‘Q’ FOR
APPROXIMATELY
LEVEL DEMAND
ASSUMPTIONS
n

The demand rate is constant and deterministic

n

There are no maximum and minimum restrictions on size and...
FEW NOTATIONS
n

Q - Replenishment order quantity

n

A - Fixed cost component (ordering cost) in rs. or in dollars

n

v ...
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) =...
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...
Economic Production Lot Size
A detergent bar soap is produced on a production line
that has an annual capacity of 60000 ca...
EOQ

n

Consider a 3-ohm resistor used in the assembly of a
electronic item. The demand for this item has been relatively
...
COST PENALTY FOR USING Q’
PCP = 50(p2 / 1+p) 24

20
16

Q’ = (1 + p)EOQ

12
8
4

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0....
PCP

n

Suppose EOQ for previous problem was used as 550 instead
of 400. Then what is PCP?

n

PCP = 50 [p2 / (1+p)]
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
100...
Tabular Aid [A = 3.20 & r = 0.24]
For Annual Dollar Usage
(Dv) in This Range

Use This Number of
Months of Supply

30720 ≤...
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 be...
Quantity Discounts

Quantities under 1000 may cost $12, those over 1000
but under 4000 may cost $10 and those from 4000
up...
QUANTITY DISCOUNTS
v0

0 ≤ Q ≤ Qb

n

v=

n

v0(1 - d)

n

Total Relevant Cost for Quantity without discount

n

TRC(Q) = ...
QUANTITY DISCOUNTS
n

Efficient algorithm for finding the best value of Q are:

n

Step 1: Evaluate EOQ when discount is a...
in the assembly of a electronic product. The supplier offers

Quantity Discount

the same discount structure for each of t...
Planned Shortages

Q-B

-B

t1

t2
QUANTITY DISCOUNTS

Suppose that Higley Radio components company has a
product for which the assumptions of the inventory ...
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)
...
Correction Factor

Cost penalty for using basic EOQ

FINITE REPLENISHMENT RATE

Correction Factor

0.5
D/m

Cost penalty

...
Selection of ‘r’. Ratio A/r based
on Aggregate Considerations
n

For a population of inventoried items, management may
imp...
n

For all inventoried items, demand rate, unit variable cost and
order quantity of item ‘i’ is given by Di, vi and Qi res...
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)...
Limits on Order Sizes

n

Maximum Time Supply or Capacity Restriction or Shelf-life
Constraint

n

Even without shelf-life...
n

SPECIAL OPPORTUNITY TO
PROCURE
Let the current unit cost be v1 and future unit cost be v2, then
the EOQ after the price...
SPECIAL OPPORTUNITY TO
PROCURE
n

n

Select optimum Q so as to maximize the improvement in total
for time Q/D by ordering ...
SPECIAL OPPORTUNITY TO
PROCURE

n

Consider a medical product that is being sold at $28.00 will
be sold at $30.00 very soo...
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Level demand1

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Level demand1

  1. 1. ‘Q’ FOR APPROXIMATELY LEVEL DEMAND
  2. 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. 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. 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. 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. 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. 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.
  8. 8. COST PENALTY FOR USING Q’ PCP = 50(p2 / 1+p) 24 20 16 Q’ = (1 + p)EOQ 12 8 4 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 p
  9. 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. 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. 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. 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. 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 $8. What should be optimal order qty?.
  14. 14. QUANTITY DISCOUNTS v0 0 ≤ Q ≤ Qb n v= n v0(1 - d) n Total Relevant Cost for Quantity without discount n TRC(Q) = Qv0r/2 + AD/Q + Dv0 n Total Relevant Cost for Quantity with discount n TRC(Q) = Qv0(1 - d)r/2 + AD/Q + Dv0(1 - d) Qb ≤ Q
  15. 15. QUANTITY DISCOUNTS n Efficient algorithm for finding the best value of Q are: n Step 1: Evaluate EOQ when discount is applicable n EOQ(d) = √2 A D / v0(1 - d)r n n Step 2: If EOQ(d) ≥ Qb, then EOQ(d) is optimal (case c); if EOQ(d) < Qb, go to step 3 Step 3: Evaluate TRC(EOQ) = √2ADv0r + Dv0 (without discount) and evaluate TRC(Qb). n If TRC(EOQ) < TRC(Qb), then EOQ (no discount) = √2AD/v0r is the best order qty. (case b) n & if TRC(Qb) < TRC(EOQ), then Qb is the best point. (case a)
  16. 16. in the assembly of a electronic product. The supplier offers Quantity Discount the same discount structure for each of the items and discounts are based on replenishment sizes of the individual items. The relevant characteristics of the items are given below: n Item D (units/yr) v0 ($/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.
  17. 17. Planned Shortages Q-B -B t1 t2
  18. 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. 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
  20. 20. Correction Factor Cost penalty for using basic EOQ FINITE REPLENISHMENT RATE Correction Factor 0.5 D/m Cost penalty 1.0
  21. 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. 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. 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. 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. 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. 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. 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.

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