Level demand1

‘Q’ FOR APPROXIMATELY LEVEL DEMAND

‘Q’ FOR APPROXIMATELY LEVEL DEMAND

ASSUMPTIONSASSUMPTIONSn 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

FEW NOTATIONSFEW 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’

EOQEOQ

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

EOQEOQ

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)

Economic Production Lot SizeEconomic 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?

EOQEOQ

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.

COST PENALTY FOR USING Q’COST PENALTY FOR USING Q’

p0-0.5 0.1 0.2 0.3 0.4 0.5-0.4 -0.3 -0.2 -0.1

4

8

12

16

20

24PCP = 50(p2 / 1+p)

Q’ = (1 + p)EOQ

PCPPCP

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 EOQGraphical Aid for EOQ

A = $1.50

A = $3.20

A = $10

1000 2000 3000 40003200

1.11

2

3

4

5

r = 0.24 $ / $ / year

TEO

Q (

mon

ths o

f su

pp

ly)

Dv ($ / year)

Tabular Aid [A = 3.20 & r = 0.24]

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

TEOQ Graph and Tabular Aid

TEOQ Graph and Tabular Aid

n 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?

Quantity DiscountsQuantity 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?.

QUANTITY DISCOUNTSQUANTITY DISCOUNTS

n v = v0 0 ≤ Q ≤ Qb

n v0(1 - d) Qb ≤ Q

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)

QUANTITY DISCOUNTSQUANTITY DISCOUNTSn 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 Step 2: If EOQ(d) ≥ Qb, then EOQ(d) is optimal (case c); if EOQ(d) < Qb, go to step 3

n 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)

Quantity DiscountQuantity Discountn Consider items E010, E012 & E014, three components used in

the assembly of a electronic product. The supplier offers 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 Because of convenience in mfg and shipping, the supplier

offers a 2% discount on any replenishment of 100 units or

higher of a single item. Find optimal qty in all the three cases.

Item D (units/yr) v0 ($/unit) A($) r($/$/yr)

E010 416 14.2 1.50 0.24

E012 104 3.10 1.50 0.24

E014 4160 2.40 1.50 0.24

Planned ShortagesPlanned Shortages

-B

Q-B

t1 t2

QUANTITY DISCOUNTSQUANTITY 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.

FINITE REPLENISHMENT RATE


n 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 Slope = m - d

Max inventory = Q(1 - D/m)



Cost penalty

Cost

pen

alt

y f

or

usin

g b

asic

EO

Q

D/m0.5 1.0

Correction FactorCorr

ecti

on

Facto

r

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

Considerations

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 inventory cannot exceed certain 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

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

P (A/r = 11.07)

Q (A/r = 2.21)

Current Operating Point

Number of Replenishments Per Year (N)

Tota

l A

vera

ge C

ycle

Sto

ck (

in $

00

0s)

1

2

3

4

5

6

7

8

9

10

250 500 750 1000 1250 1500 1750 2000

A = $2.10

Limits on Order SizesLimits 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

SPECIAL OPPORTUNITY TO PROCURE


n Let the current unit cost be v1 and future unit cost be v2, then the EOQ after the price rise is given by,

n EOQ2 = √2AD/v2r

n 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



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:

n 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



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.

Level demand1

Business

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