Modeling and Analysis of Manufacturing and Service Enterprises

Why Manage Inventory? n

n

Inventory Management Introduction and EOQ

n n n n

Total $ investment in inventories is $1.37 trillion (last quarter of 1999) 34% in Manufacturing 82% of the total 26% in Retail 22% in Wholesale 8% in Farm 10% in Other

Sources: plants vendors ports

Why Manage Inventory? n

n

Total investment in inventory is 20-25% of the total annual GNP. What would happen if you control inventories and achieve 1% improvement overall? n

You can improve the efficiency of the overall economy

Regional Warehouses: stocking points

Field Warehouses: stocking points

Customers, demand centers sinks

Supply

Inventory & warehousing costs Production/ purchase costs

Transportation costs Inventory & warehousing costs

Transportation costs

1

Inventory n

Where do we hold inventory? n n n

n

Types of Inventory n n n

n

suppliers and manufacturers warehouses and distribution centers retailers WIP and subassemblies raw materials finished goods

Why do we hold inventory? (Short answer) n n

Economies of scale Uncertainty in supply and demand

Decisions to Make n

We have to decide n n

n

How often we review the inventory When we should issue a (replenishment/production) order How large the order should be

Why do we hold inventory? n n n n n n n

Economies of scale Uncertainty in supply and demand Speculation Transportation Smoothing production/purchasing Logistics Cost of controlling inventory

A Classification of Inventory n

Functional categories of inventories n n n n n n

Cycle stock Congestion stock Safety stock Anticipation inventory Pipeline (or work-in-process) inventory Decoupling stock

2

ABC classification n

n

Observe that 20% of the stocked parts make up 80% of the total inventory investment Focus on the “important” parts that have significant effect on the bottom line

ABC classification Cont. n

A items: n

n

B items n

n

n

Demand n n

n n

Constant (level) or variable Deterministic (known) or Stochastic (random or uncertain)

Lead Time Review Time n

Continuous or periodic review

n

Excess Demand

n

Changing inventory

n

Next 50% of the products (around remaining 45-50% of total investment)

C items n

Characteristics of Inv. Systems

Usually First 5 to 10% of the products (around 50% of the total investment)

Many, but low value

Relevant Costs n

Unit value or unit variable cost (v) n

Cost of making a part available for usage n

n n

Purchase + Freight + Mfg. Costs

Usually different from “accounting” cost Should include more than just book value

Backordered or lost

3

Relevant Costs n

Relevant Costs

Holding cost (cost of carrying in inv.) n n

n n

Opportunity costs of the money tied to inv. Warehousing and Handling (cost of providing space to store items, counting and moving items in the WH) Deterioration, damage, obsolescence Insurance and taxes

Relevant Costs

n

Holding cost (cont’d) n

n

n

n

h = ic

Total annual holding cost = Ih =Iic

Relevant Costs n

Inv.

Ordering or Setup Cost (A) n

Avg. inv. level

Dec

Time, t

Fixed cost n

n

Jan

i: total annual interest charge (or carrying charge) h: holding cost, $ per item per year

Independent of the size of the replenishment or production order

Ordering forms, phone calls, other communication costs, receiving, inspection, cost of interrupted production, opportunity cost of lost time, etc.

4

EOQ Model n

Book Store Mug Sales n n n

n

n

EOQ Model: Assumptions D

Demand is constant, at 20 units a week Fixed order cost of $12.00, no lead time Holding cost of 25% of inventory value annually A Mugs cost $1.00, sell for $5.00

Question n

How many, when to order? Q

T?

i c

EOQ Model

Inventory

n

Production is instantaneous

n

Delivery is immediate

n

Demand is deterministic

n

Demand is constant over time

n

A production run incurs a fixed setup cost

n

Products can be analyzed singly

n

No shortages are allowed

n

Anything else?

EOQ Model

Note: • No Stockouts • Order when no inventory • Order size determines policy

n n n

Time unit: A year Purchase Cost Constant Annual Holding Cost: n

Order Quantity Q

n

Annual Ordering (Setup) Cost:

n

Goal:

n

Avg. Inventory

(Avg. Inventory) * (Holding Cost)

Number of Orders in a year * Order Cost

Find the order quantity that minimizes total costs

5

n n n n n

Average Inventory: Q/2 Annual holding costs: Qh/2 Time between replenishments: Q/D Number of replenishments per year: D/Q Replenishment costs: n

n

EOQ Model 160 140 120 Cost

EOQ Model

Total Cost

100

Holding Cost

80 60

Order Cost

40

(A+Qh)D/Q=AD/Q + Dc

20

Total Relevant Costs: TRC(Q)=AD/Q+Qh/2

0 0

500

Optimal Order Quantity, Q*

EOQ Model

EOQ Model

nTotal

n

Relevant Costs: TRC(Q)=AD/Q+Qh/2 dTRC (Q ) h A = − =0 dQ 2D Q 2

n n n

2 AD EOQ = Q * = h

n n

1000

1500

Order Quantity

Optimal Quantity (EOQ) = (2*Setup Cost*Demand)/holding cost = SQRT(2AD/h) What is the EOQ for our problem? The corresponding time between orders? What is the number of months of demand that EOQ will satisfy, T(EOQ)? Annual relevant cost, TRC(EOQ)? The corresponding Inv. Turnover Ratio?

6

EOQ Model n

n

EOQ Model: Sensitivity

Tradeoff between set-up costs and holding costs when determining order quantity. In fact, we order so that these costs are equal per unit time Total Cost is not particularly sensitive to Q=1.10EOQ the optimal order quantity

Order Quantity 50% Cost Increase

80%

90%

100% 110% 120% 150% 200%

n

TRC( Q′) = n

n

TRC(Q)=1.0045*TRC(EOQ)

n

n

Batching causes inventory (i.e., larger lot sizes translate into more stock). Under specific modeling assumptions the lot size that optimally balances holding and setup costs is given by the square root formula: Q* =

n

2 AD h

Total cost is relatively insensitive to lot size (so rounding for other reasons, like coordinating shipping, may be attractive).

hQ′ AD + 2 Q′

Compare it to EOQ Costs: TRC( Q′ ) h Q′ 2 + AD Q′ 1  Q′ Q*  = =  *+ TRC(Q *) 2 Q Q′  2 ADh

125% 103% 101% 100% 101% 102% 108% 125%

EOQ Observations

Total Relevant Cost for Q’

Example: If Q ' = 2Q*, then the ratio of the actual to optimal cost is (1/2)[2 + (1/2)] = 1.25

EOQ Extensions n

Finite Replenishment/Production Rate: n

n n n n n

Economic Production Lot (EPL) or Economic Production Quantity (EPQ)

Nonzero constant lead time Quantity Discounts Considering Inflation Limits on Order Size Exchange Curves

7

Economic Production Lot (EPL) n

What happens when there is finite production rate? n n

n

P: production rate (in units per year) P>D (demand rate per year). Why?

What happens if you keep producing? n

The inventory will keep growing forever with a rate of P-D.

EPL

EPL n n

n

n

EPL Slope=P-D

Inv

n

Slope=-D

n

T1

Stop Prod.

T2

Production time = T1 n

n

H

Start Prod.

When do we stop production: T1 T1 also determines our production lot size or run size, Q. Until the next production run, we satisfy demand from the inventory that we built during T1. Total cycle: T=T1+T2

Time

Q=PxT1 or T1=Q/P

Demand during production time that uses up inventory: DxT1 = DxQ/P The inventory we built up (H) is the leftover inventory during production: n

H = Q - DxT1 = Q (1-D/P)

Start Prod.

8

EPL n

Annual Total Relevant Cost for EPL: n n

n

TRC(EOQ) = AD/Q + hQ/2

Rename h(1-D/P) as adj. holding cost, h’, and find the optimal order quantity (lot size): EPL = Q* =

2 AD = h'

2 AD h(1 − D / P )

Nonzero Order Lead Times n n

n

n

TRC(EPL) = AD/Q + hH/2 or TRC(EPL) = AD/Q + h(1-D/P)Q/2

Compare this to TRC(EOQ): n

n

EPL

Recall EOQ’s assumption: zero lead time What would you do if you have nonzero lead time, i.e., it takes some time for your orders to arrive, shipping and handling, etc? Say, the lead time in our bookstore mug example is 1 month. What would you do differently?

H should be enough to cover the demand during T2: n

n n n

H = DxT2 or T2=H/D

Overall production cycle: T=T1+T2 Total demand during cycle, T: DxT Check if production lot is enough to cover this demand: Q=DxT?

EOQ with Positive Lead Times Inv Q*=316

LT=1mo

Reorder Point=? T=3.8 mo

Time

Place Order Order arrives

9

Quantity Discounts n

EOQ’s assumption: v is constant What would you do if the cost of the part depends on the order quantity?

n

One easy case to analyze:

n

n

n

Quantity Discounts Slope=c0

Ordering Cost

Price discounts if you order more all-units discount

Qb Breakpoint quantity

Quantity Discounts n

n n

Slope=c0 (1- d)

If you order less than Qb, then the unit cost is c0 If you order more, then it is c0(1-d) When you calculate total relevant costs, include variable order cost (or purchase cost), Dc 0 or Dc 0(1-d).

Q, Order Quantity

Quantity Discounts n

n

n

Calculate EOQ w/o discount: EOQ using c0 for c Calculate EOQ w/ discount: EOQ(d) using c0(1-d) for c Three cases: n n n

1. If EOQ(d)>=Q b, then Q*=EOQ(d) 2. If TRC(EOQ)TRC(Q b)], order Q b.

10

Quantity Discounts Case 2

Case 1

EOQ with Inflation Case 3

n n

n

Qb

Q*

Q* Q b

Example: n

n

Q* =

Q*=Q b

EOQ with Inflation n

Inflation rate: r Calculate the present value of costs that are occurring over time EOQ w/ inflation:

How many mugs should the bookstore order if the annual inflation rate is 5%? Compare it with EOQ w/o inflation.

2 AD 1 = EOQ v(i − r) 1− r / i

Limits on Order Sizes n

n

n

Sometimes EOQ leads to a really large order quantity and in turn long time supply (time between orders). If EOQ is greater than what you can store, order/produce less, Q*=C when EOQ>C If Time Supply (order interval or cycle) is longer than the allowed shelf life (SL), then you may want to order less: n

n

Q*(SL)=DxSL when T(EOQ)>SL

Uncertainty into the future

11

EOQ Trade-off n

Two interpretations: n

n

n

Aggregate Considerations n

If you order more (larger Q), you incur higher inventory cost, but less setup cost If you order less frequently, you incur larger inventory cost, but less setup cost

Instead of determining A and i, n

n

n

The trade-off is not linear!

n

Aggregate Considerations n

Two major assumptions n

n

n n

Assume that fixed setup cost for each item cannot be determined explicitly Assume a common value A holds (approx.) for all items

The number of items: n Define Dj, Qj, and c j for item j

Average total inventory cannot exceed a certain dollar value or volume Number of replenishments in a year must be less than certain value The maximum allowed backorder delay may not exceed a certain value Trade-off btw avg. inv. and cost (or the number) of replenishments in a year should be at certain prescribed value

Aggregate Considerations n

Total Avg. Cycle Stock (in $): n

TACS = ∑ Q j c j / 2 j =1

n

Total number of replenishments: N = ∑ D /Q Use EOQ for each item n

j

j

j =1

n

Qj = *

2 AD j c ji

12

Aggregate Considerations Put Qj * into TACS and N:

n n

ADj c j

j =1

2i

TACS = ∑

=

A n ∑ Djc j 2i j =1

n

N =∑

j =1

Resource Constrained Multiple Product Systems n

Djc ji

i n = ∑ Djc j 2A 2 A j =1

n

n

n

Multiply TACS and N: TACS × N =

n

 1  n ∑ D j c j  2  j =1 

n n

Plot TACS versus N

n

Example: n

D_j c_j A_j

n

n

2 1150 350 150

3 800 85 50

Limit on the maximum inventory investment: $30,000 n

So Total Q jc j