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## Stochastic Models

by: Mafalda Ebert

19

0

5

# Stochastic Models OR 542

Mafalda Ebert
Mason
GPA 3.9

Staff

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COURSE
PROF.
Staff
TYPE
Class Notes
PAGES
5
WORDS
KARMA
25 ?

## Popular in Operations And Info Mgmt

This 5 page Class Notes was uploaded by Mafalda Ebert on Monday September 28, 2015. The Class Notes belongs to OR 542 at George Mason University taught by Staff in Fall. Since its upload, it has received 19 views. For similar materials see /class/215268/or-542-george-mason-university in Operations And Info Mgmt at George Mason University.

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Date Created: 09/28/15
CHAPTER 16 SOLUTIONS SECTION 162 1a1b and 1c D 4000 gallonsmonth h 025gallon month K 50 q 4000 gallons 025 OrdersMonth Dq 40004000 1 ordermonth 12 ordersyear There will be one month between orders 1d Demand is usually random and varies with the day of the week If this is the case then the EOQ model should not be used see Section 166 1e If Leadtime 2 weeks then Leadtime demandltEOQ and order should bei placed when 25248000 1846 gallons are in stock If Leadtime 10 weeks then Leadtime demandgtEOQ Assume that an order arrives at time39ri 0 Then an order must have been placed at T 10 We know that an order is placed every 5212 133 weeks Thus an order was placed at T 173 T 43 T 3 etc Since at T 0 we will have 4000 gallons in stock we know that at the reorder point T 3 and any other reorder point we will have 4000 35248000 1231 gallons in stock 2 K 1410 250 D 10000year h 10dollar year 225010000 2a and 2b q 70711 should be withdrawn on 10 each trip to the bank Trips 10000 1414 tripsyear Year 707 11 4 2c Since Dq hD2K12 an increase in D will increase Dq Trips to bankyear 2d This increases h which causes q to decrease This will increase the number of trips to thetbank each year 2e Increasing the number of tellers should reduce the time we spend waiting in line This reduces K which leads to a decrease in q Since q decreaseswe will go to the bank more often 3a This is like the bus problem D 30 callshour K 10 and h 12pizza hr Then 21030 q 707 and 30707 424 trucks should be 12 656 qn wksa I 39 I sent out each hour 3b Total cost as a function of q looks like Figure 4 Thus we want the lowest point on the Total Cost curve that has qs5 Since the total cost curve attains its minimum for a value of q exceeding 5 q 5 will since qu minimize the total cost Thus each truck should deliver 5 pizzas and 305 6 trucks per hour will be sent out 4a Yes A large turnover ratio indicates that the average amount of onhand inventory is relatively low This is indicative of a relatively efficient inventory system 4b Let p unit purchase cost Then Annual Cost of Goods Sold pD KD Average Value of On hand Inventory pq2 p 2h 2Dh Thus TR K 4c The answer to 4b shows that an incease in D increases TRthis is because the quantity of goods sold is proportional to Dbut q is only proportional to DVZ 5 We know that HCq OCq We also know that HC8q h28q 8HCq and OC8q KD8q 1250Cq 125HCq Thus HC8q OC8q 205HCq and HC 8q OC8q 205HCq z 1025 HCq OCq 2HCq Thus a 20 deviation on the low sidein the EOQ increases controllable costs by only 25 6 If q items are used during a cycle then an average item is in inventory for 12 of a cycle or 12qD K2Dh Thus K2Dh will be small for a faster moving item while K2Dh will be large for a slower moving item 7 K lOorder h 2bottleyear D 3052 156O bottlesyear EOQ 2lO1560 2 This order quantity would satisfy 1249030 416 weeks of demand Thus this order quantity will result in some drug spoilage Effectively we are looking for minimium point on total cost curve having q530 Since total cost curve is shaped like Figure 4 and minimum point on TC curve occurs for qgt30 we know that since qs30 minimum point on total cost curve occurs for q 30 Thus each order should be for 30 bottlesone week39s demand of the drug CHAPTER 17 SOLUTIONS SECTION 172 SOLUTIONS la q 6 lb Since E2 Ellt0 and E3 E2gt0 marginal analysis indicatestmati q 2 is optimal i 1c Marginal analysis fails in this example because Eq is not a convex function of q Thus while q 2 is a local minimum it need not minimizeq v over all values of q a SECTION 173 SOLUTIONS 1 For dsq Total Cost 10000q 15000d 9000q d 1000q 6000d For dzq 1 Total Cost 10000q 15000d 12000d q 2000q 3000d Thus c01000 and cu2000 Hence we should order q where q is the smallest value of q satisfying 2000 2 PDsq2 667 3000 3 Since PDs3060 and PDs3580 35 cars should be ordered in August 2 For dsq Total Cost 15q 30d For daq 1 Total Cost 15q 30q 15q Thus co15 and cu15 Hence we should order q papers where q is the smallest value of q satisfying Since PDs70 45 and PDs90 7090 papers should be ordered 3 Increasing co decreases cucocu Thus a smaller value of q will be the first value of q that makes PDsq exceed 39 cuco cu Thus q will decrease 4 An increase in cu will increase cucfux Thus a larger value of q will be needed to make PDsq exceed cuco cg Hence q will increase 5a For dsq Total Cost 20 q For d2ql Total Cost 20q 30d q 30d qu Thus co 20 and cu 10 Since the smallest value of q satisfying PDsq2 1020 10 333 is q 60 PDs60 35 60 cells should be ordered 5b The opportunity cost of the capital tied up in unneeded cells is being neglectedalso leftover cells might be usable in the future 6 For dsq cdq 30q and for dzq cdq 30q 54d q This yields co 30 and cu 24 Then q is smallest value of q satisfying Fq2242430 444 This yields q 450 7 q Number of books produced cdq 50000 35d 15q 5q 2000 d for dsq 2000 cdq 50000 35q 2000 15q for dzq 2001 co 10 and cu 20 Produce to smallest point q where Fq20002 202010 67 Thus produce 5000 books Expected Profit is 3535000 26000 57000 155000 50000 532000 21000 96000 Producing no books we earn profit of 200035 70000 Thus 5000 books should be produced If we have 4000 books then if we produce at all 5000 2000 3000 books should be produced and expected profit 96000 200015 126000 But now producing no books yields an expected profit of 354000 140000 so no books should be produced 8 q week you begin class and d week of birth cdq 100 5d q 5 for q55d cdq 100 20q5 d for q52d1 We want Fq525520 2 Thus we want to finish by week 37 so we begin at end of week 32 8 Let I income before any taxes or child care expenses S State taxes paid without any child care adjustment d actual child care expenses and q amount put into child care account We are maximizing income left after taxes and child care expenses Cdq I q 4I q S for dsq cdq I d 4I q S 25d q for dzq 1 Thus C 6 and cu 15 Choose q so that Fq215156 2 Thus put only 3000 in child care account SECTION 174 SOLUTIONS 1a In all likelihood each ticketed customer has a fixed probabilitysay 05 of not boarding the flight If this is the casethen the number of noshows would have a mean of 05qwhich depends on the number of tickets sold 1b Nobecause the newsboy problem assumes that fd is given and does not depend on q 2 Let dAmount of funds actually needed and qAmount of funds borrowed from bank If dsq Total Cost qu If dzq Total cost 10q 25d q 25d 15q Thus co 10 and cu 15 q dollars should be borrowed where q satisfies 15 PDsq 60 15 10 Standardizing we obtain q 700000 PZs 60 300000 q 700000 Since F2560 25 300000 CHAPTER 19 SOLUTIONS SECTION 192 SOLUTIONS 1 Letting the state for a day be the day s weather yields a two stag Markov Chain with the following transition matrix Sunny Cloudy Sunny 90 10 Cloudy 20 80 2 O l 2 3 4 O O 0 13 13 13 P l O O 13 13 13 2 13 13 13 0 0 3 O 13 13 13 0 4 O 0 l3 l3 13 3 0 working 1 working 2 working 0 working 0 O l 1 working K O 13 23 2 working 19 49 49 For example if 0 machines are working at the beginning of the currentdai then two machines must have broken down during the previous day amiatt beginning of the next day two machines will be working If two machinesa working at the beginning of a day let Wi event that machine i doemr break down during current day Then next day begins with 0 machines wo dg with probability a P mow 2 132 19 Next day begins with 1 machine working with probability PW1nW2 Pw1rTw2 213 23 49 Next day begins with 2 machines working with probability Pw1nw2 232 49 4 For any day the state is the weather pattern that occurred during today and yesterday For exampleif the state is SC 3 yesterday was sunny and today was cloudy With this definition we own39 the following transition matrix SS SC CS CC SS 95 05 O 0 SC 0 O 40 60 CS 70 30 O 0 CC 0 O 20 80 If the last three days weather were the statethere would be 238 possible states corresponding to the 8 possible three day weather patterns ie SSS etc 5 The state is the location before a dice roll of a player39s t token on the board 1Go40 Boardwalk41 In Jail for O turns42f In Jail for 1 turn 43 In Jail for 2 turns and the number of times an My mu

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