Supply Chain Management Formulas

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NEWSVENDOR MODEL
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...
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