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The order quantity where the TSC is at a minimum (EOQ) can be found using calculus (take the first derivative, set it equal to zero and solve for Q)
27.
Example: Basic EOQ
Zartex Co. stocks fertilizer to sell to retailers. One item – calcium nitrate – is purchased from a nearby manufacturer at Rs. 22.50 per ton. Zartex estimates it will need 5,750,000 tons of calcium nitrate next year.
The annual carrying cost for this material is 40% of the acquisition cost, and the ordering cost is Rs. 595.
a) What is the most economical order quantity?
b) How many orders will be placed per year?
c) How much time will elapse between orders?
28.
Example: Basic EOQ
Economical Order Quantity (EOQ)
D = 5,750,000 tons/year
C = .40(22.50) = Rs. 9.00/ton/year
S = Rs. 595/order
= 27,573.135 tons per order
29.
Example: Basic EOQ
Total Annual Stocking Cost (TSC)
TSC = (Q/2)C + (D/Q)S
= (27,573.135/2)(9.00)
+ (5,750,000/27,573.135)(595)
= 124,079.11 + 124,079.11
= Rs.248,158.22
Note: Total Carrying Cost equals Total Ordering Cost
30.
Example: Basic EOQ
Number of Orders Per Year
= D/Q
= 5,750,000/27,573.135
= 208.5 orders/year
Time Between Orders
= Q/D
= 1/208.5
= .004796 years/order
= .004796(365 days/year) = 1.75 days/order
Note: This is the inverse of the formula above.
31.
Model II: EOQ for Production Lots
Used to determine the order size, production lot, if an item is produced at one stage of production, stored in inventory, and then sent to the next stage or the customer
Differs from Model I because orders are assumed to be supplied or produced at a uniform rate (p) rate rather than the order being received all at once
. . . more
32.
Model II: EOQ for Production Lots
It is also assumed that the supply rate, p, is greater than the demand rate, d
The change in maximum inventory level requires modification of the TSC equation
TSC = (Q/2)[(p-d)/p]C + (D/Q)S
The optimization results in
33.
Example: EOQ for Production Lots
Highland Electric Co. buys coal from Cedar Creek Coal Co. to generate electricity. CCCC can supply coal at the rate of 3,500 tons per day for $10.50 per ton. HEC uses the coal at a rate of 800 tons per day and operates 365 days per year.
HEC’s annual carrying cost for coal is 20% of the acquisition cost, and the ordering cost is $5,000.
a) What is the economical production lot size?
b) What is HEC’s maximum inventory level for coal?
34.
Example: EOQ for Production Lots
Economical Production Lot Size
d = 800 tons/day; D = 365(800) = 292,000 tons/year
p = 3,500 tons/day
S = $5,000/order C = .20(10.50) = $2.10/ton/year
= 42,455.5 tons per order
35.
Example: EOQ for Production Lots
Total Annual Stocking Cost (TSC)
TSC = (Q/2)((p-d)/p)C + (D/Q)S
= (42,455.5/2)((3,500-800)/3,500)(2.10)
+ (292,000/42,455.5)(5,000)
= 34,388.95 + 34,388.95
= $68,777.90
Note: Total Carrying Cost equals Total Ordering Cost
36.
Example: EOQ for Production Lots
Maximum Inventory Level
= Q(p-d)/p
= 42,455.5(3,500 – 800)/3,500
= 42,455.5(.771429)
= 32,751.4 tons
Note: HEC will use 23% of the production lot by the time it receives the full lot.
37.
Key Points from EOQ Model
In deciding the optimal lot size, the tradeoff is between setup (order) cost and holding cost.
If demand increases by a factor of 4, it is optimal to increase batch size by a factor of 2 and produce (order) twice as often. Cycle inventory (in days of demand) should decrease as demand increases .
If lot size is to be reduced, one has to reduce fixed order cost. To reduce lot size by a factor of 2, order cost has to be reduced by a factor of 4.
38.
Model III: EOQ with Quantity Discounts
Under quantity discounts, a supplier offers a lower unit price if larger quantities are ordered at one time
This is presented as a price or discount schedule, i.e., a certain unit price over a certain order quantity range
This means this model differs from Model I because the acquisition cost (ac) may vary with the quantity ordered, i.e., it is not necessarily constant
. . . more
39.
All-Unit Quantity Discounts
Pricing schedule has specified quantity break points q 0 , q 1 , …, q r , where q 0 = 0
If an order is placed that is at least as large as q i but smaller than q i+1 , then each unit has an average unit cost of C i
The unit cost generally decreases as the quantity increases, i.e., C 0 >C 1 >…>C r
The objective for the company (a retailer in our example) is to decide on a lot size that will minimize the sum of material, order, and holding costs
40.
Model III: EOQ with Quantity Discounts
Under this condition, acquisition cost becomes an incremental cost and must be considered in the determination of the EOQ
The total annual material costs (TMC) = Total annual stocking costs (TSC) + annual acquisition cost
TSC = (Q/2)C + (D/Q)S + (D)ac
. . . more
41.
All-Unit Quantity Discounts: Example Cost/Unit Rs. 3 Rs. 2.96 Rs.2.92 Order Quantity 5,000 10,000 Order Quantity 5,000 10,000 Total Material Cost
42.
Model III: EOQ with Quantity Discounts
To find the EOQ, the following procedure is used:
1. Compute the EOQ using the lowest acquisition cost.
If the resulting EOQ is feasible (the quantity can be purchased at the acquisition cost used), this quantity is optimal and you are finished.
If the resulting EOQ is not feasible, go to Step 2
2. Identify the next higher acquisition cost.
43.
Model III: EOQ with Quantity Discounts
3. Compute the EOQ using the acquisition cost from Step 2.
If the resulting EOQ is feasible, go to Step 4.
Otherwise, go to Step 2.
4. Compute the TMC for the feasible EOQ (just found in Step 3) and its corresponding acquisition cost.
5. Compute the TMC for each of the lower acquisition costs using the minimum allowed order quantity for each cost.
TC2 < TC1 The optimal order quantity Q* is q2 = 10001
46.
Example: EOQ with Quantity Discounts
A-1 Auto Parts has a regional tire warehouse in Atlanta. One popular tire, the XRX75, has estimated demand of 25,000 next year. It costs A-1 $100 to place an order for the tires, and the annual carrying cost is 30% of the acquisition cost. The supplier quotes these prices for the tire:
Q ac
1 – 499 $21.60
500 – 999 20.95
1,000 + 20.90
47.
Example: EOQ with Quantity Discounts
Economical Order Quantity
This quantity is not feasible, so try ac = $20.95
This quantity is feasible, so there is no reason to try ac = $21.60
Continually review ordering practices and decisions
Modify to fit the firm’s demand and supply patterns
Constraints, such as storage capacity and available funds, can impact inventory planning
Computers and information technology are used extensively in inventory planning
51.
Inventory cycle is the central focus of independent demand inventory systems
Production planning and control systems are changing to support lean inventory strategies
Information systems electronically link supply chain
52.
Planning Supply Chain Activities Anticipatory - allocate supply to each warehouse based on the forecast Response-based - replenish inventory with order sizes based on specific needs of each warehouse
53.
Integrated planning at Shell
“ The most successful e-supply initiatives so far have been in industries where the components converge to create a product and where prices are not volatile. Energy is different. The supply chain is divergent; there are more products than raw materials and prices are highly volatile. Shell understands this and is aiming to create a reliable, real time, multi-point-optimized, an overview of entire supply chain…..
Across the globe this initiative has the potential to generate new value and drive savings to the tune of multiple of million US dollars a day.”
- A vice president of Shell Oil Products
54.
Integrated planning at Shell
Requirements for IT toolsets:
Complete horizontal supply chain integration
Convergence of strategy, planning and scheduling
Modularity to enable phased implementation and customization
Scalability
Interactive
Convenient User-interfacing
Real time results
Direct links to online refinery / plant optimization
55.
56.
Determining Order Points
Basis for Setting the Order Point
DDLT Distributions
Setting Order Points
57.
Basis for Setting the Order Point
In the fixed order quantity system, the ordering process is triggered when the inventory level drops to a critical point, the order point
This starts the lead time for the item.
Lead time is the time to complete all activities associated with placing, filling and receiving the order.
. . . more
58.
Basis for Setting the Order Point
During the lead time, customers continue to draw down the inventory
It is during this period that the inventory is vulnerable to stockout (run out of inventory)
Customer service level is the probability that a stockout will not occur during the lead time
. . . more
59.
Basis for Setting the Order Point
The order point is set based on
the demand during lead time (DDLT) and
the desired customer service level
Order point (OP) = Expected demand during lead time (EDDLT) + Safety stock (SS)
The amount of safety stock needed is based on the degree of uncertainty in the DDLT and the customer service level desired
60.
DDLT Distributions
If there is variability in the DDLT, the DDLT is expressed as a distribution
discrete
continuous
In a discrete DDLT distribution, values (demands) can only be integers
A continuous DDLT distribution is appropriate when the demand is very high
61.
Setting Order Point for a Discrete DDLT Distribution
Assume a probability distribution of actual DDLTs is given or can be developed from a frequency distribution
Starting with the lowest DDLT, accumulate the probabilities. These are the service levels for DDLTs
Select the DDLT that will provide the desired customer level as the order point
62.
Example: OP for Discrete DDLT Distribution
One of Sharp Retailer’s inventory items is now being analyzed to determine an appropriate level of safety stock. The manager wants an 80% service level during lead time. The item’s historical DDLT is:
DDLT (cases) Occurrences
3 8
4 6
5 4
6 2
63.
OP for Discrete DDLT Distribution
Construct a Cumulative DDLT Distribution
Probability Probability of
DDLT (cases) of DDLT DDLT or Less
2 0 0
3 .4 .4
4 .3 .7
5 .2 .9
6 .1 1.0
To provide 80% service level, OP = 5 cases
.8
64.
OP for Discrete DDLT Distribution
Safety Stock (SS)
OP = EDDLT + SS
SS = OP EDDLT
EDDLT = .4(3) + .3(4) + .2(5) + .1(6) = 4.0
SS = 5 – 4 = 1
65.
The Role of Safety Inventory in a Supply Chain
Forecasts are rarely completely accurate
If average demand is 1000 units per week, then half the time actual demand will be greater than 1000, and half the time actual demand will be less than 1000; what happens when actual demand is greater than 1000?
If you kept only enough inventory in stock to satisfy average demand, half the time you would run out
Safety inventory: Inventory carried for the purpose of satisfying demand that exceeds the amount forecasted in a given period
66.
Role of Safety Inventory
Average inventory is therefore cycle inventory plus safety inventory
There is a fundamental tradeoff:
Raising the level of safety inventory provides higher levels of product availability and customer service
Raising the level of safety inventory also raises the level of average inventory and therefore increases holding costs
Very important in high-tech or other industries where obsolescence is a significant risk (where the value of inventory, such as PCs, can drop in value)
Compaq and Dell in PCs
67.
Two Questions to Answer in Planning Safety Inventory
What is the appropriate level of safety inventory to carry?
What actions can be taken to improve product availability while reducing safety inventory?
68.
Determining the Appropriate Level of Safety Inventory
Measuring demand uncertainty
Measuring product availability
Replenishment policies
Evaluating cycle service level and fill rate
Evaluating safety level given desired cycle service level or fill rate
Impact of required product availability and uncertainty on safety inventory
69.
Determining the Appropriate Level of Demand Uncertainty
Appropriate level of safety inventory determined by:
supply or demand uncertainty
desired level of product availability
Higher levels of uncertainty require higher levels of safety inventory given a particular desired level of product availability
Higher levels of desired product availability require higher levels of safety inventory given a particular level of uncertainty
70.
Measuring Demand Uncertainty
Demand has a systematic component and a random component
The estimate of the random component is the measure of demand uncertainty
Random component is usually estimated by the standard deviation of demand
Notation:
D = Average demand per period
D = standard deviation of demand per period
L = lead time = time between when an order is placed and when it is received
Uncertainty of demand during lead time is what is important
71.
Measuring Demand Uncertainty
P = demand during k periods = kD
= std dev of demand during k periods = R Sqrt(k)
Coefficient of variation = cv = = mean/(std dev) = size of uncertainty relative to demand
72.
Measuring Product Availability
Product availability: a firm’s ability to fill a customer’s order out of available inventory
Stockout: a customer order arrives when product is not available
Product fill rate (fr): fraction of demand that is satisfied from product in inventory
Order fill rate: fraction of orders that are filled from available inventory
Cycle service level: fraction of replenishment cycles that end with all customer demand met
73.
Replenishment Policies
Replenishment policy: decisions regarding when to reorder and how much to reorder
Continuous review: inventory is continuously monitored and an order of size Q is placed when the inventory level reaches the reorder point ROP
Periodic review: inventory is checked at regular (periodic) intervals and an order is placed to raise the inventory to a specified threshold (the “order-up-to” level)
74.
Continuous Review Policy: Safety Inventory and Cycle Service Level
L : Lead time for replenishment
D: Average demand per unit time
D: Standard deviation of demand per period
D L : Mean demand during lead time
L : Standard deviation of demand during lead time
CSL : Cycle service level
ss : Safety inventory
ROP : Reorder point
Average Inventory = Q/2 + ss
75.
Auto Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed. The store manager is concerned that sales are being lost due to stockouts while waiting for an order. It has been determined that lead time demand is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons.
The manager would like to know the probability of a stockout during lead time.
Example: OP - Continuous DDLT Distribution
76.
Example: OP - Continuous DDLT Distribution
EDDLT = 15 gallons
DDLT = 6 gallons
OP = EDDLT + Z( DDLT )
20 = 15 + Z(6)
5 = Z(6)
Z = 5/6
Z = .833
77.
Example: OP - Continuous DDLT Distribution
Standard Normal Distribution
0 .833 Area = .2967 Area = .5 Area = .2033 z
78.
Example: OP - Continuous DDLT Distribution
The Standard Normal table shows an area of
.2967 for the region between the z = 0 line and the
z = .833 line. The shaded tail area is .5 - .2967 =.2033.
The probability of a stock-out during lead time is .2033
79.
Setting Order Point for a Continuous DDLT Distribution
The resulting DDLT distribution is a normal distribution with the following parameters:
EDDLT = LT(d)
DDLT =
80.
Setting Order Point for a Continuous DDLT Distribution
The customer service level is converted into a Z value using the normal distribution table
The safety stock is computed by multiplying the Z value by DDLT .
The order point is set using OP = EDDLT + SS, or by substitution
81.
Example
Q = 5; σ d = 1.5; SL = 95%
R = d + Z σ d = 5 + 1.645*1.5 = 5 + 2.5
= 7.5
Order 5 (Q) whenever the inventory level is below 7.5 (8).
So, what does this mean?
82.
Example A: Estimating Safety Inventory (Continuous Review Policy)
D = 2,500/week; D = 500
L = 2 weeks; Q = 10,000; ROP = 6,000
D L = D L = (2500)(2) = 5000
ss = ROP - R L = 6000 - 5000 = 1000
Cycle inventory = Q/2 = 10000/2 = 5000
Average Inventory = cycle inventory + ss = 5000 + 1000 = 6000
Average Flow Time = Avg inventory / throughput = 6000/2500 = 2.4 weeks
83.
Example B: Estimating Cycle Service Level (Continuous Review Policy)
D = 2,500/week; D = 500
L = 2 weeks; Q = 10,000; ROP = 6,000
Cycle service level, CSL = F(D L + ss, D L , L ) =
= NORMDIST (D L + ss, D L , L ) = NORMDIST(6000,5000,707,1)
= 0.92 (This value can also be determined from a Normal probability distribution table)
84.
Impact of Supply Uncertainty
D: Average demand per period
D: Standard deviation of demand per period
L : Average lead time
s L : Standard deviation of lead time
85.
Impact of Supply Uncertainty
D = 2,500/day; D = 500
L = 7 days; Q = 10,000; CSL = 0.90; s L = 7 days
D L = DL = (2500)(7) = 17500
ss = F -1 s (CSL) L = NORMSINV(0.90) x 17550
= 22,491
86.
Impact of Supply Uncertainty
Safety inventory when s L = 0 is 1,695
Safety inventory when s L = 1 is 3,625
Safety inventory when s L = 2 is 6,628
Safety inventory when s L = 3 is 9,760
Safety inventory when s L = 4 is 12,927
Safety inventory when s L = 5 is 16,109
Safety inventory when s L = 6 is 19,298
87.
Information Centralization
Virtual aggregation
Information system that allows access to current inventory records in all warehouses from each warehouse
Most orders are filled from closest warehouse
In case of a stockout, another warehouse can fill the order
The ability of a supply chain to delay product differentiation or customization until closer to the time the product is sold
Goal is to have common components in the supply chain for most of the push phase and move product differentiation as close to the pull phase as possible
Examples: Dell, Benetton
89.
Impact of Replenishment Policies on Safety Inventory
Continuous review policies
Periodic review policies
90.
Estimating and Managing Safety Inventory in Practice
Account for the fact that supply chain demand is lumpy
Adjust inventory policies if demand is seasonal
Use simulation to test inventory policies
Start with a pilot
Monitor service levels
Focus on reducing safety inventories
91.
Setting Order Point for a Continuous DDLT Distribution
Assume that the lead time (LT) is constant
Assume that the demand per day is normally distributed with the mean (d ) and the standard deviation ( d )
The DDLT distribution is developed by “adding” together the daily demand distributions across the lead time
. . . more
92.
Customer Service Criterion
The number of units short in one year (time period) is equal to the percentage short times the annual demand.
(1 – SL) * D
This is equal to the number of units short per order ( σ d E(Z)) times the number of orders per year (time period).
σ d E(Z) [D/Q]
SL = σ d E(Z)/Q
E(Z) = Q(1- SL)/ σ d
93.
Example Text
Q = 5; σ d = 1.5 SL = 95%
E(Z) = Q(1 - .05)/ σ d = 5*.05/1.5
= .167
From the tables Z = 0.6
R = d + Z σ d = 5 + 0.6*1.5 = 5 + 0.9 = 5.9
Order 5 when the inventory level reaches 5.9
94.
Example
A service station is located right across campus. His gas sales have been going down. To improve his sales he is considering utilizing some available space to place some soda vending machines. When he orders, he usually orders 10 cases (240 cans). He estimates that the daily demand can be approximated by a Normal distribution with a mean of 75 cans and a standard deviation of 10 cans. He also feels that an 85% (very sophisticated gas station owner) service level would be adequate. His soda supplier promises that his lead time will be exactly 4 days.
What should his reorder point be?
What is the safety stock?
95.
Solution in terms of probability of stock-out
a) N(75, 10) Time period correction factor
N(75*4, 10√4)
R = d + Z σ d
= 4*75 + (1.04)*(10 √4)
= 300 + 20.8 = 321
b) Safety Stock
SS = 20.8
96.
Another Example
D = annual demand 1000 units LT=15days
Q = 200 units Service Level = 95 % (.95)
Working days/yr = 250 σ d =50 units
Average demand/day=1000/250 = 4 units/day
R = d + Z σ d = 4*15 + Z(50)
E(Z)=(1-.95)200/50 = 0.2 from tables
Z = 0.49
R = 4(15) + 0.49*50 = 84.5
R = 4(15) + 1.645*50 = 142.25
97.
Example (Cont.)
Policy:
When inventory level gets to 85 or less then order 200.
What is the expected number of units short per order?