1. Operations Research
Lecture 14

2. Operations Research
Inventory Control

3. Inventory
Physical stock of items held in any business for the production or sales

4. Type of Inventory Finished Goods ! Finished Product Raw Material
In-Process
Finished Goods !

5. Inventory Costs Shortage Cost Rs C1 /item/unit time
Item Cost, Rs C1/item Ordering Cost, Rs C2 /order
Holding Cost Rs C /item/unit time
Shortage Cost Rs C /item/unit time

6. Inventory Models (EOQ Models)
The inventory control model can be broadly classified into two categories:
Deterministic inventory problems
Probabilistic inventory problems

7. Model 1
Purchasing Model
with no Shortages

8. Assumptions Demand (D) is at a constant rate.
Replacement of items is instant
The C1, C2, and C3 are constant.
No shortage cost, C4 = 0.

9. Cont !
Quantity
Time

10. D is the demand per year

11. Model 2
Purchasing Model
with Shortages

13. Example

14. The demand for an item is 18000 units/year
The demand for an item is units/year. The cost of one purchase is Rs 400. The holding cost is Rs 1.2 per unit per year. The item cost is Rs 1 per item. The shortage cost is Rs 5 per unit per year. Determine:
(a)     The optimum order quantity.
(b)     The time between orders.
(c)     The number of orders per year.
(d)     The optimum shortages.
(e)     The maximum inventory.
(f)      The time of items being held.
(g)     The optimum annual cost.

15. = 1500 units / month C1 = Rs 1.0 / item C3 = Rs 1.2 / year/ item
Data
D = units / year
= 1500 units / month
C1 = Rs 1.0 / item
C3 = Rs 1.2 / year/ item
C2 = Rs 400 /order
C4 = Rs 5.0 / year / item

16. Optimum Order Quantity

17. Time Between Orders

18. Number of Orders
per year
Number of orders
per year
= 12 / 2.57
= 4.66

19. Optimum Shortages

20. Maximum Inventory

21. The Time when
Items being held

22. Order cost = Rs 400 per order
Optimum Annual Cost
Annual cost = Item cost
+ Ordering cost
+ Holding cost
+ Shortage cost
Item cost = Rs 3857 per order
Order cost = Rs 400 per order

23. Holding Cost

24. Shortage Cost

25. Total Cost per order

26. Model 3
Manufacturing Model
with no Shortages

27. Demand rate (D) is constant
 Assumptions:
Demand rate (D) is constant
All cost coefficients (C1, C2, C3) are constants
There is no shortage cost, or C4 = 0
The replacement rate is finite and greater than the demand rate. This is also called replenishment rate or manufacturing rate, denoted by R

28. Demand rate (D) is constant
 Assumptions:
Demand rate (D) is constant
All cost coefficients (C1, C2, C3) are constants
There is no shortage cost, or C4 = 0

29.  Cont !
The replacement rate is finite and greater than the demand rate. This is also called replenishment rate or manufacturing rate, denoted by R

31. Slope = D
Slope = (R-D) Q Im t t2 t1
t = t1 + t2

32. Q T Q Im s t t2 t1
t = t1 + t2

33. The total cost of inventory per period is the sum of three components:
The total cost of inventory per period is the sum of three components: item cost, order cost, and items holding cost.

34. Let Im be the maximum inventory, t1 be the time of manufacture and t2 be the time during which there is no supply.
In this model, all items required for a cycle are not stored at the beginning as in Wilson’s Model.
The items are manufactured at a higher rate than the demand so that the difference (R–D) is the existing inventory till the items are exhausted.

35. Item cost/period = C1Q
Order cost/period = C2
Item holding cost/period =

39. Differentiating C with respect to Q and setting equal to zero for minimum C, we get,

40. Example

41. The demand for an item in a company is units/year and the company can produce at the rate of 3000 per month. The cost of one set up is Rs. 500 and the holding cost of 1 unit per month is 15 paisa. Determine:
(a)     The optimum manufacturing quantity.
(b)     The maximum inventory.
(c)     The time between orders.
(d)     The number of orders/year.
(e)     The time of manufacture.
(f)      The optimum annual cost if the cost of the item per unit is Rs. 2.
Assume no shortages.

42. C2 = Rs 500 per order. C3 = Rs 0.15 / item per month
Data
C1 = Rs 2 per item.
C2 = Rs 500 per order.
C3 = Rs 0.15 / item per month
D = / year = 1500/ month
R = 3000/month

43. Operations Research
Lecture 14