1. Production and Service Systems Operations
Spring
Inventory Control – II Deterministic Demand: EOQ Extensions
Slide Set #4

2. Lead time
Inventory Q
Reorder point
R= τλ
Order
given
Arrives τ
Time

3. In-Class Exercise: Nahmias 4.15
David Gold orders salamis from New York and sells them
The demand for salamis is pretty steady at 175 per month
The salamis cost David 1.85 each
The fixed cost of ordering salamis from New York is $200
It takes three weeks to receive an order
The annual cost of capital is 22%, the cost of shelf space is 3% of value and the cost of taxes and insurance is 2% of value
Find the optimal order quantity of salamis and how often David should order
How many salamis should David have on hand when he orders?
Suppose that salamis sell $3 each. What is David’s annual profit from this business?

4. In-Class Exercise: Nahmias 4.15

5. Sensitivity What is the cost of using a suboptimal Q?
At the optimal solution,
G(Q*) = Kλ/ Q* + h Q*/2
For any other Q

6. Sensitivity Let Q*=250. What if we order Q=300?
G(Q) /G(Q*) = 0.5 ( ) =
We can conclude that G(Q) is relatively insensitive to errors in Q

7. Finite Production Rate
Inv. Level H -λ
P-λ
Time T1 T2 T

8. Example: Nahmias 4.3
Read-only memory (EPROM) producer. Demand flat at 2,500 units per year. Production rate is 10,000 units per year. Costs $50 to initiate a production run. Each unit costs $2 to manufacture. Annual interest rate is 30%.
Optimal size of a production run?
Length of a production run?
Average annual cost of holding and setup?
Maximum level of on-hand inventory?
maximum dollar investment in inventory?

9. Quantity Discount Models

10. Relaxing EOQ: Quantity Discounts
When the unit ordering cost depends on the order size
Two most popular types:
All-units: the discount is applied to all units in an order
Incremental: the discount is applied only to the items beyond the breakpoint

11. All-Units Discounts: Example
Somewhat irrational
499 bags cost $149.70
500 bags cost $145.00
What is the optimal order quantity, if
λ=600/year, K=$8, I=20% ?
Can we use EOQ directly?

12. All-Units Discounts
Copyright © 2001 by The McGraw-Hill Companies, Inc

13. The Average Annual Cost Function
500
1000
G0(Q)
G1(Q)
G2(Q)
G(Q) Q

14. All-Units Discounts: Solution Technique
Determine the largest realizable EOQ value
compute the EOQ for the lowest price first, and continue with the next higher price
stop when the EOQ value is realizable
Compare the value of the average annual cost at the largest realizable EOQ and at all the price breakpoints that are greater than the largest realizable EOQ

15. In-Class Exercise: Nahmias 4.24
In the calculation of an all-units discount schedule, you first compute the EOQ values for each of the three order costs, and you obtain:
Q0=800, Q1=875, Q2=925
The all-units discount schedule has breakpoints at 750 and 925. Based on this information only, can you the optimal order quantity?

16. Incremental Quantity Discounts

17. Incremental Quantity Discounts
The first 500 units cost 30 cents each,
the second 500 units cost 29 each,
the remaining cost 28 cents each

18. Incremental Quantity Discounts
Copyright © 2001 by The McGraw-Hill Companies, Inc

19. The Unit Cost
The unit cost is a function of Q (for Q>=500). Hence, we cannot use the EOQ formula
Instead, we use C(Q)/Q. The average annual cost function is:

20. Finding the Minima for Each Interval
Next, for each of the three intervals:
we substitute the relevant C(Q)/Q expression into G(Q) expression
find the Q value that minimizes G(Q) and check if it is realizable
For the interval Q < 500 , we have
which is minimized at
This value is realizable as it satisfies Q < 500

21. Finding the Minima for Each Interval
For the interval 500<= Q <1000, we have
which is minimized at
This value is realizable as it satisfies 500<= Q <1000
Note that

22. Finding the Minima for Each Interval
For the interval <= Q, we have
which is minimized at
This value is NOT realizable
Then,

23. Average Annual Cost Function
Copyright © 2001 by The McGraw-Hill Companies, Inc

24. Incremental Discounts: Solution Technique
Determine an algebraic expression for C(Q) corresponding to each price interval. Use that to determine an algebraic expression for C(Q)/Q
Substitute the expression derived for C(Q)/Q into the defining equation for G(Q). Compute the minimum value of Q corresponding to each price interval separately.
Determine which minima computed in (2) are realizable (that is, fall into the correct interval). Compare the values of the average annual costs at the realizable EOQ values and pick the lowest.

25. Resource-Constrained Multiple Product Systems

26. Resource-Constrained Multiple Product Systems
There is more than one product and the budget to invest in inventory is limited
Suppose that we are selling three different kinds of goods with the following data
Suppose we don’t want more than $30,000 invested in inventory at any time. Interest rate is 25%. What should be the ordering quantities?
Item 1
Item 2
Item 3
Demand
1,850
1,150
800
Variable Cost 50
350 85
Set-up Cost
100
150

27. Budget Constraint
If the budget constraint is satisfied when the EOQ values are used, then the EOQs are optimal
The EOQ values are calculated to be
EOQ1=172, EOQ2= 63, EOQ3=61
The budget is exceeded when using the EOQs: 172*50+63*350+61*85 = $35,835 > 30,000
Hence, we need to reduce the lot sizes…
but, how much?

28. Budget Constraint In general, the budget constraint is expressed as
c1Q1+ c2Q2+…+ cnQn<=C
If the constraint is not active, than EOQ is optimal
If the constraint is active:
if we include the following assumption,
c1/h1= c2/h2=…= cn/hn
(which requires the same I to be used for all products)
then the optimal solution is to scale the EOQ values so that the budget constraint holds

29. Space Constraint

30. Space Constraint Suppose that each item occupies an area of wi
The space constraint is
w1Q1+ w2Q2+…+ wnQn <= W
Mathematically similar to the budget-constraint
If the constraint is not active, the EOQ values are optimal

31. Space Constraint If the constraint is active
the simplifying condition requires
which is not realistic
When the condition is not met, solve the problem using the Lagrangian function. The optimal lot sizes are found as
where θ is the Lagrange multiplier

32. An EOQ Model for Production Planning

33. An EOQ Model for Production Planning
Determine the optimal procedure to produce n products on a common machine
to minimize the total cost of holding and setups
to guarantee that no stock outs occur
Assumptions:
Rotation cycle policy
Setup costs are not sequence-dependent
The following relation holds for feasibility:

34. An EOQ Model for Production Planning
Why not produce the EOQ for each item?
Let T be the cycle time.
The lot size for product j satisfies
The objective is to minimize

35. An EOQ Model for Production Planning
Optimization yields
If setup times (sj) need to be considered,
which leads to the constraint
In this case, the cycle time would be the larger of T* and Tmin