Too Much/Too Little Problem

Maximize Expected Profit

STEP 1: overage/underage costs

Co = overage cost per unit

Co = Variable Cost – Salvage value

Cu = underage cost per unit

Cu = Price – Cost (+ Future Cost)

STEP 2: Find Critical Ratio

F(Q*) = Probability demand < Q.

Cu /Cu+ C0 = critical ratio/srvc level

STEP 3: Calculate z from table.

φ(Ζ) = Critical ratio

STEP 4: Calculate Q*

= optimal order quantity

Ζ= from above (table)

μ = mean

σ = standard deviation

aka measure of uncertainty

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NEWSVENDOR VARIATIONS

(Co/Cu) Ratio Given

1. Critical Ratio

F(Q) =

2. Find φ(Ζ) from table

3. then

Service Level is given.

1. Find φ(Ζ) for service level given, 2.

Finding SERVICE LEVEL given Q.

1.

2. find φ(Ζ)(expressed as %)

Comparing Stocking Quantities

Actual costs not given.

Holding cost in 2d year:

Yr1 C – Yr2 C + %holding cost

e.g. C – C + 0.25C

Salvage costs in 2d year:

Cost (C)– % Salvage value

e.g. C - 0.75C

ECONOMIC ORDER QUANTITY (EOQ MODEL)

How much to Order Problem

TC = total annual inventory cost

D = annual demand (units/year)

Q = order quantity (units)

K = cost of placing an order or

setup cost ($)

h = annual inventory carrying cost

($/unit/year)

may include:

warehousing cost

avg rtn for working capital

P = price of goods ($/unit)

STEP 1: Convert in D, h, K into same time units (mo, yr, etc…)

STEP 2: Determine EOQ, Q*

Optimal quantity

EOQ =

STEP 3: Find total annual inventory cost (ordering + holding)

annual ordering cost (D/Q)*K

annual holding cost(Q/2)*h

annual purchasing costP*D

annual inventory costTC(Q)

STEP 4: Other Measures

# of Orders per year = D/Q*

Length of Order Cycle T= Q*/D

(Time between two orders)

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EOQ VARIATIONS

Take new quantity discount?

(new Q* & new purchase cost)

1. Calculate original TC

2. Calculate new TC w/ new Q*

3. Compare TC for inc/dec in inventory cost. (TC1 – TC2)

4. Calculate savings w/ discount.

D x (old price – new price)

(TC1 – TC2)/D x 100 = %

Under/over estimate demand by %

1. Calculate new Demand. Add/minus % of D to orig. D.

2. Calculate EOQ with new D.

3. Calculate TC with new Q* & D.

4. Find difference in cost by comparing (subtract) TCs.

BASE STOCK MODEL (s)

How much inventory is needed during lead-time?

D occurs during replenishment

AVG = average demand

STD = standard deviation

L = lead time

convert to time period

i.e. 4.33 weeks/mo

AVGL = average lead time

Z = φ(Ζ) from distribution table

Computing Lead Time Demand

Mean (μ) of lead time demand

= AVG x L

Standard dev of lead time demand

= sqrt (STD2 x L)

Base Stock Formulas

Lead Time is CONSTANT

s = μ + Ζσ

s = base stock level

μ = pipeline inventory

Ζσ = safety stock (SS)

s =(AVG x L) +Ζ x √(STD2x L)

AVG x L = pipeline inventory (μ)

Ζ x √(STD2x L) or

Ζ x STD x √(L) = safety stock

Lead Time is UNCERTAIN

s = AVG x AVGL + Z√(STDL2 x AVG2 + STD2 x AVGL)

AVG x AVGL = mean demand (μ)

Minimizing costs w/o service level

1. F(s) = Cu/(Cu + C0)

Cu = backorder penalty

C0= converted holding cost

2. Find Ζ of critical ratio (table)

3. Calculate base stock level (s) via formulas above

Base Stock Policy

Inventory Position (IP)

= on-hand + on-order – backorder

Reorder when IP drops below s (base stock level)

Amt to order = s – IP

(s, S) Policy: Fixed Ordering Cost

Little s = reorder point(when)

S = order-up-to level (how much)

1. Compute s: base stock model

s = (AVG x AVGL) + z x √(STDL2 x AVG2 + STD2 + AVGL)

Pipeline Inventory = AVG + AVGL

ss = z x √(STDL2 x AVG2 + STD2 + AVGL)

2. Compute Q using EOQ formula

3. Find order up to level

S = s + EOQ

When IP drops below s, order enough so IP after ordering is S.

INVENTORY MODELS

1. EOQ MODEL

CONSTANT demand rate & lead time

-how much: Q...