INVENTORY THEORY

I N V E N TO RY P L A N N I N G A N D C O N T R O L

S C I E N T I F I C I N V E N TO RY M A N A G E M E N T

INVENTORY (STOCKS OF MATERIALS) Because there is difference in the timing or rate of supply and demand

What would happen if • Supply>Demand? •Demand>Supply?

INVENTORY IN INPUT-OUTPUT PROCESS

Δ𝑉Δ𝑡 = 𝐼 − 𝑟(𝑉

See MS Excel file…

Δ𝑉Δ𝑡 = 𝐼 − 𝑘

Model 1 (k is constant):

Model 2:

INVENTORY IN INPUT-OUTPUT PROCESS

Δ𝑉Δ𝑡 = 𝐼 − 𝑟(𝑉

Costs not yet included

Satisfy DemandMinimize VWhat is the value of I?

How much to order? When to order?

INVENTORY COSTS• Cost of placing the order (affects “when to order”)

• Price discount costs (affects “how much to order”)

• Stock-out or shortage costs (e.g., cost of backlogging and cancellation of order; affects customer satisfaction)

• Working capital costs (this is due to the lag between paying the suppliers and receiving payment from customers; a type of opportunity cost and liquidity risk cost)

• Storage costs

• Obsolescence costs (e.g., salvage cost, discounted sale, etc.)

• Operating inefficiency costs (large inventory prevents seeing operational inefficiency)

THERE ARE DIFFERENT KINDS OF INVENTORY SYSTEMS

EOQEconomic Order Quantity

Economic Lot-size

PRELIMINARIESAssumptions:

• One particular stock item in retail operation

• Q items are supplied per order and they arrive in one batch instantaneously at reorder point

• Demand is deterministic, predictable and fixed (at a rate D items per period of time)

• When stocks are all depleted, new batch of Q items instantaneously arrives (a continuous review)

“Period” here means a time of reference (e.g., 1 year, 1 month, 1 day, etc.). This DOES NOT pertain to the time interval between orders or deliveries.

LEAD TIME between ordering and delivery = 0

Continuous review: an order is placed as soon as the stock level falls down to the prescribed level

Periodic review: inventory is checked at discrete scheduled intervals even if inventory level dips below prescribed level before the review schedule

Assumptions:

• One particular stock item in retail operation

• Q items are supplied per order and they arrive in one batch instantaneously at reorder point

• Demand is deterministic, predictable and fixed (at a rate D items per period of time)

• When stocks are all depleted, new batch of Q items instantaneously arrives (a continuous review) INVENTORY PROFILE (CHART)

Notes:

• Average inventory = Q/2 because two shaded areas are equal

• Time interval between deliveries = Q/D

• Frequency of deliveries per period = D/Q

WE CAN HAVE DIFFERENT INVENTORY PLANS

METHOD #1: EOQ MODELECONOMIC ORDER QUANTITY

• Most common approach in deciding how much and when to order a particular item (with replenishing)• Finds the best balance between advantages and disadvantages

of holding stock • Total cost of stocking the item includes Holding Costs and

Order Costs • Holding Costs (per period) may include working capital costs,

storage costs, insurance cost and obsolescence costs• Order Costs may include administrative cost of placing the

order, transportation cost of supplies and price discount costs

METHOD #1: EOQ MODELECONOMIC ORDER QUANTITY

• Holding Costs = holding cost per item x average inventory

• Order Costs = cost per order x number of orders per period + fixed administrative costØPrice discount cost not included

𝐻𝑜𝑙𝑑𝑖𝑛𝑔𝑐𝑜𝑠𝑡𝑠 = 𝐶5𝑄2

𝑂𝑟𝑑𝑒𝑟𝑐𝑜𝑠𝑡𝑠 = 𝐶(𝐷𝑄 + 𝐾

METHOD #1: EOQ MODELECONOMIC ORDER QUANTITY

𝑇𝑜𝑡𝑎𝑙𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦𝑐𝑜𝑠𝑡𝑠 𝐶A = 𝐶5𝑄2 + 𝐶(

𝐷𝑄 + 𝐾

Q=200 is the EOQ

METHOD #1: EOQ MODELECONOMIC ORDER QUANTITY

𝑇𝑜𝑡𝑎𝑙𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦𝑐𝑜𝑠𝑡𝑠 𝐶A = 𝐶5𝑄2 + 𝐶(

𝐷𝑄 + 𝐾

𝑑𝐶A𝑑𝑄 =

𝐶52 +

−𝐶(𝐷𝑄B

𝐶52 −

𝐶(𝐷𝑄∗B

= 0

Finding the EOQ (Q*)

𝑄∗ =2𝐶(𝐷𝐶5

�

METHOD #1: EOQ MODELECONOMIC ORDER QUANTITYRemarks:When using the EOQ, the optimal inventory policy is• Order quantity = Q*• Time between orders = Q*/D

ØThis is also the time interval between deliveries

• Order frequency = D/Q* per periodØ This is also the frequency of deliveries

per period 𝑄∗ =2𝐶(𝐷𝐶5

�

METHOD #1: EOQ MODELSENSITIVITY OF EOQ

Small errors (e.g., round-off error and estimation error) may still give near-optimal solution

METHOD #1: EOQ MODELEXAMPLE

A building materials supplier obtains its bagged cement from a single supplier. Demand is reasonably constant throughout the year, and last year the company sold 2 million kg. of this product. It estimates the costs of placing an order at around PhP1,700 each time an order is placed, and calculates that the annual cost of holding inventory is 20 per cent of purchase cost. The company purchases the cement at PhP4,000 per 1,000 kg.. How much should the company order at a time?Also, compare computed EOQ with a more convenient order quantity.

METHOD #1: EOQ MODELWITH QUANTITY DISCOUNTS

• Order Costs = cost per order x number of orders per period + fixed administrative cost

𝑂𝑟𝑑𝑒𝑟𝑐𝑜𝑠𝑡𝑠F = 𝐶(,F𝐷𝑄 + 𝐾, 𝑖 = 1,2, … ,𝑚

𝐶𝑜𝑠𝑡𝑝𝑒𝑟𝑜𝑟𝑑𝑒𝑟 =

𝐶(,L 𝑖𝑓𝑄 < 𝑞L𝐶(,B 𝑖𝑓𝑞L ≤ 𝑄 < 𝑞B⋮ ⋮

𝐶(,R 𝑖𝑓𝑄 ≥ 𝑞R

METHOD #1: EOQ MODELECONOMIC ORDER QUANTITY

𝑇𝑜𝑡𝑎𝑙𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦𝑐𝑜𝑠𝑡𝑠 𝐶A,F = 𝐶5𝑄F∗

2 + 𝐶(,F𝐷𝑄F∗

+ 𝐾

𝑄F∗ =2𝐶(,F𝐷𝐶5

�

𝑖 = 1,2, … ,𝑚

Compare the computed Ct,i’s. Choose the Qi* that gives minimum total inventory costs.

EBQEconomic Batch Quantity

Economic Manufacturing Quantity

Production Order Quantity

METHOD #2: EBQ MODELECONOMIC BATCH QUANTITY

In EOQ, order (Q items) arrives at one point in time, i.e., one lot or one batch).

In EBQ, we have similar assumptions as EOQ except that order (Q items) arrives in more than one batch, i.e., we do gradual replacement.

METHOD #2: EBQ MODELECONOMIC BATCH QUANTITY

Batches arrive (% of Q items) for some period of time at rate P, and at the same time demand is removing stocks in the inventory at a rate D.

METHOD #2: EBQ MODELECONOMIC BATCH QUANTITY

If P>D, inventory is increasing.If P<D, such as when all Q items already arrived, inventory decreases.

METHOD #2: EBQ MODELECONOMIC BATCH QUANTITY

Maximum stock level = MSlope inventory build-up = P – D (derivation: use rates)Slope inventory build-up = M➗ Q/P (derivation: use slope)

METHOD #2: EBQ MODELECONOMIC BATCH QUANTITYSlope inventory build-up = P – D

Slope inventory build-up = M➗ Q/P

𝑀𝑃𝑄 = 𝑃 − 𝐷

𝑀 =𝑃 − 𝐷 𝑄𝑃

Averageinventory:𝑀2 =

𝑃 − 𝐷 𝑄2𝑃

METHOD #2: EBQ MODELECONOMIC BATCH QUANTITYLike in EOQ: Total inventory costs = holding costs + order costs

𝑇𝑜𝑡𝑎𝑙𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦𝑐𝑜𝑠𝑡𝑠 𝐶A = 𝐶5𝑀2 + 𝐶(

𝐷𝑄 + 𝐾

𝑇𝑜𝑡𝑎𝑙𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦𝑐𝑜𝑠𝑡𝑠 𝐶A = 𝐶5𝑃 − 𝐷 𝑄2𝑃 + 𝐶(

𝐷𝑄 + 𝐾

METHOD #2: EBQ MODELECONOMIC BATCH QUANTITY𝑇𝑜𝑡𝑎𝑙𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦𝑐𝑜𝑠𝑡𝑠 𝐶A = 𝐶5

𝑃 − 𝐷 𝑄2𝑃 + 𝐶(

𝐷𝑄 + 𝐾

Finding the EBQ (Q*)

𝑄∗ =2𝐶(𝐷

𝐶5 1 − 𝐷/𝑃�

𝐶5𝑃 − 𝐷2𝑃 − 𝐶(

𝐷𝑄∗B

= 0

𝑑𝐶A𝑑𝑄 = 𝐶5

𝑃 − 𝐷2𝑃 − 𝐶(

𝐷𝑄B

Note: P>D

METHOD #2: EBQ MODELEXAMPLE

The manager of a bottle-filling plant which bottles soft drinks needs to decide how long a ‘run’ of each type of drink to process. Demand for each type of drink is reasonably constant at 80,000 per month (a month has 160 production hours). The bottling lines fill at a rate of 3,000 bottles per hour, but take an hour to clean and reset between different drinks. The cost (of labor and lost production capacity) of each of these changeovers has been calculated at PhP7,000 per hour. Stock holding costs are counted at PhP7 per bottle per month.

METHOD #2: EBQ MODELEXAMPLE

The staff who operate the lines have devised a method of reducing the changeover time from 1 hour to 30 minutes. How would that change the EBQ?

PROBLEMS WITH EOQ AND EBQ:• The model and solution are as good as the assumptions!

The assumptions are over-simplistic. Factors in inventory management are generally NOT

– deterministic and fixed– linear– instantaneous– accurately known– zero-lead-time

• For products that deteriorates or go out of fashion, a periodic review is more likely to be used.

PROBLEMS WITH EOQ AND EBQ:• EOQ and EBQ methods are usually descriptive and not

prescriptive. However, sometimes they can be good approximates of reality.

• From the perspective of Japanese-inspired “lean” and Just-In-Time (JIT) philosophies: EOQ and EBQ methods are reactive approaches. They fail to ask the right question: “How can I change the operation in some way to reduce the overall level of inventory?”

• Inventory costs minimization is not the only objective for inventory management.