1 Inventory Models18slds

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MaterialsManagement andInventory Control

1/26/12 1Prepared by Merlin Sem-7 Civil Engg



Inventory is essential to provide flexibility in operating a system.

Inventory classified as

– Raw materials inventory

– In-process inventory

– Finished goods inventory

Functions of inventory

– Smoothing out regularities in supply

– Minimizing the production cost

– Allowing organizations to cope with perishable materials

Inventory Decisions

– When to replenish the inventory of that item.

– How much of an item to order when the inventory of that is to be replenished.

INVENTORY CONTROL




INVENTORY MODELS

Deterministic Probabilistic

Purchase Model Manufacturing Model

With Shortages

With

outShortages

With Shortages

Without Shortages

(InstantaneousReplenishment)




Inventory models are classified as deterministic and probabilistic models.

Various deterministic models are:-

● Purchase model with instantaneous replenishment and withoutshortages

● Manufacturing model without shortages

● Purchase model with instantaneous replenishment and with shortages

● Manufacturing model with shortages.

MODELS OFINVENTORY




Orders of equal size are placed at periodical intervals Items against an order are replenishedinstantaneously and the items are consumed at aconstant rate.

The purchase price per unit is the same irrespectiveof order size.

◦ Let, D be the annual demand in units.◦ Co be the ordering cost/order ◦ Cc be the carrying cost/unit/year ◦

P be the purchase price per unit◦ Q be the order size.

Purchase Model with Instantaneous

Replenishment and withoutShortages




The corresponding model is shown

Fig.1. Purchase model without stockout1/26/12 6Prepared by Merlin Sem-7 Civil Engg



Therefore,

The number of orders/year = D/Q

Average inventory =Q/2

Cost of ordering/year = D/Q ×Co

Cost of carrying/year = Q/2 ×Cc

Purchase cost/year = D × P

The total inventory cost (TC)/year = D/Q ×Co + Q/2 Cc + D × P

Differentiating w.r.t. Q yields

d/dQ (TC) = –D/Q2 Co + Cc/2




The second derivative = +2D/Q3 Co

Since the second derivative is positive, we canequate the first derivative to zero to get theoptimal value for Q.

-D/Q2 Co + Cc/2 = 0

Q2 = 2CoD

Q* = √2CoD/Cc

No. of orders = D/Q*

Time between orders = Q*/D




Ø A company manufactures components required for

its main product, then the corresponding model of inventory is called "Manufacturing model".

Ø This model will be with shortages or without shortages.

Ø The rate of consumption of items is uniform throughoutthe year.

Ø The cost of production per unit is same irrespective of production lot size

Manufacturing Model withoutShortages




Let

r __annual demand of an item.

k __production rate of item (No. of units

produced per year).

Co __cost per set up.

Cc __carrying cost per unit per period.

P --cost of production per unit.

EBQ be Economic Batch Quantity




Ø During period t1, the item is produced at the rate of k unitsper period and simultaneously it is consumed at the rate of

r units per period.

Ø So, during this period, the inventory is built at the rate of units per period.

Ø During period t2, the production of item is discontinued butthe consumption of that item is continued.

Ø Hence, the inventory is decreased at the rate of r units per period during this period.




Fig.2 Manufacturing model without shortage




Ø The various formulas for this

situation are given below.

Ø EBQ = √2Cor/Cc(1-r/k

Ø t*1 = Q*/k

Ø t*1 = Q*[1–r/k]/r

Ø Cycle time = t*1 + t*2




Ø Items on order will be received instantaneously

and they are consumed at a constant rate.

Ø Purchase price per unit remains sameirrespective of order size.

Ø

If there is no stock at the time of receiving a request forthe items, it is assumed that it will be satisfied at alater date with a penalty. This is called backordering.

Purchase Model withShortages (InstantaneousSupply)




Ø The variables which are used in thismodel are given below:

Ø D − Demand/period

Ø Cc − Carrying cost/unit/period.

Ø Co − Ordering cost/order.Ø Cs − Shortage cost/unit/period



http://slidepdf.com/reader/full/1-inventory-models18slds 16/351/26/12 16Prepared by Merlin Sem-7 Civil Engg



Ø Q represents Economic order quantity,

Ø Q1, Maximum inventory,

Ø Q2, Maximum stockout

Ø Q* = EOQ = √2CoD/Cc (Cs+Cc)/Cs

Ø

Q*1 = √2CoD/Cc (Cs+Cc)/CsØ Q*2 = Q* – Q1

Ø t* = Q*/D

Ø t*1 = Q*1/D

Ø t*2 = Q*2/D




Manufacturing Model with Shortages

Ø

Items are produced and consumed simultaneously for aportion of the cycle time.

Ø Rate of consumption of items is uniform throughout the year.

Ø Cost of production per unit is same irrespective of productionlot-size.

Ø Stockout is permitted. These units will be satisfied from theunits which will be produced at a later date with a penalty. This

is called backordering.




Ø The variables which are used in this model are given

below. Let,

Ø r --annual demand of an item

Ø k --production rate of the item (No. of units

Ø produced/year)Ø Co --cost/set up.

Ø Cc --carrying cost/unit/period

Ø Cs --shortage cost/unit/period

Ø P -- cost of production/unit




Fig. Manufacturing model of inventory with stockout




Ø Q − Economic Batch Quantity

Ø Q1 − Maximum inventory

Ø Q2 − Maximum stockoutØ Q* = EBQ = √2Co/Cc kr/(k–r) (Co+Cs)/Cs

Ø Q*1 = √2Co/Cc kr/(k–r) (Co+Cs)/Cs

Ø

Q*2 = √2CoCc/Cs(Co+Cs) r(k–r)/kØ Q*1 = (k–r/k Q*) – Q*2

Ø t* = Q*1/r

Ø t* = Q*1/(k–r)

Ø t*2 = Q*1/r

Ø t*4 = Q*2/(k–r)




Consider the purchase model of inventory system whichis as shown in Fig.

Fig. Operation of inventory system

OPERATION OF INVENTORY SYSTEM




Ø Q* is the economic order size and t* is the cycle time.

Ø

Ø Consider a model with constant demands and constant lead time,

Ø we will have to place order well before the end of cycle time, sothat the items are received exactly at the end of the present cycleor at the beginning of the next cycle.

Let DLT be the demand during lead time.

DLT = demand rate × lead time period

= (d/day) × (LT in days)




Ø If there is no variation in lead time and demand, then it issufficient to have a stock of DLT at the time of placingorder.

Ø Reorder level (ROL) = DLT

Fig. Inventory system with constant demand and constantlead time1/26/12 24Prepared by Merlin Sem-7 Civil Engg



Ø Reorder level is the stock level at which an order isplaced so that we receive the items against theorder at the beginning of the next cycle.

Ø If the demand is varying, then the ROL is as givenbelow

ROL = DLT + SS

Ø

Where, SS is the safety stock, which acts as acushion to absorb the variation in demand.

SS = Kσ

Ø Where, σ is the standard deviation of demand

K is the standard normal statistic value for agiven service level.




The corresponding chart is shown in Fig.

Fig. Inventory system with safety stock for variation in lead time demand.




When items are purchased in bulk, buyers are usuallygiven discount in the purchase price of goods. Theprocedure to compute the optimal order size for thissituation is given in the following steps.

Step 1:Find EOQ for the nth (last) price break.Q*n = √2CoD/iPn

If it is greater than or equal to bn–1, then the optimalorder size otherwise go to Step 2.

QUANTITY DISCOUNT




Step 2: Find EOQ for the n − 1th price break.

Q*n–1 = √2CoD/iPn–1

If it is greater than or equal to bn–2, then compute thefollowing, and select least cost purchase quantity asthe optimal order size; otherwise go to Step 3:

(i) Total cost, TC(Q*n–1)

(ii) Total cost, TC(bn–1)

Step 3: Find EOQ for the n − 2th price break.

Q*n–2 = √2CoD/iPn–2




If it is greater than or equal to bn–3, then computethe following and select least cost purchase quantity;otherwise go to Step 4.

(i) Total cost, TC(Q*n–2)(ii) Total cost, TC(b*n–2)

(iii) Total cost, TC(bn–1 )

Step 4: Continue in this manner until Q*n–1 > bn–i-1 .Then compare total costs TC(Q*n–1) TC(bn–1), TC(bn–i+1),..., TC(bn–1) corresponding to purchasequantities Q*n–i, bn–1, bn–i+1, ..., bn–1, respectively.

Finally, select the purchase quantity w.r.t. theminimum total cost.




The practical version of purchase model of inventory canbe classified into

Fixed Order Quantity System (Q System)

Fixed Period Quantity System (P System).The following cases exist in each of the systems:

Varying demand and constant lead time.

Constant demand and varying lead time.Varying demand and varying lead time.

IMPLEMENTATION OFPURCHASE INVENTORY

MODEL




Ø Fixed Order Quantity System (Q System)

Ø Whenever the stock level touches the reorder level, anorder is placed for a fixed quantity which is equal to EOQ.




Fig. Q system of inventory




Ø The average demand during the lead time (average leadtime) is known as the demand during lead time (DLT).

Ø The variation in demand during lead time (average leadtime) is known as safety stock.

Ø The average demand during delivery delays is calledreserve stock.

Ø The Reorder level is computed as the sum of the demandduring lead time (DLT), the variation in demand during leadtime (safety stock) and the average demand duringdelivery delays (reserve stock).




Periodic Review System (P System)

Ø Stock position is reviewed once in a fixed period and an order isplaced depending on the stock position, unlike a fixed quantity inthe Q System of inventory.

Ø Review period is approximately equal to EOQ/D.

Ø Desired Maximum Inventory Level is fixed as the sum of theaverage demand during average lead time plus review period,variation in demand during average lead time plus review period,and the average demand during delays in supply.




A schematic representation of this model is

shown in Fig.

Fig. P-system of inventory

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1 Inventory Models18slds