1. Operations Management Inventory Management Chapter 12 - Part 2

2. Outline Functions of Inventory. ABC Analysis. Inventory Costs.
Inventory Models for Independent Demand.
Economic Order Quantity (EOQ) Model.
Production Order Quantity (POQ) Model.
Quantity Discount Models.
Probabilistic Models for Varying Demand.
Fixed Period Systems.

3. Production Order Quantity Model
Material is not received instantaneously.
For example, it is produced in-house.
Other EOQ assumptions apply.
Model provides production lot size (like EOQ amount) for one product.
Similar to EOQ with setup cost rather than order cost.
One way to approach this is as an EOQ model with the instantaneous replenishment assumption relaxed. The following slide (EOQ Model modified to show changes for POQ) allows you to do this if you wish. Otherwise, skip it and move on.

4. Production Order Quantity Model
Consider one product at a time.
Produce Q units in a production run; then switch and produce other products.
Later produce Q more units in 2nd production run (Q units of product of interest).
Later produce Q more units in 3rd production run, etc.
One way to approach this is as an EOQ model with the instantaneous replenishment assumption relaxed. The following slide (EOQ Model modified to show changes for POQ) allows you to do this if you wish. Otherwise, skip it and move on.

5. POQ Model Inventory Levels
Production portion of cycle
Demand portion of cycle with no production (of this product)
Given that students recognize that production takes place for only a portion of the cycle, you might ask how one determines the appropriate length of the production period. If they understand the model, they will perceive that the production period is determined by the POQ.
Time
Production Begins
Production Run Ends

6. POQ Model Inventory Levels
Production rate = p = 20/day
Demand rate = d = 7/day
Inventory Level
Slope = p-d = 13/day
Slope = -d = -7/day
Given that students recognize that production takes place for only a portion of the cycle, you might ask how one determines the appropriate length of the production period. If they understand the model, they will perceive that the production period is determined by the POQ.
Time
Production Begins
Production Run Ends
Note: Not all of production goes into inventory

7. POQ Model Inventory Levels
Production rate = p = 20/day
Demand rate = d = 7/day
Inventory Level
Slope = p-d = 13/day
Inventory increases by 13 each day while producing
Slope = -d = -7/day
Inventory decreases by 7/day after producing
Given that students recognize that production takes place for only a portion of the cycle, you might ask how one determines the appropriate length of the production period. If they understand the model, they will perceive that the production period is determined by the POQ.
Time
Production Begins
Production Run Ends
Note: 1-(d/p) = fraction of production that goes into inventory

8. POQ Model Equations Given D = Q D = S Q (average inventory level)  H
D = Annual demand (relatively constant)
S = Setup cost per setup
H = Holding (carrying) cost per unit per year
d = Demand rate (units per day, units per week, etc.)
p = Production rate (units per day, units per week, etc.)
Determine: Q = Production run size (number of items per production run)
Given D
Number of Production Runs per year = Q
For some students, it is most important at this point to explain in detail the meaning and significance of each equation. It might be helpful to actually work through a numerical example.
Setup Cost per year = S D Q
Holding Cost per year =
(average inventory level)  H

9. POQ Model Inventory Levels
Maximum Inventory = Q(1-(d/p))
Production Portion of Cycle
Time
Demand portion of cycle with no supply

10. POQ Model Equations Given D = Q = S D Q Q = (ave. inventory level)  H
D = Annual demand (relatively constant)
S = Setup cost per setup
H = Holding (carrying) cost per unit per year
d = Demand rate (units per day, units per week, etc.)
p = Production rate (units per day, units per week, etc.)
Determine: Q = Production run size (number of items per production run)
Given D
Number of Production Runs per year = Q
For some students, it is most important at this point to explain in detail the meaning and significance of each equation. It might be helpful to actually work through a numerical example.
Setup Cost per year = S D Q Q
Holding Cost per year =
(ave. inventory level)  H =
H [1-(d/p)] 2

11. POQ Model Equations Given = × Q* D S H[1-(d/p)] 2 H 2DS p-d p =
D = Annual demand
S = Setup cost per setup
H = Holding (carrying) cost per unit per year
d = Demand rate
p = Production rate
Given = × Q* D S
H[1-(d/p)] 2 H
2DS
p-d p
Optimal Production Run Size =
For some students, it is most important at this point to explain in detail the meaning and significance of each equation. It might be helpful to actually work through a numerical example.
Maximum inventory level
= Q [1- (d/p)] D Q
Total Cost =
S +
H [1-(d/p)] Q 2

12. Run Length & Cycle Length
Production Run length (time) =
Q /p
Time
Inventory Level
For some students, it is most important at this point to explain in detail the meaning and significance of each equation. It might be helpful to actually work through a numerical example.
Cycle length (time) = Q /d

13. POQ Example 117.36 Demand = 1000/year (of product A)
Setup cost = $100/setup
Holding cost = $20 per year per item
Production rate = 10/day
365 working days per year
Demand rate = d = 1000/365
= 2.74/day
2 ×1000×100 =
Qp*
20×[1-(2.74/10)]
= units/run
Maximum inventory level
= [1- (2.74/10)] = 85.2 units
One way to approach this is as an EOQ model with the instantaneous replenishment assumption relaxed. The following slide (EOQ Model modified to show changes for POQ) allows you to do this if you wish. Otherwise, skip it and move on.
1000
117.36
Total Cost =
100 +
20 [1-(2.74/10)]
117.36 2
= = $ /year

14. POQ Example Demand = 1000 units/year Production rate = 10 units/day
Qp* = units per run
42.8
Demand rate = d = 1000/365
= 2.74/day
11.74
Production run length = /(10/day) = days
One way to approach this is as an EOQ model with the instantaneous replenishment assumption relaxed. The following slide (EOQ Model modified to show changes for POQ) allows you to do this if you wish. Otherwise, skip it and move on.
Cycle length = /(2.74/day) = 42.8 days
Number of production runs per year = 1000/ = 8.52

15. Robustness of POQ POQ is robust (like EOQ):
Can adjust production run size.
Useful even when parameters are uncertain.
A large (20%) change in parameters or operations will cause a small (~2%) change in total costs.
Production run size (Q) and run length (Q/p) can be adjusted to fit normal business cycles.
One way to approach this is as an EOQ model with the instantaneous replenishment assumption relaxed. The following slide (EOQ Model modified to show changes for POQ) allows you to do this if you wish. Otherwise, skip it and move on.

16. POQ Robustness Example
Set production run length to 14 days (2 weeks) rather than days (as was optimal).
Q/p = 14 days means that: Q = 10x14 = 140 units
Q = 140 is 19% over optimal value of units.
Cycle length = Q/d = 140/2.74 = 51.1 days.
Total cost = $
Only 1.6% over minimum cost with optimal Q!
One way to approach this is as an EOQ model with the instantaneous replenishment assumption relaxed. The following slide (EOQ Model modified to show changes for POQ) allows you to do this if you wish. Otherwise, skip it and move on.

17. POQ & Multiple Products
POQ computes a production run size for a single product.
For multiple products made on the same equipment:
Compute POQ, run time, and cycle time for each product.
Find a common cycle time for all products.
Recalculate run time and cycle time, so the common cycle time is a multiple of each product’s cycle time.
Fit production runs into largest cycle time.
One way to approach this is as an EOQ model with the instantaneous replenishment assumption relaxed. The following slide (EOQ Model modified to show changes for POQ) allows you to do this if you wish. Otherwise, skip it and move on.

18. Multiple Products Example
Example: Company makes 3 products: A, B, C
A: Optimal run time = 3 days; Optimal cycle time = 10 days
B: Optimal run time = 6 days; Optimal cycle time = 18 days
C: Optimal run time = 10 days; Optimal cycle time = 33 days 3 7 A 6 12 B
One way to approach this is as an EOQ model with the instantaneous replenishment assumption relaxed. The following slide (EOQ Model modified to show changes for POQ) allows you to do this if you wish. Otherwise, skip it and move on. 12 10 23 C

19. Multiple Products Example
Optimal run time and cycle time:
A: Run time = 3 days; Cycle time = 10 days (1 run/10 days)
B: Run time = 6 days; Cycle time = 18 days (1 run/18 days)
C: Run time = 10 days; Cycle time = 33 days (1 run/33 days)
Use 30 days as a common cycle; adjust run & cycle times:
A: Run time = 3 days; Cycle time = 10 days (3 runs/30 days)
B: Run time = 5 days; Cycle time = 15 days (2 runs/30 days)
C: Run time = 9 days; Cycle time = 30 days (1 run/30 days)
One way to approach this is as an EOQ model with the instantaneous replenishment assumption relaxed. The following slide (EOQ Model modified to show changes for POQ) allows you to do this if you wish. Otherwise, skip it and move on. 3 5 9 A B C
2 days idle time

20. Quantity Discount Model
Variation of EOQ (not POQ).
Allows quantity discounts.
Reduced price for purchasing larger quantities.
Other EOQ assumptions apply.
Trade-off lower price to purchase item & increased holding cost from more items.
Total cost must include annual purchase cost.
Total Cost = Order cost + Holding cost + Purchase cost

21. Quantity Discount Model - Holding Cost
Depends on price.
Usually expressed as a % of price per unit time.
20% of price per year, 2% of price per month, etc.
I = Holding cost percent of price per year
P = Price per unit
H = Holding cost = IP

22. Quantity Discount Equations
D = Annual demand
S = Order cost per order
H = Holding (carrying) cost = IP
I = Inventory holding cost % per year
P = Price per unit 2 × D × S
Order Quantity = Q* = IP
Annual purchase cost
For some students, it is most important at this point to explain in detail the meaning and significance of each equation. It might be helpful to actually work through a numerical example. D Q
Total Cost ($/yr) =
S +
IP + PD Q 2

23. Quantity Discount Model
Q P IP
< $100 $20
$ 95 $19
 $ 90 $18
D = 1000/year
S = $100/order
I = 20% per year
To solve:
1. Find EOQ amount for each discount level.
2. If EOQ is not in range for discount level, adjust to the nearest end of range.
3. Calculate total cost for each discount level.
4. Select lowest cost and corresponding Q.

24. Quantity Discount Example
Q P IP
< $100 $20
$ 95 $19
 $ 90 $18
D = 1000/year
S = $100/order
I = 20% per year
1. P = $100 IP = $20
EOQ = 100 in range!
Total Cost = 1, , ,000 = $102,000/year
2. P = $95 IP = $19
EOQ = not in range ( )!
Adjust to Q = 500
Total Cost = , ,000 = $99,950/year

25. Quantity Discount Example - cont.
Q P IP
< $100 $20
$ 95 $19
 $ 90 $18
D = 1000/year
S = $100/order
I = 20% per year
3. P = $90 IP = $18
EOQ = not in range (>1000)!
Adjust to Q = 1000
Total Cost = , ,000 = $99,100/year
Q Total costs
< $102,100
$ 99,950
 $ 99,100
Lowest cost, so order 1000

26. Stockouts
In basic EOQ model, demand and lead time are known and constant, so there should never be a stockout.
If demand or lead time vary, then may have a stockout:
Due to larger than expected demand.
Due to longer than expected lead time.
One point to stress here is that this is simply an extension of the original EOQ model where we are now allowing the demand to vary. Students should become accustomed to seeking such extensions as the need arises.
The next slide presents a graphical view of this model.

27. Probabilistic Models Inventory Level Reorder Point (ROP) Place order
Average demand
Reorder Point (ROP)
Place order
Receive order
Time
Lead Time 32

28. Probabilistic Models - Stockout
Inventory Level
If demand is greater than average - then stockout
Reorder Point (ROP)
Place order
Receive order
Time
Lead Time 32

29. Safety Stock to Reduce Stockouts
Inventory Level
Safety stock
New ROP
Old ROP
Place order
Receive order
Time
Lead Time 32

30. Safety Stock & Service Level
Safety stock is inventory held to protect against stockout.
Service level = 1 - Probability of stockout
Service level of 95% means 5% chance of stockout.
Higher service level means more safety stock.
More safety stock means higher ROP.
ROP = Expected demand during lead time + Safety stock
One point to stress here is that this is simply an extension of the original EOQ model where we are now allowing the demand to vary. Students should become accustomed to seeking such extensions as the need arises.
The next slide presents a graphical view of this model.

31. Probabilistic Models Demand follows normal distribution.
d = Average demand rate per day.
 = Standard deviation of demand.
ROP = d L + safety stock
safety stock = ss = Z 
Z is from Standard Normal Table in Appendix I.
One point to stress here is that this is simply an extension of the original EOQ model where we are now allowing the demand to vary. Students should become accustomed to seeking such extensions as the need arises.
The next slide presents a graphical view of this model.

32. Time between 1st & 2nd order Time between 2nd & 3rd order
EOQ-based Models
Order same amount every time = Q.
Time between orders varies.
ROP
Time
Time between 1st & 2nd order
Time between 2nd & 3rd order
This represents a model in which orders are based upon time, not the quantity needed. The following slide provides a graphical representation.

33. Fixed Period Model Order at fixed intervals (e.g., every 2 weeks).
Order different amounts each time, based on amount on hand.
If large amount on hand, then order small amount.
If small amount on hand, then order large amount.
Useful when vendors visit routinely.
Example: P&G representative calls every 2 weeks.
This represents a model in which orders are based upon time, not the quantity needed. The following slide provides a graphical representation.

34. Fixed Period Model
Compute optimal order interval, T (equation is similar to EOQ).
For example, days
Compute maximum inventory level, M (equation is similar to ROP).
Adjust order interval to a convenient length.
For example, one month.
Then, adjust M correspondingly.
Order M - inventory on hand every T time units.
This represents a model in which orders are based upon time, not the quantity needed. The following slide provides a graphical representation.

35. Fixed Period Models Order at constant interval.
Order amount Q varies: M - amount on hand.
On-hand for order 2
On-hand for order 1
This represents a model in which orders are based upon time, not the quantity needed. The following slide provides a graphical representation.
1st order
2nd order
3rd order
Time