Inventory Management Subject To Uncertain Demand

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Esma Gel, Pınar Keskinocak, 2007Inventory Management Subjectto Uncertain DemandISYE 3104 - Fall 2012Inventory Control Subject to Uncertain Demand In the presence of uncertain demand, the objectiveis to minimize the expected cost or to maximize theexpected profitTwo types of inventory control models Fixed time period - Periodic review One period (Newsvendor model)Multiple periodsFixed order quantity - Continuous review (Q,R) models1

Types of Inventory Control Policies Fixed order quantity policies The order quantity is always the same but the timebetween the orders will vary depending on demandand the current inventory levelsInventory levels are continuously monitored and anorder is placed whenever the inventory level dropsbelow a prespecified reorder point.Continuous review policyTypes of Inventory Control Policies Fixed time period policies The time between orders is constant, but thequantity ordered each time varies with demand andthe current level of inventoryInventory is reviewed and replenished in given timeintervals, such as a week or month (i.e., reviewperiod)Depending on the current inventory level an ordersize is determined to (possibly) increase theinventory level up-to a prespecified level (i.e., orderup-to level)Periodic review policy2

Inventory Control - c OrderQuantity (EOQ) –Tradeoff betweenfixed cost and holdingcostLot size/Reorder point(Q,R) or (s,S) models –Tradeoff between fixedcost, holding cost, andshortage costAggregate Planning –Planning for capacitylevels given a forecastMaterials RequirementsPlanning (MRP)Very difficult problem!Newsvendor – singleperiodREAD THE APPENDIX ONPROBABILITY REVIEW3

Esma Gel, Pınar Keskinocak, 2007Newsvendor ModelsISYE 3104 – Fall 2012Example - INFORMS INFORMS (The Institute for Operations Research andManagement Science, www.informs.org) will hold its annualmeeting in Washington D.C. in 2008. Six months before themeeting begins, INFORMS must decide how many roomsshould be reserved at the conference hotel. At this time,rooms can be reserved at a cost of 50 per room. It isestimated that the demand for rooms is normally distributedwith mean 5000 and standard deviation 2000. If the numberof rooms required exceeds the number of rooms reserved,extra rooms will have to be found at neighboring hotels at acost of 80 per room. The inconvenience of staying atanother hotel is estimated at 10. How many rooms shouldbe reserved to minimize the expected cost?4

Example – Fashion Bags The buyer for What-a-Markup Fashion Bags mustdecide on the quantity of a high-priced woman’shandbag to procure in Italy for the followingChristmas Season. The unit cost of the handbag tothe store is 28.50 and the handbag will sell for 150. A discount firm purchases any handbags notsold by the end of the season for 20. In addition,the store accountants estimate that there is cost of 0.40 for each dollar tied up in inventory at the endof the season (after all sales have been made).Newsvendor model - Properties One-time decision Current decisions only impact the “next period” butnot future periods Retail: fashion/seasonal itemsOne-time events5

Tradeoff in inventory decisionsDemandSupplySupply DemandShortageLost sales / Lost profitSupply DemandExcess inventoryInventory costSupplyDemandNewsvendor model - Properties One-time decision Relevant costs Co: Unit cost of excess inventory (cost of overage)Cu: Unit cost of shortage (cost of underage)Demand D is a random variable with Current decisions only impact the “next period” but not futureperiods Retail: fashion/seasonal items One-time eventsProbability density function (pdf), f(x)cumulative distribution function (cdf), F(x)Objective: Choose the ordering quantity Q (before you knowthe demand) to minimize Total expected overage and underage costs6

Newsvendor model – Finding the optimal orderquantity First, write the cost function:G (Q, D ) : total overage underage cost (if Q is ordered and demand is D) c ( D Q ) if D QG (Q, D ) : u co (Q D ) if D QG (Q ) expected overage underage cost if Q is ordered G (Q ) E[G (Q, D ] G (Q, x) f ( x)dxCritical Ratio0Q 0Q co (Q D) f ( x)dx cu ( D Q) f ( x)dx F (Q ) cucu coDerivation of the Critical Ratio - 1G (Q ) expected overage underage cost if Q is orderedQ 0QG (Q ) co (Q D ) f ( x)dx cu ( D Q) f ( x)dx coG1 cu G2To take the derivative of G(Q), use Leibniz' s Ruleddyf2 ( y) h( x, y)dx f1 ( y )In G1 :f2 ( y) h( x, y )dx h( f 2 ( y ), y ). f 2' ( y ) h( f1 ( y ), y ). f1' ( y ) yf1 ( y ) y Qf 2 ( y) Qf1 ( y ) 0h( x, Q) (Q x) f ( x)7

Derivation of the Critical Ratio - 2QG1 (Q) (Q D) f ( x)dxIn G1 :y Qf 2 ( y) Qf1 ( y ) 0h( x, Q) (Q x) f ( x)0To take the derivative of G(Q), use Leibniz' s Ruleddyf2 ( y)f2 ( y) h( x, y )dx h( f 2 ( y ), y ). f 2' ( y ) h( f1 ( y ), y ). f1' ( y )y f1 ( y ) h( x, y)dx f1 ( y ) h( x, y ) [(Q D ) f ( x)] f ( x) y Qh( f 2 ( y ), y ) h(Q, Q ) (Q Q) f ( x) 0f 2' ( y ) 1h( f1 ( y ), y ) h(0, Q) (Q 0) f ( x) Qf ( x)f1' ( y ) 0 Q ddG1 (Q ) f ( x)dx Similarly,G2 (Q) f ( x)dxdQdQQ0Derivation of the Critical Ratio - 3Q dG (Q ) co f ( x)dx cu f ( x)dx co F (Q ) cu (1 F (Q))dQ0Q (co cu ) F (Q) cu 0 F (Q) cucu co Recall : f ( x)dx 10 Q 0f ( x)dx F (Q) P( D Q ) f ( x)dx 1 F (Q) P( D Q)QAlso need to check if G(Q) is convex. Second derivative is (co cu)f(Q) 08

Example - INFORMS D number of rooms actually requiredQ number of rooms reservedWhat are Co and Cu ?Example - INFORMS D number of rooms actually requiredQ number of rooms reservedD Q: cost 50QD Q: cost 50Q 80(D-Q) 10(D-Q) 90D-40QCost ofCost ofCost ofreserving reserving inconvenienceextra roomsin other hotels9

Example - INFORMS D number of rooms actually requiredQ number of rooms reservedCo D Q: cost 50QCuD Q: cost 50Q 80(D-Q) 10(D-Q) 90D-40QCost ofCost ofCost ofreserving reserving inconvenienceextra rooms inother hotels F (Q ) 404cu cu co 40 50 9What is Q?Example - INFORMS Recall: D Normal(5000,2000) We have lookup tables for Standard Normal distribution( 0, 1) Convert to Standard Normal! F (Q) P ( D Q) 4 0.4449 D Q P ( D Q ) P ( D Q ) P P( Z z ) ( z ) Standard normal ZzFind z from the lookup table Q z 10

Example - INFORMS Recall: D Normal(5000,2000) We have lookup tables for Standard Normal distribution( 0, 1) Convert to Standard Normal! F (Q) P ( D Q) 4 0.4449 D Q P ( D Q ) P ( D Q ) P P( Z z ) ( z ) SafetyStandard normal ZzstockExpecteddemandFind z from the lookup table Q z Standard Normal Distribution - 1Density function ( z ) e z222 Symmetric around the meanArea under the entirecurve 1 P(Z )11

Standard Normal Distribution - 2Density function ( z ) e z222 Symmetric around the meanArea under the entireWe will use ɸ(z) and F(z)interchangeablycurve 1 P(Z )F(z) P(Z z)zStandard Normal Distribution - 2Density function ( z ) e z222 Symmetric around the meanArea under the entireWe will use ɸ(z) and F(z)interchangeablycurve 1 P(Z )F(z) P(Z z)F(-z) P(Z -z) 1-F(z)-zz12

Example - INFORMSP( Z z ) ( z ) Area under the curve 0.444 z -0.14Q* z 2000( 0.14) 5000Q* 4720z - 0.14Example - INFORMSWhy is Q* 5000,i.e., less than theexpected demand? Expectedtotal costExpectedcost ofunderageExpectedcost ofoverageQ* 472013

Example - INFORMS What if co 150?Expectedtotal costcu40 F (Q ) cu co 40 150 0.21 z 0.805 Q* z (2000)( 0.805) 5000 3390Expectedcost ofoverageExpectedcost ofunderageQ* 3390Example - INFORMS What if co 150?F (Q ) cu40 cu co 40 150 0.21 z 0.805 Q* z (2000)( 0.805) 5000 3390Expectedtotal costCo Q* Expectedcost ofoverageCu Q* Expectedcost ofunderageQ* 339014

When do we have Q* Expected demand?Q* z If Q* then z 0 !!!P( Z z ) ( z ) Area under the curve 0.5 z 0 For Q* we needF (Q ) cu 0.5, i.e., cu co !!!cu coz 0Optimal quantity versus expected demandcu co F (Q) cu 0.5 z 0 Q* cu coIf shortages are more costly, order more than expected demandcu co F (Q) cu 0.5 z 0 Q* cu coIf excess inventory is more costly order less than expected demandcu co F (Q) cu 0.5 z 0 Q* cu coIf shortages and excess inventory cost the same, order expected demandNote: Assuming the demand distribution is symmetric around its mean15

The impact of standard deviation What happens to the optimal quantity as thestandard deviation increases?The impact of standard deviation What happens to the optimal quantity as the standarddeviation increases? (Assuming the demand distribution is symmetric around its mean)cu co z 0 Q* z increases in If shortages are more costly, Q * increases in std. dev.cu co z 0 Q* z decreases in If excess inventory is more costly , Q * decreases in std. dev.cu co z 0 Q* does not change in If shortages and excess inventory cost the same,order expected demand regardless of standard deviation16

Example - INFORMScu 40 co 50 z -0.14cu 70 co 50 z 0.21Q*Q*Standard deviationStandard deviationExample – Fashion BagsInput Unit costc 28.50Selling pricep 150Salvage values 20Cost of inventory 0.40 for each dollar tied up ininventory at the end of the season17

Example – Fashion BagsInput Unit costc 28.50Selling pricep 150Salvage values 20Cost of inventory 0.40 for each dollar tied up ininventory at the end of the seasonComputed input: Holding cost per bagcu co F (Q) h cu cu coExample – Fashion BagsInput Unit costc 28.50Selling pricep 150Salvage values 20Cost of inventory 0.40 for each dollar tied up ininventory at the end of the seasonComputed input: Holding cost per bagcu p – c 121.5co c – s h 19.9F (Q) h (0.40)(28.50) 11.4cu121.5 0.86cu co 121.5 19.918

Example – Fashion Bags – Normally distributeddemandDemand Normal( 150, 20)F (Q) 0.86 z 1.08Area under the curve Critical ratio 0.86 Q* z (20)(1.08) 150Q * 172z 1.08Example – Fashion Bags – Uniformly distributeddemand Demand Uniform between 50 and 250 150 Same expected demand as in Normal distribution0.86 (Q * 50)Q* 2221 200Uniform densityArea under the curve Critical ratio 0.861/200Q25050Q* 22219

Example – Fashion Bags Even though both the Normal and the Uniform distributionshave the same mean ( 150), why did we get differentquantities? Normal distribution Q* 172Uniform distribution Q* 222Example – Fashion Bags Even though both the Normal and the Uniform distributionshave the same mean ( 150), why did we get differentquantities? Normal distribution Q* 172Uniform distribution Q* 222Because of the variance (equivalently, standard deviation) andthe shape of the distribution !!! Normal 20Uniform 57.7NormalUniform20

Types of Inventory Control Policies Fixed order quantity policies The order quantity is always the same but the time between the orders will vary depending on demand and the current inventory levels Inventory levels are conti nuously monitored and an order is placed whenever the inventory level drops below a prespecified reorder point.

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