Inventory Management 2 DSC 335 Zhibin Yang Assistant Professor, Decision Sciences

Review – EOQ Model Three types of cost in EOQ model The objective: to find order quantity, Q, that minimizes total annual inventory holding and ordering (setup) costs Five assumptions Demand rate is constant and known with certainty No constraints are placed on the size of each lot The only two relevant costs are the inventory holding cost and the fixed cost per lot for ordering or setup Decisions for one item can be made independently of decisions for other items The lead time is constant and known with certainty

Three types of cost in EOQ model

The objective: to find order quantity, Q, that minimizes total annual inventory holding and ordering (setup) costs

Five assumptions

Demand rate is constant and known with certainty

No constraints are placed on the size of each lot

The only two relevant costs are the inventory holding cost and the fixed cost per lot for ordering or setup

Decisions for one item can be made independently of decisions for other items

The lead time is constant and known with certainty

(cont’d) EOQ model is insensitive to calculation error Because the total (holding and setup) cost curve is flat at the bottom At EOQ, holding cost = setup cost Sensitivity to demand, setup cost and holding cost When there is a order lead time We place an order of Q when inventory level drops to ROP ROP = d*L

EOQ model is insensitive to calculation error

Because the total (holding and setup) cost curve is flat at the bottom

At EOQ, holding cost = setup cost

Sensitivity to demand, setup cost and holding cost

When there is a order lead time

We place an order of Q when inventory level drops to ROP

ROP = d*L

Exercise - EOQ A cloth item is held in stock at a retail store Variable cost, c = $0.1 per yard; Inventory holding cost, H = $0.75 per yard/yr; Ordering (setup) cost: S= $150 per order Annual demand: D = 10,000 yards/yr. Demand lead time: L=10 days Note: There are 311 operating days per year for the store Calculate EOQ, annual total cost TC (variable, holding and setup), time between two orders (TOB), and reorder point (ROP)

A cloth item is held in stock at a retail store

Variable cost, c = $0.1 per yard;

Inventory holding cost, H = $0.75 per yard/yr;

Ordering (setup) cost: S= $150 per order

Annual demand: D = 10,000 yards/yr.

Demand lead time: L=10 days

Note: There are 311 operating days per year for the store

Calculate EOQ, annual total cost TC (variable, holding and setup), time between two orders (TOB), and reorder point (ROP)

Solution

Outline of Inventory Management 2 Continuous review systems Periodic review systems ABC analysis

Continuous review systems

Periodic review systems

ABC analysis

Two types of Inventory Control Systems Continuous review system ( Q -system) When to order: when inventory declines to ROP Event-trigger restocking Also known as: Reorder Point (ROP) system How much: a fixed quantity is ordered every time EOQ is a continuous review system with uncertain demand Periodic review system ( P -system) When to order: an order is placed after a fixed period of time Time-triggered restocking How much: An order of variable amount

Continuous review system ( Q -system)

When to order: when inventory declines to ROP

Event-trigger restocking

Also known as: Reorder Point (ROP) system

How much: a fixed quantity is ordered every time

EOQ is a continuous review system with uncertain demand

Periodic review system ( P -system)

When to order: an order is placed after a fixed period of time

Time-triggered restocking

How much: An order of variable amount

Continuous Review, Certain Demand – EOQ Slope = D (units/yr) = d (units/day) Q Time Reorder Point (ROP) Receive order Place order Receive order Lead time: L (days) Reorder Point: ROP = d  L Inventory

When Demand is Uncertain Average demand rate:  d Is ROP =  d  L good enough? Time 0 Stockout!  d  L Inventory Stockout may occur, only during delivery lead time!!! Place order Receive order L Place order L Receive order slope  d slope  d

Average demand rate:  d

Is ROP =  d  L good enough?

Adding Cushion – Safety Stock (SS) Time 0 d  L Inventory L L Increase ROP by SS SS ROP ROP = d  L + SS

Safety Stock (SS) When demand has unpredictable variability, stockout occur when actual demand during lead time exceeds ROP Safety stock is held to cushion against uncertainties ROP = average demand during lead time + safety stock What determines safety stock? Service level= Probability of no stockout during lead time = 1 – (Probability of stockout during lead time) Variability of the demand (rate) Lead time

When demand has unpredictable variability, stockout occur when actual demand during lead time exceeds ROP

Safety stock is held to cushion against uncertainties

ROP = average demand during lead time + safety stock

What determines safety stock?

Service level= Probability of no stockout during lead time = 1 – (Probability of stockout during lead time)

Variability of the demand (rate)

Lead time

What Determines Safety Stock? Probability of stockout during lead time Probability of no stockout in lead time (= service level) Demand during lead time Probability distribution of demand during lead time 0 Distribution of demand of higher variability d  L SS ROP

What Determines Safety Stock? d  L SS Inventory ROP L L Service Level Distribution of demand during lead time Increased demand variability

Compute ROP for a Given Service Level Service level Demand during lead time Probability distribution of demand during lead time 0 SS = z  dLT ROP =  d L + z  dLT If demand has a normal distribution d  L SS ROP

Compute StdDev of Lead Time Demand, σ dLT  dLT =  d  L Lead time L = 2 Two-day demand __  dLT = 3  2 Demand Daily demand variability  d Lead time L days One-day demand   d = 3  d  d  dLT  dLT _ d=10 20

“ z ” Value for a Service Level using Normal Distribution Table The area under standard normal distribution function represents the service level . Find the corresponding “ z ” value for the given service level. Example: a service level of 0.95 has a “ z ” value of 1.6 + (0.4+0.5)/2 = 1.645 Service level

Sensitivity Analysis: Safety Stock Safety stock increases (decreases) with an increase in: demand (per day) variability or forecast error service level delivery lead time

Safety stock increases (decreases) with an increase in:

demand (per day) variability or forecast error

service level

delivery lead time

Example: Inventory of Bird-feeders A museum of natural history opened a gift shop which operates 52 weeks per year. Managing inventories has become a problem. Top-selling SKU is a bird feeder . The average sales ( d ) are 18 units per week with a standard deviation of 5 units. The supplier charges $60 per unit ( c ). Ordering cost ( S ) is $45. Annual holding cost is 25% ( i ) of a feeder’s value. The lead time is constant at two (2) weeks. Management wants a 90% service level Determine the order quantity (Q), safety stock (SS) and reorder point (ROP)

A museum of natural history opened a gift shop which operates 52 weeks per year. Managing inventories has become a problem. Top-selling SKU is a bird feeder .

The average sales ( d ) are 18 units per week with a standard deviation of 5 units.

The supplier charges $60 per unit ( c ).

Ordering cost ( S ) is $45.

Annual holding cost is 25% ( i ) of a feeder’s value.

The lead time is constant at two (2) weeks.

Management wants a 90% service level

Determine the order quantity (Q), safety stock (SS) and reorder point (ROP)

Solution D= 18*52/year, S=$45/order, H=i*c=25%*60=$15/year Q = EOQ = 75 units σ d = 5,  d = 18 units/week, and L = 2 weeks σ dLT = σ d ( L ) 1/2 = 5 (2) 1/2 = 7.07. 90% service level: z = 1.28 Safety stock (SS) = z σ dLT = 1.28(7.07) = 9 units Reorder point (ROP) =  d L + SS = 2(18) + 9 = 45 units

D= 18*52/year, S=$45/order, H=i*c=25%*60=$15/year

Q = EOQ = 75 units

σ d = 5,  d = 18 units/week, and L = 2 weeks

σ dLT = σ d ( L ) 1/2 = 5 (2) 1/2 = 7.07.

90% service level: z = 1.28

Safety stock (SS) = z σ dLT = 1.28(7.07) = 9 units

Reorder point (ROP) =  d L + SS = 2(18) + 9 = 45 units

Exercise Weiss’s paint store uses a reorder point system to control the stock level of its white latex paint product in the gallon size. Demand is observed to be normally distributed with a monthly mean of 28 gallons and a standard deviation of 8. Re-supply lead time is 2.5 months. Costs associated with replenishment are $15 per order and carrying cost is 30 percent per year. The store manager targets a 98 percent in-stock probability during the lead time. A gallon of paint costs the store $6. When should the manager plan for replenishment of this item and what should the reorder quantity be?

Weiss’s paint store uses a reorder point system to control the stock level of its white latex paint product in the gallon size. Demand is observed to be normally distributed with a monthly mean of 28 gallons and a standard deviation of 8. Re-supply lead time is 2.5 months. Costs associated with replenishment are $15 per order and carrying cost is 30 percent per year. The store manager targets a 98 percent in-stock probability during the lead time. A gallon of paint costs the store $6.

When should the manager plan for replenishment of this item and what should the reorder quantity be?

Solution

a) The hospital orders bandages in lot sizes of 900 boxes. What extra costs does the hospital incur, which it could save by applying the EOQ concept? b) Demand is normally distributed with a standard deviation of weekly demand being 100 boxes. The lead-time is 1/2 week. What safety stock is necessary if the hospital uses continuous review inventory system, and a 97% service level? What should the reorder point be? Extra Exercise Wood County Hospital consumes 500 boxes of bandages per week. The price of bandages is $70 per box. The cost of processing an order is $60, and the cost of holding one box for a year is 15% of the value of the material. Assume 52 weeks per year.

a) The hospital orders bandages in lot sizes of 900 boxes. What extra costs does the hospital incur, which it could save by applying the EOQ concept?

b) Demand is normally distributed with a standard deviation of weekly demand being 100 boxes. The lead-time is 1/2 week. What safety stock is necessary if the hospital uses continuous review inventory system, and a 97% service level? What should the reorder point be?

Solution

Summary of Continuous Review Model Optimal Order Quantity: Q , use EOQ Reorder Point = Avg lead time demand + Safety stock ROP =  d L + SS Safety Stock is determined based on Service level; Variability of demand; lead time For normal demand z can be compute for a given service level  d = demand variability L = lead time SS = z  dLT = z  d  L

Optimal Order Quantity: Q , use EOQ

Reorder Point = Avg lead time demand + Safety stock ROP =  d L + SS

Safety Stock is determined based on

Service level; Variability of demand; lead time

For normal demand

z can be compute for a given service level

 d = demand variability

L = lead time

SS = z  dLT = z  d  L

Two types of Inventory Control Systems Continuous review system ( Q -system) When to order: when inventory declines to ROP Event-trigger restocking Also known as: Reorder Point (ROP) system How much: a fixed quantity is ordered every time EOQ is a continuous review system with uncertain demand Periodic review system ( P -system) When to order: an order is placed after a fixed period of time Time-triggered restocking How much: An order placed for a variable amount

Continuous review system ( Q -system)

When to order: when inventory declines to ROP

Event-trigger restocking

Also known as: Reorder Point (ROP) system

How much: a fixed quantity is ordered every time

EOQ is a continuous review system with uncertain demand

Periodic review system ( P -system)

When to order: an order is placed after a fixed period of time

Time-triggered restocking

How much: An order placed for a variable amount

Periodic Review Systems Key differences compared to continuous review model: When to order: every T days (review cycle/period) How much to order: variable, raising inventory up to a target position Objective: to minimize inventory cost with service level guarantee

Key differences compared to continuous review model:

When to order: every T days (review cycle/period)

How much to order: variable, raising inventory up to a target position

Objective: to minimize inventory cost with service level guarantee

Periodic Review Model T T T L L L Target Inventory Position Stockout Safety Stock Inventory Level Time ( t ) Place order Place order Place order Place order Order Arrival Order Arrival Order Arrival Order Arrival Exposure period (or protection interval): time between placing order n and receiving order n +1

At time t n , you review inventory and find I n units on hand. You decide to order q n . Once you’ve placed an order at t n , no other later orders can be received until t n + L+T . Therefore, I n + q n must be sufficient to cover demand through a period of length L+T ( Exposure Period or Protection Interval ). I n + q n is referred to as Target Inventory Position . Exposure Period (Protection Interval) L L T T T Time ( t ) q 1 q 2 q 3 L t 1 I 1 I 2 I 3 Exposure Period t 2 t 3

At time t n , you review inventory and find I n units on hand. You decide to order q n .

Once you’ve placed an order at t n , no other later orders can be received until t n + L+T .

Therefore, I n + q n must be sufficient to cover demand through a period of length L+T ( Exposure Period or Protection Interval ).

I n + q n is referred to as Target Inventory Position .

Periodic review models: Target inventory position: ( L+T )  d + z σ L+T Current inventory level: I Order quantity: ( L + T )  d + z σ L+T – I z is computed from Normal Distribution Table In comparison, Continuous review models: ROP: L  d + z σ dLT Order quantity: Q=EOQ Order Quantity to Achieve a Service Level

Periodic review models:

Target inventory position: ( L+T )  d + z σ L+T

Current inventory level: I

Order quantity: ( L + T )  d + z σ L+T – I

z is computed from Normal Distribution Table

In comparison, Continuous review models:

ROP: L  d + z σ dLT

Order quantity: Q=EOQ

Continuous Review vs. Periodic Review Continuous review: SS= Periodic review: SS= What if you run out of inventory?! Continuous review  Order more immediately Periodic review  have to wait till next order time z  σ dLT z  σ L+T Why?

Continuous review: SS=

Periodic review: SS=

What if you run out of inventory?!

Continuous review  Order more immediately

Periodic review  have to wait till next order time

z  σ dLT

z  σ L+T

Why?

Example Bird feeder example. Recall that demand for the bird feeder is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units. The lead time is 2 weeks, and the business operates 52 weeks per year. The Q system developed in Example 12.4 called for an EOQ of 75 units and a safety stock of 9 units for a cycle-service level of 90 percent. What is the periodic review ( P) system? What’s the order cycle (T) What’s target inventory level?

Bird feeder example. Recall that demand for the bird feeder is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units. The lead time is 2 weeks, and the business operates 52 weeks per year. The Q system developed in Example 12.4 called for an EOQ of 75 units and a safety stock of 9 units for a cycle-service level of 90 percent.

What is the periodic review ( P) system?

What’s the order cycle (T)

What’s target inventory level?

We first define average annual demand, D, and T, the time between reviews (weeks) Solution D = (18 units/week)(52 weeks/year) = 936 units T = (52) = EOQ D (52) = 4.2 or 4 weeks 75 936

We first define average annual demand, D, and T, the time between reviews (weeks)

(cont’d) We now find the standard deviation of demand over the protection interval ( L+T ) = 6: Before calculating Target Inventory Position , we also need a z value. For a 90 percent cycle-service level z = 1.28. The safety stock, SS = z σ L+T =1.28(12.25) = 15.68 or 16 units Target Inventory Position = Average demand during the protection interval + Safety stock =  d(L+T) + SS = (18 units/week)(6 weeks) + 16 units = 124 units

We now find the standard deviation of demand over the protection interval ( L+T ) = 6:

Before calculating Target Inventory Position , we also need a z value. For a 90 percent cycle-service level z = 1.28.

The safety stock, SS = z σ L+T =1.28(12.25) = 15.68 or 16 units

Target Inventory Position

= Average demand during the protection interval + Safety stock

=  d(L+T) + SS

= (18 units/week)(6 weeks) + 16 units = 124 units

Exercise Weiss’s paint store uses a reorder point system to control the stock level of its white latex paint product in the gallon size. Demand is observed to be normally distributed with a monthly mean of 28 gallons and a standard deviation of 8. Re-supply lead time is 2.5 months. Costs associated with replenishment are $15 per order and carrying cost is 30 percent per year. The store manager targets a 98 percent in-stock probability during the lead time. A gallon of paint costs the store $6. What are the target inventory position and replenish cycle?

Weiss’s paint store uses a reorder point system to control the stock level of its white latex paint product in the gallon size. Demand is observed to be normally distributed with a monthly mean of 28 gallons and a standard deviation of 8. Re-supply lead time is 2.5 months. Costs associated with replenishment are $15 per order and carrying cost is 30 percent per year. The store manager targets a 98 percent in-stock probability during the lead time. A gallon of paint costs the store $6.

What are the target inventory position and replenish cycle?

Summary * Depending on the ownership of the pipeline inventory. Continuous Review Periodic Review Service level Prob. of no stockout during the lead time (L) Prob. of no stockout during the exposure period (L+T) When to order Inventory hits ROP ROP =  d L + SS Every T days How much to order Fixed quantity: Order EOQ q = Target IP – On-hand Target IP = (L+T)  d + SS Safety Stock (normal demand) SS = z σ dLT SS = z σ L+T Average Cycle Inventory EOQ / 2  d T / 2 Average Inventory EOQ / 2 + SS (+  d L) *  d T / 2 + SS (+  d L) *

Aggregate Control of Inventories Manages the entire inventory or broad groups of products collectively Simplifies administration when thousands of items are involved Two tools: Turnover ratio ABC classification of items

Manages the entire inventory or broad groups of products collectively

Simplifies administration when thousands of items are involved

Two tools:

Turnover ratio

ABC classification of items

Turnover Ratio A fruit grower stocks its dried fruit products in 12 warehouses around the country. What is the turnover ratio for the distribution system?

ABC Classification Large number of stock-keeping units (SKU) Demand volume and value of items vary Classify SKUs into 3 classes, based on their aggregate values The breakdown between classes is arbitrary Use a Pareto chart Number of parts (percentage of total) Value of all parts (percentage of total) A 5% to 15% 75% B 30% 15% C 50% to 60% 10%

Large number of stock-keeping units (SKU)

Demand volume and value of items vary

Classify SKUs into 3 classes, based on their aggregate values

The breakdown between classes is arbitrary

Use a Pareto chart

ABC Classification Using a Pareto Chart 10 20 30 40 50 60 70 80 90 100 Percentage of SKUs Percentage of dollar value 100 — 90 — 80 — 70 — 60 — 50 — 40 — 30 — 20 — 10 — 0 — Class C Class A Class B

ABC Classification Step 1: Classify items according to $ value Dollar value=dollar cost of one unit * annual demand Step 2: Determine level of inventory for items in each classification Class A items want “right” inventory – to achieve this need improved forecasts, reduced lead times, etc. Careful! Cost is not only reason to classify as “A” Some items may be cheap but crucial to production of more expensive items Parts are scarce Difficult to supply

Step 1: Classify items according to $ value

Dollar value=dollar cost of one unit * annual demand

Step 2: Determine level of inventory for items in each classification

Class A items want “right” inventory – to achieve this need improved forecasts, reduced lead times, etc.

Careful! Cost is not only reason to classify as “A”

Some items may be cheap but crucial to production of more expensive items

Parts are scarce

Difficult to supply

ABC Classification Example Class Items % Value % Parts A 9,8,2 70.9 15 B 1, 4, 3 16.5 28 C 6, 5, 10, 7 12.6 57 A B C Set different service levels for each class Unit cost Unit sales Part no. Value $510 60 9 $30,600 320 50 8 16,000 350 40 2 14,000 60 90 1 5,400 80 60 4 4,800 30 130 3 3,900 20 180 6 3,600 30 100 5 3,000 20 120 10 2,400 10 170 7 1,700 Value 35.8% 18.7 16.4 6.3 5.6 4.6 4.2 3.5 2.8 2.1 Part 10.0% 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 Part Cumulative 6.0% 11.0 15.0 24.0 30.0 43.0 61.0 71.0 83.0 100.0 85,400 $ Value Cumulative 35.8% 54.5 70.9 77.2 82.8 87.4 91.6 95.1 97.9 100.0