2. ASSUMPTIONS
n
The demand rate is constant and deterministic
n
There are no maximum and minimum restrictions on size and
also need not be in integers
n
Unit variable cost doesn’t depend on the replenishment
quantity meaning no discounts in bulk purchase or bulk
transportation
n
An item is treated independently of other items. Benefits from
joint replenishment do not exist
n
Replenishment lead time is of zero duration
n
No shortages allowed and
n
Entire replenishment quantity is delivered at the same time
3. FEW NOTATIONS
n
Q - Replenishment order quantity
n
A - Fixed cost component (ordering cost) in rs. or in dollars
n
v - Unit variable cost of an item. Expressed in rs./unit or $/unit
n
r - Carrying charge, cost of having one rupee/$ of the item
tied up in the inventory for a unit time interval (rs./rs/unit time
or $/$/unit time)
n
TRC (Q) - Total relevant costs per unit time - Sum of those
costs per unit time which can be influenced by the order
quantity ‘Q’
4. EOQ
n
Total Carrying Cost = Īvr where Ī = Average inventory =
(Q+0)/2 = Q/2
n
So, Carrying cost = Q/2 . vr
n
TRC(Q) = Total carrying cost + Total ordering cost
n
TRC(Q) = Qvr / 2 + AD / Q
n
Tangent or slope of the curve is zero i.e. dTRC(Q)/dQ = 0
n
So, vr / 2 - AD / Q2 = 0
n
Qopt or EOQ = √2AD / vr
5. EOQ
Annual Demand D = 10000 units
No. of days in an year = 250
C = $10
h = 0.4 * 10 = 4
S = $500
Find EOQ.
p = 100 units/day (production rate)
6. Economic Production Lot Size
A detergent bar soap is produced on a production line
that has an annual capacity of 60000 cases. The
annual demand is estimated at 26000 cases, with
the demand rate essentially constant throughout the
year. The cleaning, preparation, and setup of the
production line cost approx $135. The
manufacturing cost per case is $4.5 and the annual
holding cost is figured at a 24% rate. Thus hC =
0.24(4.5) = $1.08. What is the recommended
production lot size?
7. EOQ
n
Consider a 3-ohm resistor used in the assembly of a
electronic item. The demand for this item has been relatively
level over time at a rate of 2400 units/yr. The unit variable
cost of the resistor is $0.40/unit and the fixed cost per
replenishment is estimated to be $3.20. Suppose further that
an r-value of 0.24$/$/year is appropriate to use. Find the
EOQ and Total relevant costs.
9. PCP
n
Suppose EOQ for previous problem was used as 550 instead
of 400. Then what is PCP?
n
PCP = 50 [p2 / (1+p)]
10. Graphical Aid for EOQ
TEOQ (months of supply)
5
4
r = 0.24 $ / $ / year
3
A = $10
2
A = $3.20
1.1
1
A = $1.50
1000
2000
3000 3200
Dv ($ / year)
4000
11. Tabular Aid [A = 3.20 & r = 0.24]
For Annual Dollar Usage
(Dv) in This Range
Use This Number of
Months of Supply
30720 ≤ Dv
1/4 (1 week)
10240 ≤ Dv < 30720
1/2 (2 weeks)
5120 ≤ Dv < 10240
3/4 (3 weeks)
1920 ≤ Dv < 5120
1
640 ≤ Dv < 1920
2
320 ≤ Dv < 640
3
192 ≤ Dv < 320
4
128 ≤ Dv < 192
5
53 ≤ Dv < 128
6
Dv < 53
12
12. T
n
EOQ
Graph and Tabular Aid
A Paper division sells 8 1/2” x 11” (A4 Size) in a box of 100
sheets. The product has been observed to have a relatively
constant demand rate (D) of 200 boxes/yr. The unit variable
cost (v) is $16/box. Also assume that it is reasonable to use
A = $3.20 and r = 0.24 $/$/year. What is EOQ using TEOQ Vs.
Dv graph and using Table as an aid?
13. Quantity Discounts
Quantities under 1000 may cost $12, those over 1000
but under 4000 may cost $10 and those from 4000
up may cost $8. What should be optimal order qty?.
14. QUANTITY DISCOUNTS
v0
0 ≤ Q ≤ Qb
n
v=
n
v0(1 - d)
n
Total Relevant Cost for Quantity without discount
n
TRC(Q) = Qv0r/2 + AD/Q + Dv0
n
Total Relevant Cost for Quantity with discount
n
TRC(Q) = Qv0(1 - d)r/2 + AD/Q + Dv0(1 - d)
Qb ≤ Q
15. QUANTITY DISCOUNTS
n
Efficient algorithm for finding the best value of Q are:
n
Step 1: Evaluate EOQ when discount is applicable
n
EOQ(d) = √2 A D / v0(1 - d)r
n
n
Step 2: If EOQ(d) ≥ Qb, then EOQ(d) is optimal (case c); if
EOQ(d) < Qb, go to step 3
Step 3: Evaluate TRC(EOQ) = √2ADv0r + Dv0 (without
discount) and evaluate TRC(Qb).
n
If TRC(EOQ) < TRC(Qb), then EOQ (no discount) = √2AD/v0r
is the best order qty. (case b)
n
& if TRC(Qb) < TRC(EOQ), then Qb is the best point. (case a)
16. in the assembly of a electronic product. The supplier offers
Quantity Discount
the same discount structure for each of the items and
discounts are based on replenishment sizes of the individual
items. The relevant characteristics of the items are given
below:
n
Item
D (units/yr) v0 ($/unit)
E010
416
14.2
E012
104
3.10
Because of convenience in mfg and
E014
4160
2.40
A($)
r($/$/yr)
1.50
0.24
1.50
shipping, the0.24
supplier
1.50
0.24
offers a 2% discount on any replenishment of 100 units or
higher of a single item. Find optimal qty in all the three cases.
18. QUANTITY DISCOUNTS
Suppose that Higley Radio components company has a
product for which the assumptions of the inventory model
with backorders are valid. Information obtained by the
company is as follows:
D = 2000 units per year; h = 0.20 of $50 = $10; S = $25 /
order; π=$30 per unit per year.
19. n
FINITE REPLENISHMENT RATE
Average Inventory = Q(1 - D/m)/2;
n
TRC(Q) = Q(1 - D/m)vr/2 + AD/Q; FREOQ = √2AD/vr(1-D/m)
n
n
Max inventory = Q(1 - D/m)
Slope = m - d
21. Selection of ‘r’. Ratio A/r based
on Aggregate Considerations
n
For a population of inventoried items, management may
impose aggregate constraint of one of the forms mentioned
below:
n
The average total
rupee/dollar value
n
The total fixed cost (or total number) of replenishments per
unit time must be less than a certain value
n
Operate at a point where the tradeoff between average
inventory and cost (or number) of replenishments per unit
time is at some reasonable prescribed value
inventory
cannot
exceed
certain
22. n
For all inventoried items, demand rate, unit variable cost and
order quantity of item ‘i’ is given by Di, vi and Qi respectively.
And ‘n’ be the number of items in the population
n
If we use EOQ for each item, Total Average Cycle Stock
TACS = √A/r . 1/√2. ∑ √Divi
n
And the total number of replenishments per unit time,
n
N = √r/A . 1/√2. ∑ √Divi
n
Multiplication of both the equations will give,
n
(TACS) (N) = 1/2 (∑ √Divi)2
n
Division of both the equation will give
n
(TACS) / (N) = A/r
23. Total Average Cycle Stock (in $ 000s)
A = $2.10
10
9
8
P (A/r = 11.07)
Current
Operating
Point
7
6
5
4
Q (A/r = 2.21)
3
2
1
250 500 750 1000 1250 1500 1750 2000
Number of Replenishments Per Year (N)
24. Limits on Order Sizes
n
Maximum Time Supply or Capacity Restriction or Shelf-life
Constraint
n
Even without shelf-life constraint, sometimes it becomes
unrealistic to order EOQ if it serves longer supply time as
demand becomes uncertain and parts obsolete. Class ‘C’
items fall into this category
n
Minimum Order Quantity restriction
25. n
SPECIAL OPPORTUNITY TO
PROCURE
Let the current unit cost be v1 and future unit cost be v2, then
the EOQ after the price rise is given by,
n
n
EOQ2 = √2AD/v2r
Total costs per unit time are given by TC(EOQ2) = √2ADv2r +
Dv2
n
If current order qty is of size Q, then Q/D is the period it
caters to, so the total costs for Q/D period only is given by
n
TC(Q) = Q/D.Qv1r/2 + Q/D.A D/Q + Dv1.Q/D
n
TC(Q) = A + v1Q + Q2v1r/2D
n
Select optimum Q so as to maximize the improvement in total
for time Q/D by ordering Q at old price and by not ordering at
old price
26. SPECIAL OPPORTUNITY TO
PROCURE
n
n
Select optimum Q so as to maximize the improvement in total
for time Q/D by ordering Q at old price and by not ordering at
old price and that is given by the total cost difference as
shown:
F(Q) = Q/D.√2ADv2r + Qv2 - A - Qv1 - Q2v1r/2D
n
To maximize the improvement in cost as said above,
differentiate the F(Q) w.r.t Q and equate with zero to get
Optimal Order Qty:
n
df(Q)/dQ = 1/D.√2ADv2r + v2 - v1 - Qv1r/D = 0
n
Qopt = √2ADv2r / v1r + (v2 - v1)D / v1r
n
Qopt = v2 / v1 . EOQ + (v2 - v1)D / v1r
27. SPECIAL OPPORTUNITY TO
PROCURE
n
Consider a medical product that is being sold at $28.00 will
be sold at $30.00 very soon by the supplier. The dealer/buyer
uses approximately 80 boxes per year and estimates the
fixed cost per order to be $1.50 and the carrying charge as
0.20 $/$/year. Find one time purchase quantity (Qopt) in this
case.
