c. Explain the basic Newsvendor model. Why is it useful for SC models? e. What additional research do you believe will be helpful in this area?

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Baldric Jacobs
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1 1. Research Methodology a. What is meat by the supply chai (SC) coordiatio problem ad does it apply to all types of SC s? Does the Bullwhip effect relate to all types of SC s? Also does it relate to SC coordiatio? b. Defie what is meat by SC icetive cotracts. I what situatios do icetive cotracts play a role? Describe the pricipal kids of icetive cotracts. Describe two research results o the performace of icetive cotracts. c. Explai the basic Newsvedor model. Why is it useful for SC models? d. Describe Shapley Value ad explai how it may be of use i SC coordiatio. What is meat by the core of a cooperative N-perso game? What properties of a game solutio does the Shapley Value solutio provide? e. What additioal research do you believe will be helpful i this area?

2 2. Stochastic Models Part A. The Newsboy Problem. Cosider the famous Newsboy Problem or the Sigle Period Ivetory problem. Let D be the demad for that oe period with pdf f(x) ad cdf F(x), ad be the order quatity (the decisio variable). For coveiece, let s assume D to be a cotiuous radom variable. Also defie the followigs: C o = the overage cost (cost per uit of positive ivetory remaiig at the ed of the period). C u = the uderage cost (cost per uit of usatisfied demad or cost per uit of egative edig Ivetory. [-D] + = max {-D, 0} = D if (D ) ad 0 otherwise. [-D] - = max {D-, 0} =D if (D ) ad 0 otherwise. Notice that [-D] + ad [-D] - are the positive ad egative remaiig ivetory at the ed of the period, respectively. Now we ca defie the expected value of the total cost (TC) to be E [TC()] = C o E[[-D] + ] + C u E[[-D] - ] (1) uestio 1: Derive the mathematical expressio for E [TC()] ad fid the optimal order quatity *. Show the ecessary ad sufficiet coditios for *. Part B. Re-iterpretatio of the meaig C o ad C u usig the stadard ivetory model parameters. Defie S = Sellig price per uit c = Variable cost per uit h = holdig cost per uit of ivetory remaiig i stock at the ed of the period. p = shortage cost per uit charge agaist the umber of back orders at the ed of the period. µ = E(D) the expected value of the demad per period. Now we ca defie a more geeral cost fuctio for oe period as: TC() = purchase cost + holdig cost + shortage cost - Reveue = c h[ D] p[ D] S mi{, D} (2)

3 uestio 2: Derive the mathematical expressio for E [TC()]of (2) ad fid the optimal order quatity * i terms of h, p, c, S. Show the ecessary ad sufficiet coditios for *. Part C. Extesio to back-order ifiite horizo problem (multi-period problem) Let D 1, D 2, D 3, be the ifiite sequece of demads i periods 1, 2, 3,. Assume D 1, D 2, D 3, to be iid radom variables with pdf f(x) ad cdf F(x). Notice that Number of uits sold i period 1 = mi {, D 1 }, Number of uits sold i period 2 = max {D 1 -, 0} + mi {, D 2 }, ( uits back orders + uits sold i period 2) Number of uits sold i period 3 = max {D 2 -, 0} + mi {, D 3 }, etc. uestio 3: Let TC () be the total cost over periods. Usig the cost parameters i part B, show that where E[TC ()] = c c S) E( D D... D ) S E{mi(, D )} L( ) ( L ( ) h ( x) f ( x) dx p ( x ) f ( x) dx. 0 uestio 4: Show that the average cost for ifiite period problem, TC ( ) E[TC(]= lim ( c S) L( ) (3) Ad hece the optimal order quatity is give by F( ) p p h (4) It is iterestig to ote that * oly depeds o p ad h. p The right had side of (4) is kow as the CRITICAL RATIO: CR p h

4 3. Supply Chai Maagemet The questio cosists of two parts aswer both parts; part 1 is weighted 60%; part 2 is weighted 40%. Part 1: Recet research by Guiffrida et al. (2007), Garg et al. (2006) ad Guiffrida ad Nagi (2006) all model delivery time performace of a sigle product to the fial customer i a serial supply chai. A commo limitatio foud i the model formulatios preseted i each of these papers is the assumptio of a make-to-order orietatio withi the serial supply chai. Clearly, ot all serial supply chais operate i a make-to-order orietatio. Hece, a geeralized model for evaluatig delivery time performace to the fial customer i a serial supply chai should accommodate both make-to-order ad make-to-stock orietatios. Cosider the followig serial supply chai: Stage Stage 2 Stage 1 Fial Customer Let the total delivery time to the fial customer (W) be defied as W N X i i1 where X i is the processig time for the i th stage of a N stage serial supply chai. I the make-to-order orietatio foud i the above cited literature the fial customer places a demad o Stage 1 of the chai which i tur places a demad o Stage 2 of the chai,, which i tur ultimately places a demad o Stage N of the chai. I a make-to-stock orietatio, ivetory held at a upstream stage of the supply chai may exist ad thus ot all stages of the supply chai are required to satisfy the customer order. Always startig with Stage 1, the umber of stages required to complete the customer order (ad hece the total delivery P N p for 1,2,... By defiig W as a radom sum of time) is a radom variable where radom variables the delivery performace models foud i Guiffrida et al. (2007), Garg et al. (2006) ad Guiffrida ad Nagi (2006) ca be geeralized to accommodate both make-to-order ad make-tostock orietatios. P N p for 1,2,... Derive the expected value ad variace of total delivery time W whe (Assume idepedece amog the X ad assume that there is o waitig time betwee stages). Part 2: Let f W w determie a approximate form for i defie the probability desity fuctio for W. Discuss how you would f W w uder the coditios described i Part 1.

6 4. Simulatio Variace reductio techiques have bee called the free luch, as they allow us to get more efficiet solutios with the same effort. The questios below all have to do with variace reductio, ad rely o the followig situatio: Ships arrive at a harbor with iterarrival times that are IID expoetial radom variables. The harbor has a dock with two berths ad two craes for uloadig the ships; ships arrivig whe both berths are occupied joi a FIFO queue. The time for oe crae to uload a ship is distributed uiformly betwee 1 ad 2 days. If oly oe ship is i the harbor, both craes uload the ship ad the (remaiig) uloadig time is cut i half. Whe two ships are i the harbor, oe crae works o each ship. If both craes are uloadig oe ship whe a secod ship arrives, oe of the craes immediately begis servig the secod ship, ad the remaiig service time of the first ship is doubled. Assume that o ships are i the harbor at time 0. We are iterested i computig the miimum, maximum, ad average time the ships are i the harbor. a. Cosider usig atithetic variates (AV) for the above problem. Specifically, which iput radom variables should be geerated atithetically, ad how could proper sychroizatio be maitaied? b. Suppose that thought is beig give to replacig the two existig craes with two faster oes. Specifically, sigle-crae uloadig times for a ship would be distributed uiformly betwee 0.5 ad 1 day; everythig else remais the same. Discuss the proper applicatio ad implemetatio of commo radom umbers (CRN) to compare the origial system to the proposed system. c. Now assume that i the proposed system (part b above), the sigle-crae uloadig times follow a ormal distributio rather tha a uiform distributio. Further, we use the Polar method to geerate these ormal radom variates. Discuss the effect of chagig the uloadig time distributio from uiform to ormal o the applicatio of CRN whe comparig this system to the origial system (part a). Hit: The Polar method requires a radom umber of UN(0,1) umbers to geerate a pair of ormal radom variates. d. Briefly, commet o the pitfalls, if ay, of applyig both AV ad CRN simultaeously, i.e., AV withi a system, ad CRN across systems, for the same simulatio experimet. Be specific ad brief i your aswers to the above questios.

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber