eoq for production lots eoq with quantity discount safety stock periodic review system ----

EOQ for production lotsEOQ with quantity discount

Safety StockPeriodic review system

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Model II: EOQ for Production Lots

• Used to determine the order size, production lot.

• Differs from Model I because orders are assumed to be supplied or produced at a uniform rate (p) rather than the order being received all at once.

Inventory Management

06 July 2012 KLE College of Pharmacy, Nipani. 2

Production Order Quantity ModelIn

vent

ory

leve

l

Time

Demand part of cycle with no production

Part of inventory cycle during which production (and usage) is taking place

t

Maximum inventory

Figure 12.6

Model II: EOQ for Production Lots

• It is also assumed that the supply rate, p, is greater than the demand rate, d

• The change in maximum inventory level requires modification of the TSC equation

• TSC = (Q/2)[(p-d)/p]C + (D/Q)S• The optimization results in

dpp

CDS2 = EOQ



Example: EOQ for Production Lots

• Highland Electric Co. buys coal from Cedar Creek Coal Co. to generate electricity. CCCC can supply coal at the rate of 3,500 tons per day for $10.50 per ton. HEC uses the coal at a rate of 800 tons per day and operates 365 days per year.




• HEC’s annual carrying cost for coal is 20% of the acquisition cost, and the ordering cost is $5,000.

a) What is the economical production lot size?

b) What is HEC’s maximum inventory level for coal?




Economical Production Lot Sized = 800 tons/day; D = 365(800) = 292,000tons/yearp = 3,500 tons/dayS = $5,000/order., C = .20(10.50)= $2.10/ton/year

= 42,455.5 tons per order

EOQ = (2DS/C)[p/(p-d)]

EOQ = 2(292,000)(5,000)/2.10[3,500/(3,500-800)]




• Total Annual Stocking Cost (TSC)TSC = (Q/2)((p-d)/p)C + (D/Q)S

= (42,455.5/2)((3,500-800)/3,500)(2.10) + (292,000/42,455.5)(5,000) = 34,388.95 + 34,388.95 = $68,777.90

Note: Total Carrying Costequals Total Ordering Cost



Model III: EOQ with Quantity Discounts

• Under quantity discounts, a supplier offers a lower unit price if larger quantities are ordered at one time

• This is presented as a price or discount schedule, i.e., a certain unit price over a certain order quantity range

• This means this model differs from Model I because the acquisition cost (ac) may vary with the quantity ordered, i.e., it is not necessarily constant

• . . . more


• Under this condition, acquisition cost becomes an incremental cost and must be considered in the determination of the EOQ

• The total annual material costs (TMC) = Total annual stocking costs (TSC) + annual acquisition cost

TSC = (Q/2)C + (D/Q)S + (D)ac

• . . . more


To find the EOQ, the following procedure is used:

1. Compute the EOQ using the lowest acquisition cost. – If the resulting EOQ is feasible (the quantity can be

purchased at the acquisition cost used), this quantity is optimal and you are finished.

– If the resulting EOQ is not feasible, go to Step 22. Identify the next higher acquisition cost.


3. Compute the EOQ using the acquisition cost from Step 2.– If the resulting EOQ is feasible, go to Step 4.– Otherwise, go to Step 2.

4. Compute the TMC for the feasible EOQ (just found in Step 3) and its corresponding acquisition cost.

5. Compute the TMC for each of the lower acquisition costs using the minimum allowed order quantity for each cost.

6. The quantity with the lowest TMC is optimal.

Example: EOQ with Quantity Discounts

A-1 Auto Parts has a regional tire warehouse in Atlanta. One popular tire, the XRX75, has estimated demand of 25,000 next year. It costs A-1 $100 to place an order for the tires, and the annual carrying cost is 30% of the acquisition cost. The supplier quotes these prices for the tire:Q ac1 – 499 $21.60500 – 999 20.951,000 + 20.90


• Economical Order Quantity

This quantity is not feasible, so try ac = $20.95

This quantity is feasible, so there is no reason to try ac = $21.60

i iEOQ = 2DS/C

3EOQ = 2(25,000)100/(.3(20.90) = 893.00

2EOQ = 2(25,000)100/(.3(20.95) = 891.93


• Compare Total Annual Material Costs (TMCs)

TMC = (Q/2)C + (D/Q)S + (D)ac

Compute TMC for Q = 891.93 and ac = $20.95TMC2 = (891.93/2)(.3)(20.95) + (25,000/891.93)100+ (25,000)20.95 = 2,802.89 + 2,802.91 + 523,750 = $529,355.80… more


Compute TMC for Q = 1,000 and ac = $20.90TMC3 = (1,000/2)(.3)(20.90) + (25,000/1,000)100+ (25,000)20.90 = 3,135.00 + 2,500.00 + 522,500 = $528,135.00 (lower than TMC2)

The EOQ is 1,000 tiresat an acquisition cost of $20.90.

SAFETY STOCKIs required to be considered in some conditionsThey Arise becauseIn practical situation

Demand of items may fluctuate at any point of time

And also suppliers always need some lead time to supply the goods

Lead time can easily be provided to supplier by placing order before inventory become zero. e.g.Lead time is 10 days , so order can be placed 10 days before it becomes zero. Let the uniform consumption of inventory be 50 units per day therefore during the 10days of lead time 500 units will be consumed . Hence ROL can be fixed at 500 units.A Stock out may occur sometime due to either excessive consumption or due to undue stretching of lead time We know stock out is undesirable for the various reasons so to avoid it extra stock is maintained throughout thr year. This is called as Safety Stock

Inventory decreases at constant rate

First Order Second Order

Re order levelQ

0

Inventory Level Lead Time

500 units

Goods Received

Lead

Lead time being provided by fixing a reorder level

10days


First Order Second Order

Re order levelQ

0


500 units

Goods Received

Lead

Excessive consumption of inventory during the lead time, leading to stock out

10days

7 days


Second Order

Re order levelQ

0


500 units

Goods Received

Safety Stock avoids a stock out caused by excessive consumption of inventory during lead time

7 days

800 units Safety Stock

Let’s do one practical

• But let’s have one understanding • A low level of safety stock can lead to a stock

out. On the other hand a high level of safety stock unnecessarily ties up capital. Therefore we need to determine the optimum level of safety stock. Which should neither be low nor high.

Safety stock involves two types of costs: 1. CC carrying cost of safety stock2. SC stock out cost These cost are inversely related to each other CC safety stock ∞ 1/Stock out costHigh the safety stock high CC low the chance of SC.

Research shows that the total cost of safety stock is minimum only when

CC safety stock = SC safety stockLet’s take an example keeping normal lead time constant . Demand during lead time may vary leading to possibility of stock

out.

Practical- CASE of NESTLE A local chocolate distributor at

Ghaziabad deals with a popular brand called chocostick. The normal lead time taken by the supplier of chocostick is 10 days . The normal consumption of inventory during the lead time is 500 units per day. there are 10 inventory cycles per year. The CC is Rs.1 per unit per year. The stock out cost is Rs.2. per unit short. Mr. Ram the sales man gives you a consumption pattern of chocostick ( based on his past 100 observations)

Find the optimum level of safety stock for chocostick.

Consumption during lead time (units)

Probability

1000 .012000 .033000 .074000 .145000 .616000 .047000 .078000 .03

1.00

• Crux of the case:• See they have given normal lead time of supplier and also

consumption during the lead time that too at normal rate.• He had analyzed the pattern of the demand during lead time .• He understand one fact very clearly stock out will cost double of

the carrying cost.• So normal consumption of chocostick during the lead time = 10

days X 500 units/day=5000 units• As per observations out of 100 consumption during lead time is

5000 unit or less 86 times.(86 = 61+14+7+3+1)• Therefore if no safety stock is maintained there is an 86%

chance that a stock out will not happen.• But still the problem is HOW MUCH safety stock.

• Let’s understand the crux of the other part• CASE 1. No safety stock: then stock out will be there only when

DDLT exceeds 5000 units. i.e 6000/7000/8000 with stock out of 1000/2000/3000

• CASE 2. 1000 units of safety stock : then stock out at DDLT exceeds 6000 units

• CASE 3. 2000 units of safety stock : then stock out at DDLT exceeds 7000 units

• CASE 4. 3000 units of safety stock: then stock out will only when DDLT exceeds to 8000 units.

• So to arrive at the optimum safety stock you need to understand that at what level of safety stock your inventory cost i.e carrying as well as stock out cost will be minimum.

Lead time consumption leading to stock out

NO. of units short

Probability Expected annual stock out cost. (RS.)

Case1.6000 1000 .04 10X1000X.04X2 = 800

7000 2000 .07 10X2000X.07X2 = 2800

8000 3000 .03 10X3000X.03X2 = 1800 5400.00

Case 2.

7000 1000 .07 10X1000X.07X2 = 1400

8000 2000 .03 10X2000X.03X2 = 1200 2600.00

Case 3.

8000 1000 .03 10X1000X.03X2 = 600 600.00

ComparisonSafety stock levels (units)

CC of safety stock (Rs.)

Stock out Cost(Rs.)

Total safety Stock Cost(Rs.)

0 0 5400.00 5400.00

1000 1000.00 2600.00 3600.00

2000 2000.00 600.00 2600.00

3000 3000.00 0 3000.00

29

Safety Stock(other method)

Use safety stock to protect against stockouts when demand or lead time is not constant.(based on service level)

Safety stock = z x s’d

z is from Standard Normal Distribution Table and is based on P = Probability of being in-stock during lead time.

ROP = expected demand during lead time + safety stock = d x LT + z x s’d , S’d= Sd * square root of LT

Average Inventory Level (AIL) = regular stock + safety stockAIL = 2

Q + z x s’d

• monthly demand forecast(d) = 11,107 units• Standard deviation (sd) = 3,099 units• Lead time (Lt) = 1.5 months• Probability of available inventory =75%• Sol:• ROP= d*LT + Z(s’d)• s’d=sd L = 3,099 * =3,795• Z=0.67 from area under the standard normal distribution

• ROP=11107 *1.5 +(0.67*3,795)= 19,203

Periodic Review System

Figure 11.8 Periodic review system: units in stock versus time

Target Level orMaximum Inventory Level

• Total of the: demand (D) during the review period (R)+ demand (D) during the lead time (L)+ safety stock (SS)

T = D(R + L) + SS

Periodic Review - Order Quantity

• When it is time to place an order– Order the difference between the target level (T)

and the quantity on hand (I ).

Q = T - I

Periodic Review System

• Used when:– there are many small issues from stock e.g.

grocery stores– many items are ordered from one source– many items are ordered together to fill a truck or

a production run

Periodic Review - Example Problem

A harware company stocks nuts and bolts and orders them from a local supplier once every two weeks (10 working days). Lead time is two days. The company has determined that the average demand for 1/2 inch bolts is 150 per week (5 working days), and it wants to keep a safety stock of three day’s supply on hand. An order is to be placed this week, and stock on hand is 130 bolts.a. What is the target level?b. How many 1/2 inch bolts should be ordered this time?Let: D= demand per unit time = 150 / 5 = 30 per working day

L = lead time = 2 daysR = review period = 10 daysSS = safety stock = 3 day’s supply = 90 unitsI = inventory on hand = 130 units

Periodic Review - Example Problem Continued

Target level T = D(R + L) + SS = 30(10 + 2) + 90 = 450 units

Order quantity Q = T - I = 450 - 130 = 320 units

Place an order now for 320 units which will arrive in 2 days.

38

Pull Inventory Control - Repetitive Ordering

• For perpetual (continual) demand.• Treat each stocking point independently.• Consider 1 product art 1 location.

Reorder PeriodicDetermine: Point System Review SystemHow much to order: Q T-I When to (re)order: ROP t

eoq for production lots eoq with quantity discount safety stock periodic review system ----

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