Ch 2.2 Inventory

7/23/2019 Ch 2.2 Inventory

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INVENTORY CONTROL



1. Introduction• Inventory is the stock of any item or

resource used in an organization.• An inventory system is the set of policies

and controls that monitor levels of inventoryand determine what levels should be

maintained, when stock should bereplenished, and how large orders should be.

• The basic purpose of inventory analysis,whether in manufacturing, distribution,retail, or services, is to specify (1 whenitems should be ordered and (! how largethe order should be.



PURPOSES OF INVENTORY

•

To maintain independence of operations. Asupply of materials at a work center allows thatcenter "e#ibility in operations.

• To meet variation in prodct demand. If the

demand for the product is known precisely, itmay be possible (though not necessarilyeconomical to produce the product to e#actlymeet the demand.

•

To allo! "e#i$ility in prodction sc%edlin&.A stock of inventory relieves the pressure on theproduction system to get the goods out.



• To provide a safe&ard for variation

in ra! material delivery time'• To ta(e advanta&e of economic

prc%ase order si)e.

•

*any ot%er domain+speci,c reasons.$epending on the situation, inventorymay need to be carried. %or e#ample, in&transit inventory is material being

moved from the suppliers to customersand depends on the order 'uantity andthe transit lead time



INVENTORY COSTS

• -oldin& .or carryin&/ costs. This broad

category includes the costs for storagefacilities, handling, insurance, pilferage,breakage, obsolescence, depreciation, ta#es,and the opportunity cost of capital.

• Setp .or prodction c%an&e/ costs. Tomake each dierent product involvesobtaining the necessary materials, arranging

speci)c e'uipment setups, )lling out there'uired papers, appropriately charging timeand materials, and moving out the previousstock of material.



• Orderin& costs. These costs refer to themanagerial and clerical costs to prepare the

purchase or production order. *rdering costsinclude all the details, such as counting itemsand calculating order 'uantities. The costsassociated with maintaining the system needed

to track orders are also included in orderingcosts.

• S%orta&e costs. +hen the stock of an item isdepleted, an order for that item must either

wait until the stock is replenished or becanceled. +hen the demand is not met and theorder is canceled, this is referred to as a stockout



INVENTORY SYSTE*S

• An inventory system provides the organizational

structure and the operating policies formaintaining and controlling goods to be stocked.

• The system is responsible for ordering andreceipt of goods timing the order placement and

keeping track of what has been ordered, howmuch, and from whom.

• The system also must follow up to answer such'uestions as -as the supplier received the

order -as it been shipped Are the datescorrect Are the procedures established forreordering or returning undesirablemerchandise



n& e+ er o nven ory*odel

• /onsider the problem that the ne!sperson has in deciding how many newspapers to put inthe sales stand outside a hotel lobby eachmorning. If the person does not put enough

papers in the stand, some customers will not beable to purchase a paper and the newspersonwill lose the pro)t associated with these sales.*n the other hand, if too many papers are

placed in the stand, the newsperson will havepaid for papers that were not sold during theday, lowering pro)t for the day.



• /onsider the person selling T&shirtspromoting a championship basketball or

football game. This is especially di0cult,since the person must wait to learn whatteams will be playing. The shirts can then beprinted with the proper team logos. *f course,

the person must estimate how many peoplewill actually want the shirts. The shirts soldprior to the game can probably be sold at apremium price, whereas those sold after the

game will need to be steeply discounted.• A simple way to think about this is to consider

how much risk we are willing to take forrunning out of inventory



• et2s consider that the newsperson sellingpapers in the sales stand had collected data

over a few months and had found that onaverage each 3onday 45 papers were soldwith a standard deviation of 15 papers(assume that during this time the papers

were purposefully overstocked in order not torun out, so they would know what 6real7demand was. +ith these data, ournewsperson could simply state a service rate

that is felt to be acceptable. %or e#ample, thenewsperson might want to be 85 percent sureof not running out of papers each 3onday.



• Assuming that the probability distribution

associated with the sales of the paper isnormal, that if we stocked e#actly 45 paperseach 3onday morning, the risk of stockingout would be 95 percent, since 95 percent

of the time we e#pect demand to be lessthan 45 papers and 95 percent of the timewe e#pect demand to be greater than 45. Tobe 85 percent sure of not stocking out, we

need to carry a few more papers.



• %rom the 6cumulative standard normaldistribution7 , we see that we need

appro#imately 5.89 standard deviation of e#trapapers to be 85 percent sure of not stockingout. A 'uick way to )nd the e#act number ofstandard deviations needed for a given

probability of stocking out is with the:*;3<I:=(probability function in 3icrosoft>#cel (:*;3<I:=(5.8 ?5.8@1!. Biven ourresult from >#cel, which is more accurate than

what we can get from the tables, the number ofe#tra papers would be 5.8@1! C15 ? 8.@1, or4 papers (there is no way to sell 5.@ paperD



• To make this more useful, it would be good

to actually consider the potential pro)tand loss associated with stocking eithertoo many or too few papers on the stand.et2s say that our newspaper person pays

E5.!5 for each paper and sells the papersfor E5.95. In this case the marginal costassociated with underestimating demand

is E5.F5, the lost pro)t. <imilarly, themarginal cost of overestimating demand isE5.!5, the cost of buying too many papers



• T%e optimal stoc(in& level, usingmarginal analysis, occurs at the point where

the e#pected bene)ts derived from carryingthe ne#t unit are less than the e#pectedcosts for that unit. Geep in mind that thespeci)c bene)ts and costs depend on the

problem.• /o ? /ost per unit of demand overestimated

• /u ? /ost per unit of demandunderestimated

• Hy introducing probabilities, the e#pectedmarginal cost e'uation becomes

• (/o J(1 K /u



• +here is the probability that the unit willnot be sold and 1 K is the probability of it

being sold because one or the other mustoccur. (The unit is sold or is not sold.

• Then, solving for , we obtain

• J /u L (/oM /u

• This e'uation states that we shouldcontinue to increase the size of the order solong as the probability of selling what we

order is e'ual to or less than the ratio /uL( /oM /u.

• Sin&le+period inventory models aresefl for a !ide variety of service and

manfactrin& applications

* lti i d I t



*ltiperiod InventorySystems

• There are two general types of multiperiod inventory

systems

• Fi#ed1order 2antity models (also called theeconomic order 'uantity, >*N, and N&model and

• Fi#ed1time period models (also referred to variously

as the periodic system, periodic review system, )#edOorder interval system, and &model.

• 3ultiperiod inventory systems are designed to ensurethat an item will be available on an ongoing basisthroughout the year.

• Psually the item will be ordered multiple timesthroughout the year where the logic in the systemdictates the actual 'uantity ordered and the timing ofthe order



• The basic distinction is that )#edOorder 'uantitymodels are 6event triggered7 and )#edOtime periodmodels are 6time triggered.7

• a ,#ed1order 2antity model initiates an orderwhen the event of reaching a speci)ed reorder leveloccurs. This event may take place at any time,depending on the demand for the items considered.

• To use the )#edOorder 'uantity model (which placesan order when the remaining inventory drops to apredetermined order point, ;, the inventoryremaining must be continually monitored. Thus, It is

a perpetual system, which re'uires that every timea withdrawal from inventory or an addition toinventory is made, records must be updated tore"ect whether the reorder point has been reached.



• T%e ,#ed1time period model is limited to placingorders at the end of a predetermined time periodQ

only the passage of time triggers the model.• In a )#edOtime period model, counting takes place

only at the review period



3' T%e Economic Order4antity *odel

• *ne of the earliest applications ofmathematics to factory managementwas the work of %ord +.-arris(141F

on the problem of settingmanufacturing lot sizes.



Assumptions about themanufacturing system

1. Prodction is instantaneos. There is no capacityconstraint, and the entire lot is produced simultaneously.!. 5elivery is immediate. There is no time lag betweenproduction and availability to satisfy demand.F. 5emand is deterministic. There is no uncertaintyabout the 'uantity or timing of demand.@. 5emand is constant overtime. In fact, it can berepresented as a straight line, so that if annual demandis F9 units, this translates to a daily demand of oneunit.

9. 0 prodction rn incrs a ,#ed setp cost.;egardless of the size of the lot or the status of thefactoryQ the setup cost is the same.. Prodcts can $e analy)ed individally. >itherthere is only a single product or there are no interactions(e.g., shared e'uipment between products.



The notation of the >*N model

•

$ ? demand rate (in units per year• c ? unit production cost, not counting setup orinventory costs (in dollars per unit

• A ? )#ed setup (ordering cost to produce(purchase a lot (in dollars

• h ? holding cost (in dollars per unit per yearQif the holding cost consists entirely of intereston money tied up in inventory, then h ? ic,where i is the annual interest rate

• N ? lot size(in unitsQ this is the decisionvariable



• The total (inventory, setup, and production costper year can be e#pressed as

• Taking the derivative of R(N and setting theresult e'ual to zero yields

• The lot size that minimizes R(N in cost function is



T%e (ey insi&%t of EO4

• The optimal order 'uantity increases with

the s'uare root of the setup cost or thedemand rate and decreases with thes'uare root of the holding cost.

• There is a trade o between lot size and

inventory.• Increasing the lot size increases the

average amount of inventory on hand,but reduces the fre'uency of ordering.

•

Hy using a setup cost to penalizefre'uent replenishments, -arrisarticulated this trade o in cleareconomic terms.



• It is worth noting that we can use theinsight that there is a trade ofbetween lot size and inventorywithout even resorting to -arrisSss'uare root formula. <ince the

average number of lots per year % is% ? $LN, and the total inventoryinvestment is

I ? cNL! ? c$L!%

Ch 2.2 Inventory

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