Review of Inventory Models

Review of Inventory Models

Recitation, Feb. 4Guillaume Roels

15.762J Supply Chain Planning

Why hold inventories?

• Economies of Scale• Uncertainties

– Demand– Lead-Time: time between order and delivery– Supply

• Transportation• Smoothing (Seasonality)• Speculation• …

Inventory Costs• Holding Cost

– Cost of Capital, Warehouse, Taxes and Insurance, Obsolescence

• Order Cost– Fixed and variable

• Penalty Cost– Lost sale vs. Backorder

Consider only costs that are relevant to the ordering decision

Outline

• Newsboy– 1-period– Random demand

(Stochastic)

– Shortages allowed– Variable costs only

– No Lead Time

• EOQ– Multiple periods– Known demand

(Deterministic)– Constant Demand– No Shortages– Fixed and variable

order costs– No Lead Time

Newsboy Example

Every week, the owner of a newsstand purchases a number of copies of The Computer Journal.

Weekly demand for the Journal is normally distributed with mean 10 and standard deviation 5.

He pays 25 cents for each copy and sells each for 75 cents.

How many copies would you recommend him to order?

Example from Nahmias, Production and Operations Analysis

Other applications…

• Short product life cycles / Long lead times– Computers– Apparel

• Fresh products– Fresh food, newspapers

• Services– Airline industry

Newsboy Model: Notations

• Random Demand: D• Ordering decision: Q• Unit Selling Price: p• Unit Purchase Cost: c

• Objective: Find Q that maximizes Expected Profit, E[π]

Review of Optimization

Max f(x)• First-Order Conditions

f’(x*)=0• Second-Order Conditions

f’’(x*) ≤ 0

≥ 0 ≤ 0

Max E[π] = p E[min{D,Q}] – c Q

• First-Order Conditions(E[π])’ = p E[(min{D,Q})’] – c

= p P(D≥Q)-c = 0

since min{D,Q}= D when D ≤Q (min(Q,D))’=0

Q when Q ≤D (min(Q,D))’=1• Second Order Conditions

One can check that (E[π])’’= p (P(D≥Q))’ ≤ 0

Order Q* such that P(D≥Q*) = c/p

Distribution FunctionSuppose that demand has cdf F(x), i.e.,F(x)=P(D≤x)Therefore,P(D≥Q*)=c/p ⇔ 1-P(D≤Q*)=c/p⇔1-F(Q*)=c/p ⇔

F(Q*)=(p-c)/p

Ratio (p-c)/p is a probability (btw 0 and 1)and is called the critical fractile

Generalization• cU: Underage Cost (when D ≥ Q)

– In the example, opportunity cost, p-c– Loss of goodwill

• cO: Overage Cost (when D ≤ Q)– In the example, c– Salvage value

Min cU E[max{D-Q, 0}] + cO E[max{Q-D, 0}]Solving for Q,

F(Q*)=cU/(cU+ cO)

How to find Q*: Graphical Representation

F(Q)

1

cU/(cU+cO)

0 Q* Q

How to find Q*: Analytical Derivation

Uniform Demand between [A,B]F(x)=(x-A)/(B-A)

Solve (Q*-A)/(B-A)=cU/(cU+ cO), i.e.Q*=A+ (B-A) cU/(cU+ cO)

xA B

How to find Q*:Excel

• Normal DemandQ*=NORMINV(µ, σ, cU/(cU+ cO))

F(Q*)=cU/(cU+ cO) ⇔ Q*=F-1(cU/(cU+ cO))

Alternatively, use standardized normal

Q*=µ + (z*) σ

where z*=NORMSINV(cU/(cU+ cO))

How to find Q*:Tables

• Example:cU=p-c=.75-.25= $.50cO=c= $.25Critical Fractile = cU/(cU+ cO) = 0.67Standardized Normal Table z*=0.43

Q*= µ + (z*) σ=10+(0.43) 5 = 12.15

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753

0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549

0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133

Service Levels

• Shortage PenaltyP(D ≥ Q*) = 1 - F(Q*) = cO/(cU+ cO) Example: 0.333

• Fill RateE[min{D,Q*}]/E[D]Example: 89% (from tables or simulation)

Extensions

• Initial Inventory IOrder Q* - I if I ≤ Q*, 0 otherwiseQ* is called the Base Stock and represents the

target inventory level• Discrete demand

Order quantity: Round Up Q*• Multiple periods• Fixed cost• Many applications in Supply Contracts

Outline


(Stochastic)


– No Lead Time




EOQ ExampleNumber 2 pencils at the campus bookstore are

sold at a fairly steady rate of 60 per week.The pencils cost the bookstore 2 cents each and

sell for 15 cents each.It costs $3 to initiate an order, and holding costs

are based on an annual interest rate of 25 percent.

Determine the optimal number of pencils for the bookstore to purchase and the time between placement of orders.


Intuition

• Trade-Off:– Spread the fixed ordering cost over many

items– Avoid high inventory costs

• Replenishment from– An outside vendor– Internal production

Application

• Steady Demand / Large Fixed Cost Industries– Manufacturing: Automobile, Electrical

Appliances, Chemical Products (Lot Sizes)– Retail: Slow-moving items (pencils, bathroom

tissue…)

EOQ Notations

• EOQ = “Economic Order Quantity”• Constant Demand Rate: λ• Fixed order cost: K• Variable order cost: c• Inventory holding cost: h• Interest rate: i• Order quantity: Q• Time between orders: T

Evolution of InventoryInventory position

Q

timeT

• Order when inventory position reaches zero• Order the same amount each time

Cost components (1)

• Inventory holding cost– h = i * c (cost of capital)

• Over a replenishment cycle:– Start from Q– Ends at 0– Decreases steadilyAverage inventory = Q/2Average inventory cost = h Q/2

Cost components (2)

• Per replenishment cycle:– Fixed cost: K– Variable cost: c Q

• Length of a cycle:– Order size: Q units– Demand rate: λ units/yearTime between orders T = Q/λ

• Average order cost = 1/T (K + cQ)= K λ/Q + c λ

Min h Q/2 + K λ/Q + c λ

• First Order Conditions:h/2 - K λ/Q2 = 0

• Second Order Conditions:2 K λ/Q3 ≥ 0

Hence, order Q*= hKλ2

Optimization

Optimal Cost:– Inventory Cost: h Q*/2 =– Fixed Order Cost: Kλ/Q*=

– Total Cost=c λ + 2

hKλ2hKλ2

hKλ2

Graphical View

0

2

4

6

8

10

12

14

1000

1200

1400

1600

1800

2000

2200

2400

Q

cost

inventory

fixed cost

total cost

Exampleλ = 60 units/week = 3,120 units/yearK= $3, c =$0.02, h=i c=(.25) (.02) = $0.005/(unit)/(year)

Q*= units

T=Q/λ=1,935/3,120=0.62 years =32 weeks

Work in the same units!

hKλ2

1935005.0

)120,3)(3)(2(==

Observations

• Very robustCan round up or down with loosing much

• Independent of selling price• Dependent of purchase cost only through

holding cost.

Extensions

• Lead-time L– same ordering quantity– Order L periods in advance, when stock

reaches L/λ.• Finite production rates• Quantity discounts• Supply Chain Application:

– Determine the lot sizes of all stages in the supply chain (global view).

Summary


(Stochastic)


– No Lead Time




OU

U

cccQF+

=*)(hKQ λ2* =

Newsboy Example (1)The buyer for Needless Markup, a famous “high end”

department store, must decide on the quantity of a high-priced women’s handbag to procure in Italy for the following Christmas season.

The unit cost of the handbag to the store is $28.50 and the handbag will sell for $150.00. Any handbags not sold by the end of the season are purchased by a discount firm for $20.00. In addition, the store accountants estimate that there is a cost of $.40 for each dollar tied up in inventory, as this dollar invested elsewhere could have yielded a gross profit. Assume that this cost is attached to unsold bags only.


Newsboy Example (2)Suppose that the sales of the bags are equally likely to be

anywhere from 50 to 250 handbags during this season. Based on this, how many bags should the buyer purchase?

cU = (150.00-28.50) = $121.50 (lost margin)

cO= (28.50 (1.4) -20.00) = $19.90 (purchase cost + inventory holding cost – salvage value)

Critical Fractile = cU/(cU+ cO) =.859Demand is Uniform between 50 and 250Q*= 50 +(250-50) *(.859) =222 units

EOQ Example (1)The Rahway, New Jersey, plant of Metalcase, a

manufacturer of office furniture, produce metal desks at a rate of 200 per month. Each desk requires 40 Phillips head metal screws purchased from a supplier in North Carolina.

The screws cost 3 cents each. Fixed delivery charges and costs of receiving and storing shipments of the screws amount to about $100 per shipment, independent of the size of the shipment. The firm uses a 25 percent interest rate to determine holding costs.

Metalcase would like to establish a standing order with the supplier and is considering several alternatives. What standing order size should they use?


EOQ Example (2)

λ = (200)(40)(12)=96,000 units/yearK=$100, h=(.25)(0.03)=.0075

Cycle time T = Q/ λ = .53 year

597,500075.

)000,96)(100)(2(2* ===hKQ λ

Review of Inventory Models

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