Managerial Decision Modeling with Spreadsheets

Chapter 12

Inventory Control Models

Learning Objectives• Understand importance of inventory control.• Use Economic Order Quantity (EOQ) model to

determine how much to order.• Compute reorder point (ROPROP) in determining when to

order more inventory.• Use EOQ with non-instantaneousnon-instantaneous receipt model to

determine how much to order or produce.• Handle EOQ problems that allow quantity discountsallow quantity discounts.• Understand use of safety stocksafety stock with known and

unknown stockout costs.• Understand importance of ABC ABC inventory analysis.

 12.1 Introduction• Inventory is one of most expensive and important

assets to many companies. • Managers have long recognized good inventory

control is crucial.

12.2 Importance Of Inventory Control

• Inventory control serves several important functions

and adds flexibility to firm’s operations.

• Five main uses of inventory are:

  1.  Decoupling function.

2.  Storing resources.

3.  Irregular supply and demand.

4.  Quantity discounts.

5.  Avoiding stockouts and shortages.

12.3 Inventory Control Decisions

• Fundamental decisions made when controlling

inventory:

  1.      How much to order

2.      When to order

•  Inventory model is utilized to determine how much

to order and when to order.

12.3 Inventory Control Decisions

• Major objective in controlling inventory is to

minimize total inventory costs:

1.      Cost of items.

2.      Cost of ordering.       

3.      Cost of carrying, or holding, inventory.

4.      Cost of stockouts.

5.      Cost of safety stock, additional inventory held

to help avoid stockouts.

Inventory Cost Factors

12.4 Economic Order Quantity (EOQ): Determining How Much To Order

Economic Order Quantity Model Assumptions:1. Demand is known and constant.2. Lead time - time between placement of and receipt of

the order is known and constant.3. Receipt of inventory is instantaneous. 4. Quantity discounts are not possible.5. Only variable costs are cost of placing an order,

ordering cost, and cost of holding or storing inventory over time, holding or carrying cost.

6. If orders are placed at right time, stockouts or shortages can be avoided completely.

Inventory Usage Over Time

Ordering and Inventory Costs

• Objective of inventory models is to minimize total to minimize total

costs. costs.

• With assumptions given, significantsignificant costs are

ordering cost and carrying cost.

Ordering and Inventory Costs

Total Cost as Function of Order Quantity

Ordering and Inventory Costs• Average inventory on hand is: 

Average inventory level = ( 0 + Q ) / 2 = Q / 2 • Other inventory parameters are:

Q* = Optimal order quantity (i.e., EOQ).

D = Annual demand in units for inventory item.

Co = Ordering cost per order.

Ch = Holding or carrying cost per unit per year.

P = Purchase cost per unit of inventory item.• Holding cost could be constant or calculated as cost of

capital:

Ch = I x P

Inventory Costs and EOQ

Total ordering cost = ( D / Q ) x Co

Total carrying cost = ( Q / 2 ) x Ch

Total cost = Total ordering cost + Total carrying cost

+ Total purchase cost

= ( D / Q ) x Co + ( Q / 2 ) x Ch + P x D

Economic Order Quantity is:* (2 /o hEOQ DC C

Plot of Costs Versus Order Quantity Sumco Pump Company

12.5 Reorder Point: Determining When To Order

• Second inventory question is when to order. • Time between placing and receipt of an order, called

lead time or delivery time, is often few days or few weeks.

• When to order decision is usually expressed in terms of reorder pointreorder point (ROP), (ROP), inventory level at which an order should be placed.

• Reorder point, ROP, is given as: ROP = (demand per day) x (lead time in days)

= d x L • Demand, dd, expressed in units demanded per day and

lead time, LL, expressed in days.

Reorder Point Curve

Reorder Point

• Recall EOQ = 200 and total cost of $5,100.

• Calculations based on annual demand of 1,000 units, ordering cost of $10 per order, annual carrying cost of $0.50 per unit, and purchase cost of $5 per pump housing.

• Assume lead time of 3 business days between time firm places an order and time order is received.

• Assume there are 250 business days in year.

• To calculate reorder point, first determine daily demand rate, d.

• Since there are 250 business days in year and annual demand is 1,000, daily demand rate is 4 (= 1,000 / 250) pump housings.

Sumco Pump Company

12.6 EOQ With Non-instantaneous Receipt

• Firm may build up inventory gradually over period of

time.

• Example, firm may receive shipments from suppliers

uniformly over period of time.

• Or firm may be producing at rate of p per day and

simultaneously selling at rate of d per day

(where (where pp > > dd).).

Average inventory levelAverage inventory level = [ 0 + Q ( 1 - d / p )] / 2 =

= = Q Q ( 1 - ( 1 - d d / / p p ) / 2) / 2

12.6 EOQ With Non-instantaneous Receipt • Or firm may be producing at rate of p per day and

simultaneously selling at rate of d per day (where p > d). Avg. inventory level = Q (1 - d / p) / 2

Finding Economic Production Quantity • Parameters are:Q* = Optimal order or production quantity (EPQ)

Cs = Setup cost per setup•  For given order quantity Q:

Total setup cost = ( D / Q ) x Cs

Total carrying cost = [ Q ( 1- d / p) / 2 ] x Ch

Total cost = Total setup cost + Total carrying cost + Total production cost

= (D / Q) x Cs + [Q (1- d / p) / 2] x Ch + P x D Calculate EPQ as: 

* 2

(1 / )s

h

DCQ

C d p

Brown Manufacturing Example• Produces mini-sized refrigeration packs in batches.

• Estimated demand for year is 10,000 units.

• Operates for 167 business days each year.

• Annual demand translates to daily demand rate of 60 units per day.

• It costs about $100 to set up manufacturing process, and carrying cost is $0.50 per unit per year.

• When production process has been set up, 80 refrigeration packs can be manufactured daily.

• Each pack costs $5 to produce.

• How many packs should Brown produce in each batch?  

EPQ Model

• Calculates and reports EPQ as well as following output measures:– maximum inventory (= Q*[1- d /p])– average inventory (= Q*[1- d /p] / 2) – number of setups (= D / Q*)

– total holding cost (= Ch x Q*[1- d /p] / 2)

– total setup cost (= Cs x D / Q*)

– total purchase cost (= P x D)

– total cost (= Ch x Q*[1- d /p] / 2 + Cs x D /Q* + P x

D)

Brown Manufacturing

EPQ Model

• Total setup cost = total carrying cost ($250 each).

• EPQ: Q* = 4,000 units.

• Total cost, including production cost of $50,000, is

$50,500.

Brown Manufacturing

Inventory Costs PlotBrown Manufacturing

12.7 Quantity Discount Models

• To increase sales, companies offer quantity discounts

to customers.

• Quantity discount is simply reduced cost for item when

purchased in larger quantities.

• It is common to have discount schedule with several

discounts for large orders.

Total cost = Total ordering cost + Total carrying cost

Total purchase cost = (D/Q) x Co + (Q/2) x Ch + P x D

• Find EOQ that incorporates cost with discount to

minimize total cost.

12.7 Quantity Discount Models• Find EOQ that incorporates cost with discount to minimize

total cost.

Four Steps to Analyze Quantity Discount Models

1. For each discount price, calculate a Q* value using EOQ formula.

2. For any discount level, if Q* computed in Step 1 is too low to qualify for discount, adjust Q* upward to lowest quantity that qualifies for discount.

3. Using total cost equation,compute total cost for every Q* determined in steps 1 and 2.

4. If Q* had to be adjusted upward because it was below allowable quantity range, be sure to use adjusted Q* value.

5. Select Q* with lowest cost as computed in Step 3.

6. It will be order quantity to minimize total cost.

Total Cost Curve for Quantity Discount Model

Brass Department Store Example• Stocks toy cars. • Store given quantity discount schedule for cars as

shown in Table 12.2. • Normal cost for cars is $5. • For orders between 1,000 and 1,999 units, unit cost is

$4.80, and for orders of 2,000 or more units, unit cost is $4.75.

• Ordering cost is $49 per order, annual demand is 5,000 race cars, and inventory carrying charge as percentage of cost, I, is 20% or 0.2.

• What order quantity will minimize total cost?

Plot of Total Cost Versus Order Quantity Brass Department Store

12.8 Use Of Safety Stock

12.9 ABC Analysis • Recognizes fact some inventory items are more

important than others. • Purpose of analysis is to divide all of company's

inventory items into three groups: A, B, and C. • Depending on group, decide how inventory levels

should be controlled.

Silicon Chips, Inc. Example

• Maker of super-fast DRAM chips, has organized its

10 inventory items on an annual dollar-volume basis.

• Parts are identified by item number, part number,

annual demands, and unit costs.

• How should company classify items into groups A, B,

and C?

Silicon Chips, Inc. Example• How should company classify items into groups A, B, and C?

Summary• Focus was to answer two questions in inventory

planning: (1) how much to order, and

(2) when to order. • EOQ makes a number of assumptions:

(1) known and constant demand and lead times.

(2) instantaneous receipt of inventory.

(3) no quantity discounts.

(4) no stockouts or shortages.

(5) only variable costs are ordering and carrying

costs.

Summary• If assumptions do not hold, more complex models are

needed:

(1) economic production quantity.

(2) quantity discount models.

• Discussed computation of safety stocks when demand during lead time was unknown for two cases:

(1) cost of stockout is known.

(2) cost of stockout is unknown.

• Presented ABC analysis to determine how inventory items should be classified based on their importance and value.