1. ISE 216 – Production Systems Analysis
Chapter 4 – Inventory Control Subject to Known Demand
McGraw-Hill/Irwin
Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

2. This Chapter
Considers inventory control policies for individual item when product demand is assumed to follow a known pattern
Assumes zero forecast error.
Is this realistic? Hardly, but it is easier.
Do not worry we will get to the more realistic cases

3. Why do we study inventory?
Investment in Inventories in the U.S. Economy (1999) Inventory is money.

4. Reasons for Holding Inventories
Economies of scale
Uncertainty in delivery lead times
Speculation (Changing costs over time)
Smoothing
Demand uncertainty
Costs of maintaining control system

5. Characteristics of Inventory Systems
Demand
May be known or uncertain
May be changing or constant in time
Lead Times (time elapses from placement of order until its arrival)
known
unknown
Review policy: Is the system reviewed
periodically
continuously

6. Characteristics of Inventory Systems
Treatment of Excess Demand
Backorder all excess demand
Lose all excess demand
Backorder some and lose some
Inventory that changes over time
perishability
obsolescence

7. Relevant Costs
Holding Costs - Costs proportional to the quantity of inventory held. Includes:
Physical space cost (3%)
Taxes and insurance (2 %)
Breakage, spoilage and deterioration (1%)
Opportunity Cost of alternative investment (18%)
(in total: 24% of all costs)
Note: Since inventory level is changing on a continuous basis, holding cost is proportional to the area under the inventory curve

8. Inventory as a Function of Time

9. Relevant Costs (continued)
Penalty or Shortage Costs (opposite of holding ): All costs that happen when the stock is insufficient to meet the demand
Loss of revenue for lost demand
Costs of bookkeeping for backordered demands
Loss of goodwill for being unable to satisfy the demand
Generally assumed cost is proportional to number of units of excess demand

10. Relevant Costs (continued)
Ordering Cost (or Production Cost) - has both fixed and variable components
slope = c K
x : order or production quantity
C(x): ordering or production cost
C(x) = K + cx for x > 0
C(x) = 0 for x = 0.

11. Simple Economic Order Quantity (EOQ) Model – Assumptions
1. Demand is constant and uniform (l units/year, l/4 units per quarter)
2. Shortages are not allowed
3. Orders are received instantaneously
4. Order quantity is fixed (Q per cycle) - can be proven to be optimal
5. Costs
Fixed and marginal ordering costs (K + cQ)
Holding cost per unit held per unit time (h)

12. Inventory Levels for the EOQ Model
First order when inventory is 0. Reorder Q units every time when inventory is 0. It must be optimal

13. The EOQ Model: Notation
Parameters
λ : the demand rate (in units per time)time
c : unit ordering/roduction cost (in dollars per unit), setup or inventory costs are not included
K : setup cost (per placed order) in dollars
h : holding cost (in dollars per unit per year)
if the holding cost consists of interest of the money tied up in inventory,
h = ic, where
i is the annual interest rate

14. The EOQ Model: Notation
Decision variable is Q Q: lot size (order size) in units T : time between two consecutive orders (cycle length) G(Q) = average cost per unit time

15. Relationships Ordering Cost per cycle: C(Q) = K + cQ
Holding Cost per cycle =
unit holding cost * area of the triangle
h * QT/2 or
unit holding cost * average inventory size * cycle length
h*Q/2*T

16. Relationships Average Inventory Size = Q/2 why ?
Time Between Orders (Cycle length)
l = Q/T
T = Q/l
Time (t)
Inventory (I(t))
Assume Constant Demand T Q
Time between orders
slope = -l
Instantaneous
Replenishment T Q
Rate of consumption l

17. Total Costs G(Q) = average total (order + holding) cost
What is the average annual cost?
G(Q) = average total (order + holding) cost
= total cost per cycle / cycle length
Average inventory
level at any time
Ordering
cost per cycle

18. Average total cost
What is the average annual cost?

19. The Average Annual Cost Function G(Q)

20. Q that minimizes the annual cost
Let Q* be the optimal Q value. Is G’(Q*)=0 ?
YES!

21. Properties of the EOQ Solution
Q* is increasing function of K and  and decreasing function of h in square roots
Q* is independent of c (except the case we calculate h = Ic), Why?

22. Properties of the EOQ Solution
This formula is well-known as economic order quantity, is also known as economic lot size
This is a tradeoff between lot size (Q) and inventory
“Garbage in, garbage out” - usefulness of the EOQ formula for computational purposes depends on the realism of the input data
Estimating setup cost is not easily reduced to a single invariant cost K

23. Example 17.5=18 working days = 0.07 years * 250 working days a/year
UVIC annually requires 3600 liters of paint for scheduled maintenance of buildings.
Cost of placing an order is $16 and the interest rate (annual) is 25%. Price of paint is $8 per liter. How many liters of paint should be ordered and how often?
17.5=18 working days =
0.07 years * 250 working days a/year
3600
240 l Q T

24. Order Point for the EOQ Modelτ
Assumption: Delivery is immediate or order lead time τ = 0

25. Order Point for the EOQ Model
Does it matter if τ < T or τ > T ?
Keep track of time left to zero inventory or set to automatic reorder at a particular inventory level, R.
Two cases:
if τ < T R = λ*τ,
if τ > T R = λ* (τ mod T)

26. Sensitivity Analysis
G(Q) : the average annual holding and set-up cost function
independent of Q and omitted
G*: the optimal average annual holding and setup cost

27. Sensitivity Analysis
Cost penalties are quite small

28. Finite Replenishment Rate: Economic Production Quantity (EPQ)
Assumptions of EOQ:
Production is instantaneous: There is no capacity constraint, and entire lot is produced simultaneously
Delivery is immediate: There is no time lag between production and availability to satisfy demand

29. Inventory Levels for Finite Production Rate Model
Assumption : production rate is P (P > λ), arriving continuously.

30. The EPQ Model: Notation
Parameters λ : the demand rate (in units per year) P : the production rate (in units per year) c : unit production cost (in dollars per unit) K : setup costs (per placed order) in dollars h : holding cost (in dollars per unit per year) if the interest rate is given calculated as h=ic, Decision variables Q : size of each production run (order) in units T : time between two consecutive production orders (cycle length) T1 : production (replenishment) time T2 : no production (down time) H : maximum on-hand inventory level

31. The EPQ Model: Formula

32. The EPQ Model: Formula
For EPQ:
For EOQ:

33. Quantity Discount Models
The assumption is “The unit variable cost c did not depend on the replenishment quantity”
In practice quantity discounts exist based on the purchase price or transportation costs
Two types of discounts:
All Units Discounts: the discount is applied to ALL of the units in the order.
Order cost function such as that pictured in Figure 4-9 in Ch. 4.7
Incremental Discounts: The discount is applied only to the number of units above the breakpoint.
Order cost function such as that pictured in Figure 4-10

34. All-Units Discount Order Cost Function

35. All-Units Discount Average Annual Cost Function
G(Q) Q
500
1,000
G0(Q)
G1(Q)
G2(Q)
Gmin(Q)

36. Incremental Discount Order Cost Function

37. Average Annual Cost Function for Incremental Discount Schedule

38. Properties of the Optimal Solutions
For all units discounts, the optimal will occur at the minimum point of one of the cost curves or at a discontinuity point
One compares the cost at the largest realizable EOQ and all of the breakpoints succeeding it
For incremental discounts, the optimal will always occur at a realizable EOQ value.
Compare costs at all realizable EOQ’s.

39. Example
Supplier of paint to the maintenance department has announced new pricing:
$8 per liter if order is < 300 liters
$6 per liter if order is ≥ 300 liters
Other data is same as before:
K = 16, i= 25%, l = 3600
Is this a case of all units or incremental discount?

40. Solution Step 1: For price 1:
Step 2: As Q(1) < 300, EOQ is realizable.
Step 3: For price 2:
Step 4: As Q(2) < 300, EOQ is not realizable.

41. Cost Function Realizable G(Q|c1) G(Q|c2) Not Realizable C(Q) 240 277
300 Q

42. Cost Function Only possible solutions G(Q|c1) G(Q|c2) C(Q) 240 277 300

43. Solution Step 5: Compare costs of possible solutions.
For $8 price, Q=240:
For $6 price, Q=300:
Q=300 is the optimal quantity.

44. Resource Constrained Multi-Product Systems
Classic EOQ model is for a single item.
If we have multiple (n) items
Option A: Treat one system with multiple items as one item
Works if there is no interaction among the items, such as sharing common resources – budget, storage capacity, or both
Option B: Modify classic EOQ to insure no violation of the resource constraints
Works if you know how to use Lagrange multipliers!

45. Resource Constrained Multi-Product Systems
Consider an inventory system of n items in which the total amount available to spend is C
Unit costs of items are c1, c2, . . ., cn, respectively
This imposes the following budget constraint on the system (Qi is the order size for product i)
Let wi be the volume occupied by product i

46. Resource Constrained Multi-Product Systems
Budget constraint
Space constraint

47. Resource Constrained Multi-Product Systems
Lagrange multipliers method: relax one or more constraints
Minimize
by solving necessary conditions:

48. Resource Constrained Multi-Product Systems: Steps to Find Optimal Solution
Single constraint case :
Solve the unconstrained problem (Find the EOQ values). If the constraint is satisfied, this solution is the optimal one.
If the constraint is violated, rewrite objective function using Lagrange multipliers
Obtain optimal Qi* by solving (n+1) equations

49. Resource Constrained Multi-Product Systems: Steps to Find Optimal Solution
Double constraints case:
Solve the unconstrained problem (Find individual EOQ values). If both constraints are satisfied, this solution is the optimal one.
Otherwise, rewrite objective function using Lagrange multipliers by including one of the constraints. Solve one-constraint problem. If the other constraint is satisfied, this solution is the optimal one.
Otherwise repeat the step 2 for the other constraint.

50. Resource Constrained Multi-Product Systems: Steps to Find Optimal Solution
Double constraints case :
If both single-constraint solutions do not yield the optimal solution, then both constraints are active, and the Lagrange equation with both constraints must be solved.
Obtain optimal Qi* by solving (n+2) equations

51. EOQ Models for Production Planning
Problem: determine optimal procedure for producing n products on a single machine λj : demand rate (in units per year) of product j Pj : producton rate of product j cj : unit production cost (in dollars per unit) of product j Kj : setup cost (per placed order) in dollars of product j hj : holding cost (in dollars per unit per year) of product j The objective is to minimize the cost of holding and setups, and to have no stock-outs. To have a feasible solution: Assumption: We apply a rotation cycle policy Exactly one setup for each product in each cycle - production sequence stays the same in each cycle.

52. The method of solution is to express the average annual cost function in terms of the cycle time, T to assure no stock-outs.
The optimal cycle time T = max{T*, Tmin}, where sj is setup time of product j
The optimal production quantities are given by