1. business and industrial
2. energy
3. oil
4. oil and gas prices

Modeling and Analysis of
Manufacturing and Service
Enterprises
Why Manage Inventory?
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Inventory Management
Introduction and EOQ
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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?
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Total investment in inventory is 20-25%
of the total annual GNP.
What would happen if you control
inventories and achieve 1%
improvement overall?
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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
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Inventory
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Where do we hold inventory?
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Types of Inventory
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suppliers and manufacturers
warehouses and distribution centers
retailers
WIP and subassemblies
raw materials
finished goods
Why do we hold inventory? (Short answer)
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Economies of scale
Uncertainty in supply and demand
Decisions to Make
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We have to decide
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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?
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Economies of scale
Uncertainty in supply and demand
Speculation
Transportation
Smoothing production/purchasing
Logistics
Cost of controlling inventory
A Classification of Inventory
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Functional categories of inventories
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Cycle stock
Congestion stock
Safety stock
Anticipation inventory
Pipeline (or work-in-process) inventory
Decoupling stock
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ABC classification
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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.
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A items:
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B items
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Demand
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Constant (level) or variable
Deterministic (known) or Stochastic (random or
uncertain)
Lead Time
Review Time
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Continuous or periodic review
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Excess Demand
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Changing inventory
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Next 50% of the products (around
remaining 45-50% of total investment)
C items
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Characteristics of Inv. Systems
Usually First 5 to 10% of the products
(around 50% of the total investment)
Many, but low value
Relevant Costs
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Unit value or unit variable cost (v)
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Cost of making a part available for usage
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Purchase + Freight + Mfg. Costs
Usually different from “accounting” cost
Should include more than just book value
Backordered or lost
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Relevant Costs
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Relevant Costs
Holding cost (cost of carrying in inv.)
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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
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Holding cost (cont’d)
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h = ic
Total annual holding cost = Ih =Iic
Relevant Costs
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Inv.
Ordering or Setup Cost (A)
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Avg. inv. level
Dec
Time, t
Fixed cost
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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.
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EOQ Model
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Book Store Mug Sales
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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
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How many, when to order?
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T?
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EOQ Model
Inventory
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Production is instantaneous
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Delivery is immediate
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Demand is deterministic
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Demand is constant over time
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A production run incurs a fixed setup cost
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Products can be analyzed singly
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No shortages are allowed
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Anything else?
EOQ Model
Note:
• No Stockouts
• Order when no inventory
• Order size determines policy
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Time unit: A year
Purchase Cost Constant
Annual Holding Cost:
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Order
Quantity
Q
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Annual Ordering (Setup) Cost:
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Goal:
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Avg.
Inventory
(Avg. Inventory) * (Holding Cost)
Number of Orders in a year * Order Cost
Find the order quantity that
minimizes total costs
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Average Inventory: Q/2
Annual holding costs: Qh/2
Time between replenishments: Q/D
Number of replenishments per year: D/Q
Replenishment costs:
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EOQ Model
160
140
120
Cost
EOQ Model
Total Cost
100
Holding Cost
80
60
Order Cost
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(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
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Relevant Costs:
TRC(Q)=AD/Q+Qh/2
dTRC (Q )
h
A
=
−
=0
dQ
2D Q 2
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2 AD
EOQ = Q * =
h
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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?
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EOQ Model
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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%
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TRC( Q′) =
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TRC(Q)=1.0045*TRC(EOQ)
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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
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Finite Replenishment/Production Rate:
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Economic Production Lot (EPL) or
Economic Production Quantity (EPQ)
Nonzero constant lead time
Quantity Discounts
Considering Inflation
Limits on Order Size
Exchange Curves
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Economic Production Lot (EPL)
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What happens when there is finite
production rate?
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P: production rate (in units per year)
P>D (demand rate per year). Why?
What happens if you keep producing?
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The inventory will keep growing forever
with a rate of P-D.
EPL
EPL
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EPL
Slope=P-D
Inv
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Slope=-D
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T1
Stop
Prod.
T2
Production time = T1
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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:
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H = Q - DxT1 = Q (1-D/P)
Start
Prod.
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EPL
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Annual Total Relevant Cost for EPL:
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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
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TRC(EPL) = AD/Q + hH/2 or
TRC(EPL) = AD/Q + h(1-D/P)Q/2
Compare this to TRC(EOQ):
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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:
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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
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Quantity Discounts
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EOQ’s assumption: v is constant
What would you do if the cost of the
part depends on the order quantity?
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One easy case to analyze:
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Quantity Discounts
Slope=c0
Ordering
Cost
Price discounts if you order more
all-units discount
Qb
Breakpoint
quantity
Quantity Discounts
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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
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Calculate EOQ w/o discount: EOQ using c0 for
c
Calculate EOQ w/ discount: EOQ(d) using
c0(1-d) for c
Three cases:
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1. If EOQ(d)>=Q b, then Q*=EOQ(d)
2. If TRC(EOQ)<TRC(Q b), order EOQ
3. Otherwise [TRC(EOQ)>TRC(Q b)], order Q b.
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Quantity Discounts
Case 2
Case 1
EOQ with Inflation
Case 3
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Qb
Q*
Q* Q b
Example:
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Q* =
Q*=Q b
EOQ with Inflation
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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
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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:
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Q*(SL)=DxSL when T(EOQ)>SL
Uncertainty into the future
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EOQ Trade-off
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Two interpretations:
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Aggregate Considerations
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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,
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The trade-off is not linear!
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Aggregate Considerations
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Two major assumptions
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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
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Total Avg. Cycle Stock (in $):
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TACS = ∑ Q j c j / 2
j =1
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Total number of replenishments:
N = ∑ D /Q
Use EOQ for each item
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j
j =1
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Qj =
*
2 AD j
c ji
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Aggregate Considerations
Put Qj * into TACS and N:
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ADj c j
j =1
2i
TACS = ∑
=
A n
∑ Djc j
2i j =1
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N =∑
j =1
Resource Constrained Multiple
Product Systems
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Djc ji
i n
=
∑ Djc j
2A
2 A j =1
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Multiply TACS and N:
TACS × N =
n

1  n
∑ D j c j 
2  j =1

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Plot TACS versus N
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Example:
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D_j
c_j
A_j
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2
1150
350
150
3
800
85
50
Limit on the maximum inventory
investment: $30,000
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So Total Q jc j <= 30000
i=25%
Procedure:
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1850
50
100
Budget
Machines
Personnel
Space, etc.
Resource Constrained Multiple
Product Systems
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3 products
Even if you have multiple items to worry, you can
analyze them separately
What happens if the items share capacitated
resources?
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Resource Constrained Multiple
Product Systems
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Another EOQ assumption:
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Calculate regular EOQ value for each
product
If the constraint is satisfied with EOQ
values, then use them.
If using EOQ values violates the budget
constraint, scale them down.
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How?
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Resource Constrained Multiple
Product Systems
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Example:
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Ratio = 30,000 / 35,835 = 0.8372
New order quantities:
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172x50 + 63x350 + 61x85 = $35,835
How do we reduce the order sizes?
Multiply each EOQ value by the ratio
(budget limit)/(needed budget)
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Needed budget:
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EOQ1 = 172
EOQ2 = 63
EOQ3 = 61
Resource Constrained Multiple
Product Systems
Q1* = 172x0.8372 = 144
Q2* = 63x0.8372 = 52
Q3* = 61x0.8372 = 51
Round them down!
Resource Constrained Multiple
Product Systems
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A major assumption for the method to
work:
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c1/h1 = c2/h 2 = … =cn /hn
If you use same i for all items, this is
automatically satisfied
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Recall hj=cjij
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