Notes

• Quiz This Friday

• Covers 13 March through today

MGTSC 352

Lecture 21: Inventory Management

A&E Noise exampleMethods for finding good inventory policies:

1) simulation2) EOQ + LTD models

Using EOQ for the Distribution Game: Multi-Echelon Systems

Why Keep Inventory?

1. Seasonality (anticipated variation)

2. Provide flexibility (unanticipated variation) a.k.a.:

3. Economies of scale

4. Price speculation (not an ops reason)

5. Something to work on

6. NDR,JP

Inventory By Where it IS

• Raw Materials

• Finished Goods

• Work in Process

• Or, with apologies to PS, “One man’s ceiling is another man’s floor.”

Time

Inventory

Approximation 1: constant demand

Therefore: We let inventory drop to zero just before an order arrives

Acquisition Costs (pg. 142)

No matter what the inventory policy,

acquisition costs = Demand X Cost

They don’t change,

So they don’t go in the model

(Unless you get quantity discounts, then it matters.)

Order Costs

• Number of orders per year (3695 VCRs / year)/(80 VCRs / order)

= 46.2 orders / year

• Total order cost per year (46.2 orders / year)($30 / order)

= $1385.63 / year

• Total Order Costs = S * D/Q

Holding Costs (pg. 143)

• Minimum inventory 0 for nowLater = Safety Stock

• Maximum inventory = Q (+SS)

• Average inventory Q/2 = (80)/2 = 40 VCRs

• Total holding cost per year (40 VCR-years)($37.5 / VCR / year) = $1500 / year

• Total Holding Costs = H*Q/2

EOQ = Economic Order Quantity Model

• Given demand is constant

• Find the Q that minimizes total cost

Total cost = acquisition cost + order cost + carrying cost + shortage cost

pg. 144

• Total relevant cost = order cost + carrying cost

No shortages, by assumption

Acquisition costs don’t depend on Q

EOQ DerivationS = order cost ($/order)

H = carrying cost ($/item/year)

D = demand (units/year)

Q = order quantity

N = number of orders per year

Iavg = average inventory

Relevant cost = order cost + carrying cost

RC = S N + H Iavg

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

Note: you can change year to day, week, or any other time unit, as long as you are consistent

Common mistake: inconsistent time units

pg. 147

To Excel

EOQ Formula

Relevant cost = ordering cost + carrying cost

RC = S N + H Iavg

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

* 2 D S

QH

pg. 147

The magic part (optional)

2

*

23

2

*

RC(Q) SD/ Q HQ / 2

dRCSD/ Q H/ 2

dQ

dRC 2DS0 Q

dQ H

d RC2SD/ Q 0,

dQ

so Q minimizes RC(Q)

Using EOQ for A&E Noise YNOS XD

D = 10.12 VCRs/day, S = $30/order, H = $0.10/VCR/day Q* = SQRT(210.1230/0.10) = 77.9 round to Q* = 78

N* = 10.12/78 = 0.13 orders/day = 47.4 orders/yearOrder every 365/47.4 = 8 days

Relevant cost:RC(Q*) = S (D/Q*) + H (Q*/2)

= 30 (10.12/78) + 0.10 (78/2) = 3.90 + 3.90 = $7.80 / day = $2,847 / year

pg. 147

Common mistake: using inconsistent time units

D = 10.12 VCRs/day, S = $30/order, H = $37.5/VCR/year

Q* = SQRT(210.1230/37.5) = 4

• Off by (77.9 – 4)/77.9 = 95%

• Will not be worth a lot of part marks

More on EOQ: Economies of Scale

The Capital Health Region* operates four hospitals. Presently each hospital orders its own supplies and manages its inventory. A common item used is a sterile intravenous (IV) kit, with a weekly demand of 600 per week at each hospital. Each IV kit costs $5 and incurs a holding cost of 30% per year. Each order incurs a fixed cost of $150 regardless of order size. The supplier takes one week to deliver an order. Currently, each hospital orders 6,000 kits at a time.

Question 1: Could costs be decreased by ordering more often?

Question 2: Would it make sense to centralize inventory management for the four hospitals?

Pg. 149

* Fictional data

Analysis for one Hospital

• D = 600 / week = (600 / week) (52 weeks/year) = 31,200 / year

• S = $150 / order• H = 0.3 5 = $1.50 / kit / year• Q = SQRT(2 D S / H) = 2,498 ≈ 2,500• Costs:

– Q = 6,000: S D / Q + H Q / 2 = $780 + $4,500 = $5,280

– Q = 2,500: S D / Q + H Q / 2 = $1,872 + $1,875 = $3,747

– 29% savings

Analysis for one Hospital

• D = 600 / week = (600 / week) (52 weeks/year)

= 31,200 / year• S = $150 / order• H = 0.3 5 = $1.50 / kit / year• Q = SQRT(2 D S / H) = 2,498 ≈ 2,500

• Close your course pack• Active Learning: How do we change the

analysis if inventory management were centralized for the four hospitals?

Analysis for four hospitals managed together

• D = 4 31,200 / year = 124,800 / year• S = $150 / order• H = $1.50 / kit / year• Q = SQRT(2 124,800 150 / 1.5) = 4,996 ≈ 5,000• Costs:

– Each hospital operated independently: 4 $3,747 = $14,988 / year

– All four together: S D / Q + H Q / 2 = $3,744 + $3,750 = $7,494 / year

– 50% savings

• Quadrupling demand doubles the optimal order quantity and doubles the total relevant cost

Four hospitals managed together

• Costs: – Each hospital operated independently:

4 $3,747 = $14,988 / year– All four together:

S D / Q + H Q / 2 = $3,744 + $3,750 = $7,494 / year– 50% savings

• Quadrupling demand doubles the optimal order quantity and doubles the total relevant cost

* 2 D S

QH

Determining ROP with EOQ model

Lead time = 5 days

Demand during lead time = (5 days) (10.12 VCRs / day) 51 VCRs

Set ROP = 51 VCRs

Time

Inventory

lead timedemand during lead time

ROP

Problem: this calculation assumes constant demand. May lead to shortages too frequently

What happens to Holding Cost when we Increase ROP?

• EOQ: constant demand, zero safety stock– ROP = avg. demand during lead time

– Iavg = (min + max)/2 = (0+Q)/2 = Q/2

– Holding cost = H Q / 2

• If we add safety stock = SS, then:– ROP = avg. demand during lead time + SS

– Iavg = Q/2 + min = SS + Q/2

– Holding cost = H (SS + Q / 2)

Pg. 149

Time

Inventory

ROPLeadtimeDemand

during leadtime Demand that was not met

How Shortages Happen

Active learning:How could we have avoided the shortage?

Pg. 152

Time

Inventory

ROP

The demand during the lead time is uncertain. Here are 4

possibilities.

We’ll see how to pick ROP so as to provide a specified fill rate

… to Excel

LTD Recap

• “LTD” worksheet in A&E Noise workbook– Purpose: vary ROP (and Q, if desired) and

see what happens to the fill rate

• “LTD-exotic version”: can vary the lead time– Useful for comparing suppliers that provide

different lead times

Simulation versus EOQ

Dimension Simulation EOQ + LTD

Ease of evaluating a policy

Need to build model – time consuming

Simple formula for RC – back of an envelope

Finding the optimum Trial and error / data table

Plug into formula for Q*

Random demand fluctuations

Taken into account Ignored in EOQ

Seasonal demand fluctuations

Can be taken into account

Ignored

Shortages Taken into account Ignored in EOQ

Likely errors

(common mistakes)

Errors in formulas Inconsistent units

pg. 151

Supplier Warehouse

Retailer

Retailer

Retailer

Back to the Distribution Game: Can we use EOQ here?

A “multi-echelon” system

Pg. 158

Using EOQ for a two-echelon system

• Upper echelon:– Use warehouse holding cost rate

• Ignore higher cost of holding inventory at retailers

– Lead time = 15 (supplier warehouse) + 5 (warehouse retailer) = 20 days

• Lower echelon:– Use incremental retailer holding cost rate– Lead time = 5 days

• Coordination: warehouse order size should be a multiple of the sum of the retailer order sizes

Data

• Supplier to warehouse transit time: 15 days• Warehouse to retailer transit time: 5 days• Demand per retailer: 500 per year• Selling price: $100/unit• Purchase price: $70/unit• Supplier to warehouse order cost: $200• Warehouse to retailer order cost: $2.75• Warehouse holding cost: $10/unit/year• Retailer holding cost: $12/unit/year

Assume open 250 days / year

… To Excel

Upper echelon

Supplier Warehouse

Retailer

Retailer

Retailer

Upper echelon:Use warehouse holding cost rate

(Ignore higher cost of holding inventory at retailers)

Lead time = 15 (supplier warehouse) + 5 (warehouse retailer) = 20 days

Lower echelon

Supplier Warehouse

Retailer

Retailer

Retailer

Lower echelon:Use incremental retailer holding cost rate = retailer holding cost rate – warehouse holding cost rate

Lead time = 5 days

Coordination

• Suppose each retailer uses QLower = 20. If all retailers order at once, the total is 60.

• Active learning: you are the warehouse manager. Knowing the retailer order sizes, how would you pick the warehouse order size?

Using EOQ for a 2-echelon system: the details

• Upper echelon:– DUpper = 3 DRetailer

– SUpper = SWarehouse

– HUpper = HWarehouse

– LTUpper = LTSupplier Warehouse + LTWarehouse Retailer

– ROPUpper = DUpper LTUpper

• Lower echelon– DLower = DRetailer

– SLower = SRetailer

– HLower = HRetailer - HWarehouse

– LTLower = LTWarehouse Retailer

– ROPLower = DLower LTLower

• Coordination: QUpper = n SUM(QLower)

• Choose n (an integer) and QLower to minimize total cost for the whole system

Data

• Supplier to warehouse transit time: 15 days• Warehouse to retailer transit time: 5 days• Demand per retailer: 500 per year• Selling price: $100/unit• Purchase price: $70/unit• Supplier to warehouse order cost: $200• Warehouse to retailer order cost: $2.75• Warehouse holding cost: $10/unit/year• Retailer holding cost: $12/unit/year

Assume open 250 days / year

… To Excel