Download - Review of Inventory
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Review of Inventory Models
Recitation, Feb. 4Guillaume Roels
15.762J Supply Chain Planning
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Why hold inventories?
Economies of Scale Uncertainties
Demand Lead-Time: time between order and delivery Supply
Transportation Smoothing (Seasonality) Speculation
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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
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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
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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
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Other applications
Short product life cycles / Long lead times Computers Apparel
Fresh products Fresh food, newspapers
Services Airline industry
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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[]
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Review of Optimization
Max f(x) First-Order Conditions
f(x*)=0 Second-Order Conditions
f(x*) 0
0 0
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Max E[] = p E[min{D,Q}] c Q First-Order Conditions
(E[]) = p E[(min{D,Q})] c= p P(DQ)-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[])= p (P(DQ)) 0
Order Q* such that P(DQ*) = c/p
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Distribution FunctionSuppose that demand has cdf F(x), i.e.,F(x)=P(Dx)Therefore,P(DQ*)=c/p 1-P(DQ*)=c/p1-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
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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)
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How to find Q*: Graphical Representation
F(Q)
1
cU/(cU+cO)
0 Q* Q
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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
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How to find Q*:Excel
Normal DemandQ*=NORMINV(, , cU/(cU+ cO))F(Q*)=cU/(cU+ cO) Q*=F-1(cU/(cU+ cO))
Alternatively, use standardized normal
Q*= + (z*) where z*=NORMSINV(cU/(cU+ cO))
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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*= + (z*) =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
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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)
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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
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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
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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.
Example from Nahmias, Production and Operations Analysis
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Intuition
Trade-Off: Spread the fixed ordering cost over many
items Avoid high inventory costs
Replenishment from An outside vendor Internal production
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Application
Steady Demand / Large Fixed Cost Industries Manufacturing: Automobile, Electrical
Appliances, Chemical Products (Lot Sizes) Retail: Slow-moving items (pencils, bathroom
tissue)
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EOQ Notations
EOQ = Economic Order Quantity Constant Demand Rate: Fixed order cost: K Variable order cost: c Inventory holding cost: h Interest rate: i Order quantity: Q Time between orders: T
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Evolution of InventoryInventory position
Q
timeT
Order when inventory position reaches zero Order the same amount each time
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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
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Cost components (2)
Per replenishment cycle: Fixed cost: K Variable cost: c Q
Length of a cycle: Order size: Q units Demand rate: units/yearTime between orders T = Q/
Average order cost = 1/T (K + cQ)= K /Q + c
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Min h Q/2 + K /Q + c First Order Conditions:
h/2 - K /Q2 = 0 Second Order Conditions:
2 K /Q3 0
Hence, order Q*= hK2
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Optimization
Optimal Cost: Inventory Cost: h Q*/2 = Fixed Order Cost: K/Q*=
Total Cost=c + 2
hK2hK2
hK2
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Graphical View
0
2
4
6
8
10
12
14
1000
1200
1400
1600
1800
2000
2200
2400
Q
c
o
s
t
inventory
fixed cost
total cost
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Example = 60 units/week = 3,120 units/yearK= $3, c =$0.02, h=i c=(.25) (.02) = $0.005/(unit)/(year)
Q*= units
T=Q/=1,935/3,120=0.62 years =32 weeks
Work in the same units!
hK2
1935005.0
)120,3)(3)(2( ==
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Observations
Very robustCan round up or down with loosing much
Independent of selling price Dependent of purchase cost only through
holding cost.
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Extensions
Lead-time L same ordering quantity Order L periods in advance, when stock
reaches L/. Finite production rates Quantity discounts Supply Chain Application:
Determine the lot sizes of all stages in the supply chain (global view).
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Summary
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
OU
U
cccQF +=*)( h
KQ 2* =
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Newsboy Example (1)The buyer for Needless Markup, a famous high end
department store, must decide on the quantity of a high-priced womens 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.
Example from Nahmias, Production and Operations Analysis
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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
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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?
Example from Nahmias, Production and Operations Analysis
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EOQ Example (2)
= (200)(40)(12)=96,000 units/yearK=$100, h=(.25)(0.03)=.0075
Cycle time T = Q/ = .53 year
597,500075.
)000,96)(100)(2(2* ===hKQ
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