How can it be that mathematics, being after all a product of

human thought independent of

experience, is so admirably adapted to the objects of reality

Albert Einstein

A typical hospital spends about 20% of its budget on medical, surgical, and pharmaceutical supplies. For all hospitals it adds up to $150 billion annually.

The average inventory in US economy about $1.13 trillion on $9.66 trillion of sales. About $430 billion in manufacturing, $230 billion in wholesaler, $411 billion in retail.

What happens when a company with a large Work In Process (WIP) and Finished Goods (FG) inventory finds a market demand shift to a new product? Two choices:

Fire-sell all WIP and FG inventories and then quickly introduce the new product Significant losses

Finish all WIP inventory and sell all output before introducing the new product Delay and reduced market response time

Importance of Inventory

Inputs inventory– Raw materials and Parts

In-process inventory– Parts and products that are being processed– Parts and products to decouple operations (line balancing inventory).– Parts and products to take advantage of Economies of Scale (batch

inventory).

Outputs inventory– To meet anticipated customer demand (average inventory and safety

stock).– To smooth production while meeting seasonal demand (seasonal

inventory). – In transit to a final destination to fill the gap between production and

demand lead times (pipeline inventory).

Inventory Classified

Poor inventory management hampers operations, diminishes customer satisfaction, and increases operating costs.

A typical firm probably has tied in inventories about – 30 percent of its Current Assets – 90 percent of its Working Capital (Current Assets – Current

Liabilities)

Understocking; lost sales, dissatisfied customers.

Overstocking; tied up funds (financial costs), storage and safe keeping (physical cost), change in customer preferences (obsolescence cost).

Inventory

At the beginning of each period, the existing inventory level is identified and the additional required volume to satisfy the demand during the period is ordered.

The quantity of order is variable but the timing of order is fixed.

Re-Order Point (ROP) is defined in terms of time.

Periodic Inventory [Counting] Systems

One-Bin System (Periodic)

Order Enough to Refill Bin

Physical count of items made at periodic intervals.

Disadvantage: no information on inventory between two counts.

Advantage: order for several items are made at the same time.

When inventory reaches ROP an order of EOQ (Economic Order Quantity) units is placed.

The quantity of order is fixed but the timing of order is variable.

ROP is defined in terms of quantity (inventory on hand).

Perpetual Inventory Systems

Two-Bin System (Perpetual)

Full Empty

Order One Bin of Inventory

Keeps track of removals from inventory continuously, thus monitoring current levels of each item.

A point-of-sales (POS) system record items at the time of sale.

ABC Analysis in terms of dollars invested, profit potential, sales or usage volume, and stockout penalties. Perpetual for class A, Periodic for class C.

A classification Approach: ABC Analysis

Item Annual Unit Annual Number Demand Cost $ Value

1 2500 330 8250002 1000 70 700003 1900 500 9500004 1500 100 1500005 3900 700 27300006 1000 915 9150007 200 210 420008 1000 4000 40000009 8000 10 8000010 9000 2 1800011 500 200 10000012 400 300 120000

Item Annual Unit Annual % of Total ClassificationNumber Demand Cost $ Value

8 1000 4000 4000000 A5 3900 700 2730000 67% A3 1900 500 950000 B6 1000 915 915000 B1 2500 330 825000 27% B4 1500 100 150000 C12 400 300 120000 C11 500 200 100000 C9 8000 10 80000 C2 1000 70 70000 C7 200 210 42000 C10 9000 2 18000 6% C

Group A: PerpetualGroup C: Periodic

The Basic Inventory Model: Economic Order Quantity

Only one productDemand is known and is constant throughout the year Each order is received in a single delivery Lead time does not vary

-Two costs Ordering Costs: Costs of ordering and receiving the order

Holding or Carrying Costs: Cost to carry an item in inventory for one year

Unit cost of product is not incorporated because we assume it is fixed. It does not depends on the ordering policy.

The Basic Inventory Model

Annual demand for a product is 9600 units.D = 9600

Annual carrying cost per unit of product is $16.H = 16

Ordering cost per order is $75. S = 75

a) How much should we order each time to minimize our total cost?

b) How many times should we order?c) What is the length of an order cycle (288 working

days/year)?d) What is the total cost?

Do NOT worry if you do not get integer numbers.

Ordering Cost

D = Demand in units / year Q = Order quantity in units / order

Q

D

Q

DS

Number of orders / year =

S = Order cost / order

Annual order cost =

Annual Ordering CostOrder Size Number of Orders Ordering Cost

50 192 14400100 96 7200150 64 4800200 48 3600250 38.4 2880300 32 2400350 27.4 2057400 24 1800450 21.3 1600500 19.2 1440550 17.5 1309600 16 1200650 14.8 1108700 13.7 1029750 12.8 960800 12 900850 11.3 847900 10.7 800

Annual Ordering CostOrder Size Number of Orders Ordering Cost

50 192 14400100 96 7200150 64 4800200 48 3600250 38.4 2880300 32 2400350 27.4 2057400 24 1800450 21.3 1600500 19.2 1440550 17.5 1309600 16 1200650 14.8 1108700 13.7 1029750 12.8 960800 12 900850 11.3 847900 10.7 800

0

2000

4000

6000

8000

10000

12000

14000

16000

0 100 200 300 400 500 600 700 800 900 1000

Order Size

Ord

erin

g C

ost

Q

DS

Time

Inve

ntor

y

The Inventory Cycle

Receive order

Quantityon hand Usage

rate

When the quantity on hand is just sufficient to satisfy demand in lead time, an order for EOQ is placedAt the instant that the inventory on hand falls to zero, the order will be received (Screencam tutorial on DVD)

Inve

ntor

y

The Inventory Cycle

Q = Order quantityAt the beginning of the period we get Q units.At the end of the period we have 0 units.

Q

0

Q/222

0 QQ

Average Inventory / Period & Average Inventory / year

Time

Time

This is average inventory / period.Average inventory / period is also known as Cycle Inventory

What is average inventory / year ?

Inventory Carrying Cost

Q = Order quantity in units / order

2

Q

2

QH

Average inventory / year =

H = Inventory carrying cost / unit / year

Annual carrying cost =

Annual Carrying Cost Order Size Average Inventory Carrying Cost

50 25 400100 50 800150 75 1200200 100 1600250 125 2000300 150 2400350 175 2800400 200 3200450 225 3600500 250 4000550 275 4400600 300 4800650 325 5200700 350 5600750 375 6000800 400 6400850 425 6800900 450 7200

0

1000

2000

3000

4000

5000

6000

7000

8000

0 100 200 300 400 500 600 700 800 900 1000

Order Size

Car

rin

g C

ost

2

QH

Total Cost

Order Size Number of Orders Ordering Cost Average Inventory Carrying Cost Total Ord&Carr. Cost50 192 14400 25 400 14800

100 96 7200 50 800 8000150 64 4800 75 1200 6000200 48 3600 100 1600 5200250 38.4 2880 125 2000 4880300 32 2400 150 2400 4800350 27.4 2057 175 2800 4857400 24 1800 200 3200 5000450 21.3 1600 225 3600 5200500 19.2 1440 250 4000 5440550 17.5 1309 275 4400 5709600 16 1200 300 4800 6000650 14.8 1108 325 5200 6308700 13.7 1029 350 5600 6629750 12.8 960 375 6000 6960800 12 900 400 6400 7300850 11.3 847 425 6800 7647900 10.7 800 450 7200 8000

0

1000

2000

3000

4000

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7000

8000

0 100 200 300 400 500 600 700 800 900 1000

Order Size

0

2000

4000

6000

8000

10000

12000

14000

16000

0 200 400 600 800 1000

Ordering Cost

Carrying Cost

Total Ord&Carr. Cost

EOQ is at the intersection of the two costs.

(Q/2)H = (D/Q)S

Q is the only unknown. If we solve it

EOQ

SQDHQTC )/()2/(

EOQ = 2DS

H =

2(Annual Demand )(Order or Setup Cost )Annual Holding Cost

Back to the Original Questions

Annual demand for a product is 9600 units.D = 9600

Annual carrying cost per unit of product is $16.H = 16

Ordering cost per order is $75. S = 75

a) How much should we order each time to minimize our total cost?b) How many times should we order?c) What is the length of an order cycle (288 working days/year)?d) What is the total cost?

What is the Optimal Order Quantity

D = 9600, H = 16, S = 75

H

DSEOQ

2

30016

)75)(9600(2EOQ

How Many Times Should We Order?

Annual demand for a product is 9600 units.D = 9600

Economic Order Quantity is 300 units.EOQ = 300

Each time we order EOQ.

How many times should we order per year?

D/EOQ

9600/300 = 32

What is the Length of an Order Cycle?

Working Days = 288/year

9600 units are required for 288 days.

300 units is enough for how many days?

(300/9600)×(288) = 9 days

What is the Optimal Total Cost

SQDHQTC )/()2/(

75)300/9600(16)2/300( TC

4800TC

24002400 TC

The economic order quantity is 300.

The total cost of any policy is computed as:

This is optimal policy that minimizes total cost.

Centura Health Hospital processes a demand of 31200 units of IV starter kits each year (D=31200), and places an order of 6000 units at a time (Q=6000). There is a cost of $130 each time an order is placed (S = $130). Inventory carrying cost is $0.90 per unit per year (H = $0.90). Assume 52 weeks per year.

What is the average inventory?Average inventory = Q/2 = 6000/2 = 3000

What is the total annual carrying cost?Carrying cost = H(Q/2) = 0.9×3000=2700

How many times do we order?31200/6000 = 5.2

What is total annual ordering cost?Total ordering cost = S(D/Q)Ordering cost = 130(5.2) = $676

Centura Health Hospital

A toy manufacturer uses approximately 32000 silicon chips annually. The Chips are used at a steady rate during the 240 days a year that the plant operates. Annual holding cost is 60 cents per chip, and ordering cost is $24. Determine the following:

a) How much should we order each time to minimize our total cost?

b) How many times should we order?c) what is the length of an order cycle (working days 240/year)?d) What is the total cost?

Assignment 12a.1

Victor sells a line of upscale evening dresses in his boutique. He charges $300 per dress, and sales average 30 dresses per week. Currently, Vector orders 10 week supply at a time from the manufacturer. He pays $150 per dress, and it takes two weeks to receive each delivery. Victor estimates his administrative cost of placing each order at 225. His inventory charring cost including cost of capital, storage, and obsolescence is 20% of the purchasing cost. Assume 52 weeks per year.

a) Compute Vector’s total annual cost of inventory system (carrying plus ordering but excluding purchasing) under the current ordering policy?

b) Without any EOQ computation, is this the optimal policy? Why?c) Compute Vector’s total annual cost of inventory system (carrying plus ordering but

excluding purchasing) under the optimal ordering policy? d) What is the ordering interval under optimal ordering policy?e) What is average inventory and inventory turns under optimal ordering policy?

Inventory turn = Demand divided by average inventory. Average inventory = Max Inventory divided by 2. Average inventory is the same as cycle inventory.

Assignment 12a.2

Complete Computer (CC) is a retailer of computer equipment in Minneapolis with four retail outlets. Currently each outlet manages its ordering independently. Demand at each retail outlet averages 4,000 per week. Each unit of product costs $200, and CC has a holding cost of 20% of the product cost per annum. The fixed cost of each order (administrative plus transportation) is $900. Assume 50 weeks per year. The holding cost will be the same in both decentralized and centralized ordering systems. The ordering cost in the centralized ordering is twice of the decentralized ordering system.

Decentralized ordering: If each outlet orders individually.Centralized ordering: If all outlets order together as a single order.

a) Compute EOQ in decentralized orderingb) Compute the cycle inventory for one outlet and for all outlets. c) Compute EOQ in the centralized orderingd) Compute the cycle inventory for all outlets and for one outlete) Compute the total holding cost + ordering cost (not including purchasing cost) for the

decentralized policyf) Compute the total holding cost plus ordering cost for the centralized policy

Assignment 12a.3