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AIBSMBA-IB, MIB 201

OPERATIONS RESEARCH

REMICAAGGARWAL



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InventoryManagement



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• Economic order quantity model• Economic production model

• Quantity discount model

Economic Order QuantityModels



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Assumptions of EOQ Model1-) Demand is known and constant.

2-) Lead time (the time between placement of order andreceipt of the order) is constant and known.

3-) Orders arrive in one batch at a time, and they arrive inone point in time.

4-) Quantity discounts are not possible.

5-) The costs include only setup cost (or ordering cost

when buying) and holding cost.6-) Orders are always placed at the right times. Therefore,

stock outs (or shortages) can be completely avoided.



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The Inventory CycleProfile of Inventory Level Over Time

Quantity

on hand

Q

Receive

order

Place

order Receive

order Place

order

Receive

order

Lead time

Reorder

point

Usage

rate

Time



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Continuous review model

1. Uniform demand, no shortages

Demand (consumption) rate: D units / month

Order (lot) size: Q units / order

Setup (ordering) cost: S rupees/ order

Production (purchase) cost: C rupees / item

Holding cost: H rupees / item / month

Problem: What is the best Q ?

Note: Q is known ordering interval = Q/D (why?)



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Total CostTotal cost = annual inventory carrying (holding) cost + annual ordering cost

TC = (Q/2)*H+ (D/Q)*S

NUMBER OF RUNS OF QUANTITY PURCHASE /PRODUCED DURING THEYEAR , n = D/Q

Everytime we place the order we incur an ordering cost S



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Cost Minimization Goal

Order Quantity

(Q)

The Total-Cost Curve is U-Shaped

Ordering Costs

QO

Annua l

Cost

(optimal order quantity)

TC Q

H D

QS = +

2



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Deriving the EOQThe total cost curve reaches its minimum

where the carrying and ordering costs areequal. i.e QH/2=DS/Qgiving

Q =2DS

H=

2(Annual Demand )(Order or Setup Cost )

Annual Holding CostOPT



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Lead time

• There is a time between placement and

receipt of an order.• This is called LEAD TIME or delivery time.



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Considering the Reorder Point

• ROP (in units) = (Demand Per Day) *(Lead time for a new order in days)

• ROP = d * L



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Numerical on EOQ model

• An Inventory model has the following characteristics:

• Annual Demand (D) = 1000 units• Ordering (Setup) cost (S)=rs10 per order;

• Holding cost per unit per year (H) = rs0.50

• Assume that there are 270 working days in a year (excluding holidays and weekends).

• LEAD TIME : 3 DAYS



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• Questions:

a) Find the Economic Order Quantity (Q*) for thisinventory model.

b) How many orders should be placed during oneyear?c) What is the expected time between two

consecutive orders?

d) What is the total annual cost of this inventorymodel?

e) what is the reorder point



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• Answers:

a) Q* = √[2(1000)10 / .50] = 200 units

b) Expected number of orders placed duringthe year (N) = D / Q* = 1000 / 200 = 5times.



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c) Expected time between orders (T) = (Workingdays in a year) / N = 270 / 5 = 54 days.

d) Total Annual Cost = Annual Setup Cost +Annual Holding Cost

= DS / Q* + (Q*)H / 2

= 1000 (10) / 200+ (200) (.50) / 2 = rs100e) Reorder point = (1000/270 )* L = (1000/270)*3

=11.1

When inventory level becomes 11 units, an Order should be placed.



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Example 2

• Annual demand for an item is D =8000/year.

• This year there will be 200 working days ina year.

• Delivery of an order for this item takes 3working days (L = 3 days).





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• Answers:

a) Demand per day for this item (d) = 8000 /200 = 40 units / day.

b) ROP = d . L = 40 . 3 = 120 units.

When inventory level becomes 120 units, anOrder should be placed.



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2. Uniform demand, shortages are allowed

In

ventory

level

Time t

Q

0

( Q - K ) / D

K

K/D

Shortage cost: p rupees / unit of unfulfilled demand / period

Why allow shortage ? Average inventory held is lower

K: inventory level at the beginning of each period tk i.e for period t1<tk

there is an inventory Q & during period t2 (t1+t2=tk) there is a shortage = Q-K



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Inventory model: Uniform demand, shortage allowed

2

2 2

h K K h K

D D=

2( ) ( ) ( )

2 2

p Q K Q K p Q K

D D

− − −=

Total cost = ordering cost + purchase cost + holding cost + shortage cost

S cQ

In

ventory

level

Time t

Q

0

( Q - K ) / D

K

K/D



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2 2( )

2 2/

hK p Q K S cQ

D DT Q D

−+ + +

=

2 2( )

( , )

2 2

DS hK p Q K T K Q Dc

Q Q Q

−= + + +

T is a function of K, Q

Total cost/unit time = (ordering + purchase + holding + shortage) cost per period

length of period

We want to minimize T(S, Q): 0, 0T T

and K Q

∂ ∂= =

∂ ∂



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2*

DS pK

h p h=

+

2*

DS p hQ

h p

+=Total cost is minimized if:

Optimum period length:* 2Q S p h

D Dh p

+=

Maximum shortage:2

* * *h aS h

Q K K

p p p h

− = =

+

Inventory

level

Time t

Q

0

( Q - K

) / D

K

K/D



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• Numerical on EOQ MODEL WITHSHORTAGES



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Derivations2 2

( )( , )2 2

DS hK p Q KS T K Q DcQ Q Q

−= + + +

( )0

T hK p Q K

K Q Q

∂ −= − =

∂

2 2

2 2 2

( ) ( )0

2 2

T DS hK p Q K p Q K

Q Q Q Q Q

∂ − −= − − + − =

∂

(Q – K) = h K / p

2 22 ( ) 2 ( )DS hK p Q K p Q K + + − = −

Q = K (h + p) / p



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(Q – K) = hK/p Q = K(h+p)/p

Solving: and

2*

DS pK

h p h=

+

2*

DS p hQ

h p

+=

2 22 ( ) 2 ( )DS hK p Q K p Q K Q+ + − = −

22 22 2 ( )

2h K h h p K

DS hK p p

++ + =

2 2 2 22 2 ( )DpS phK h K h h p K + + = +

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