal machlnes to do the Job Upon arr1v1ng at the production statlon, the sheet wdl be processed by one of the two machines to form a mug sectlon but the sheet W111 Jom a queue if both machlnes are busy The tlme requlred by a machme to form a Wlng sectlon out of the sheet (the productlon servme tlme) has an exponential d1str1but10n wrth a mean of one ( 1) hour Question 6 (The; questzon as based on Problem 2 ) Grve an overall flowchart that can be used to S1mulate thlS M /M /2 model Define any variables that you may use Question 7 (Tins question ts based on Problem 2 ) Using the random numbers U1, U2, U3, U4 (from Appendix A) sequentlally, generate the first five arrlval times of sheets at the productlon sectlon (assume that the first arrlval occurs at time 0) Use also,U5, U5, U7, U3, U9 (from Appendlx A) to generate the corresponding five production semce tunes Usmg only these five (5) arnvals, determine the average total time (1n hours) a metal sheet spends at the production stat1on You should present your answers 1n the form of Table 1 Table 1 —n —- —n —- —- Question 8 (Thzs questwn 35 based on Problem 2.) After productlon each w1ng sectlon 1s dlrectly transferred to an 1nspect10n statlon where 1t must be tested to make sure 1t meets spemfications There 1s only one server at the 1nspect10n station She has a full-tune Job of inspecting these wmg sections to make sure they meet specnfications The insPectlon tune (1n hours) has a Welbull dlstnbutlon We1bu11(o:, L?) where a = 2 and fl 2 0,3. [TURN OVER] DSC3703 5 Oct / Nov 2013 (a) Assunnng that you only have access to (ID), 1) random numbers, give an algorithm that should be used to generate inspection times [4] (b) Apply your algorithm on the random numbers U10, U11, U12 (from Appendlx A) to gen- erate the inspection times of the first three (3) wmg sect1ons arr1v1ng at the mspectmn station [3] Question 9 (Tins questzon 23 based on Problem 2 ) Assume that, after productlon, each w1ng section must be tested to make sure that 1t meets specificatlons After the test, each Wlng section 1s labelled either “good” or “bad” according to whether it passes the test or not Based on hlstoncal data of the company, 1t is estnnated that 10% of all the Wing sections fell the test and must be transferred back to the product1on stat1on Assume that you are bufldlflg a Simulation model for this system (a) Wthh probability distribution should you use to model the state of each wmg section being tested? [2] (b) Give an algorithm that can be used to label a wing section (as either good or bad). [3] (c) Assume that the 40 random numbers U1, U2, . . , U40 in appendix A are sequentially used to label the first 40 wing sections after inspection How many wing sectlons w111be labelled as bad? Is this number equal to What you should expect‘P Justify your answer [3] TOTAL: 80 © UNISA 2013 Random numbers 1 APPENDIX A Formulas and numbers DSC3703 Oct/Nov 2013 The following sequence of 40 mndom numbers was generated sequentwlly from a U[0, 1) dzstn- 3 4 5 6 7 8 9 1—; D button. U1 = 0,46 U9 = 0,35 U17 =-' 0,81 U25 = 0,69 U33 = 0,78 U2 = 0,71 U10 = 0,19 U13 = 0,63 U26 = 0,68 U34 = 0,92 U3 = 0,97 U11 = 0,70 U19 = 0,87 U27 = 0,60 U35 = 0,19 U4 = 0,80 U12 = 0,12 U20 2 0,56 U28 = 0,20 U35 = 0,55 U5 “-1 0,57 U13 = 0,85 U21 3 0,96 U29 = 0,25 U37 = 0,85 Us = 0,73 U14 2 0,56 U22 = 0,12 U30 = 0,19 U38 = 0,73 U7 2 0,05 U15 = 0,42 U23 3 0,80 U31 = 0,83 U39 = 0,42 U3 : 0,43 U15 = 0,73 U24 = 0,33 U32 = 0,08 U40 = 0,54 The chi-square test value k k n 2 _ _ _ _ 2 3:1 Percentile points of the chi-square distribution Degrees of freedom (d f ) a 1/ 0,95 0,90 0,10 0,05 The Kolmogorov—Smirnov test values ._ _1 = + _ D —llggl[Fx(m{,)) n] D max(D ,D ) T = 13M? + 0,12 + 0,11/,/E) Percentile points for T: L138 DSC3703 11 Oct / Nov 2013 Cumulative distribution functions 1 — (”J/3) ‘f > 0 Exponential distribution F(:i:) = e l :1: 0 otherWise 1 _ dab/:3)“ f 0 Weibull distribution F(.’E) = e l I > 0 otherw1se 0 if a: < a Uniform distribution F(:i:) = E if a S :1: S b 1 1f :1: > b 0 if a: < a, 22—0 2 f < < Triangular distribution F(a:) = b‘“ :33 2 1 a “h x _ c (9-9 0H2) if c < a: S b 1 if a: > 13 Confidence intervals and the t-distribution A 100(1 — a)% confidence interval for E(X) is given by _ 52 X i t(a/2,n—1) g where n n 2 — _ X1 2 (X; _ X) X _ 2 g S _ Z n — 1 1:] i=1 Percentile points of the t-distribution Degrees of freedom (d.f ) a V 0,1 0,05 0,025 1 3,078 6,314 12,706 31,821 2 1,886 2,920 4,303 6,965 3 1,638 2,353 3,182 4,541 4 1,533 2,132 2,776 3,747 5 1,476 2,015 2,571 3,365 6 1,440 1,943 2,447 3,143 7 1,415 1,895 2,365 2,998 8 1,397 1,860 2,306 2,896 9 1,383 1,833 2,262 2,821 10 1,372 1,812 2,223 2,764 Central limit theorem The sum of n independent and identically distributed random variables, each With mean p. and finite variance «:2, is apprwumately normally distributed With mean up and variance n02 ...
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