Chapter 2 - Lot sizing, EOQ, POQ, EOQ with Discounts
What is cycle stock and how is it controlled?
Give the assumptions of the EOQ problem. Work out a few problems
Give the Assumptions of the POQ problem. Work out a few problems
Give the assumptions of the EOQ with price discounts. Work out a few problems
Cycle stock and EOQ
Cycle stock, also known as inventory between order points, refers to the average amount of
inventory a business holds to meet customer demand during the regular lead time for new
orders. Economic Order Quantity (EOQ) is a mathematical formula that helps determine the
optimal order size for cycle stock. By using EOQ, businesses can minimize total inventory
costs, which include ordering costs and holding costs.
Here's how EOQ helps control cycle stock:
1. Balancing Costs: The EOQ formula considers both the demand for a product (annual
usage) and the costs associated with inventory. Ordering costs include things like
processing paperwork, shipping fees, and staff time dedicated to placing orders.
Holding costs encompass storage space, insurance, and potential product
degradation over time. EOQ helps find the order quantity that minimizes the sum of
these two costs.
2. Setting Order Points: EOQ determines the ideal order quantity, but it doesn't tell you
exactly when to order. Knowing your average daily or weekly usage rate, you can
calculate a reorder point – the inventory level at which you should trigger a new order
to ensure you don't run out of stock before the next shipment arrives. This reorder
point helps maintain your desired cycle stock level.
There are limitations to using EOQ alone for cycle stock control. For instance, EOQ
assumes constant demand and ignores factors like seasonal fluctuations or product
promotions. However, EOQ serves as a strong foundation for managing cycle stock, and
many businesses use it along with other inventory management techniques for optimal
control.
Assumptions of the EOQ framework
The EOQ model in inventory control relies on several simplifying assumptions to make the
calculations tractable. While these assumptions may not perfectly reflect real-world
situations, they provide a good starting point for determining optimal order quantities:
● Constant Demand and Lead Time: The EOQ model assumes that demand for the
product is constant throughout the year and that the lead time (time between placing
an order and receiving the inventory) is always consistent. This allows for a
predictable inventory flow.
● Fixed Costs: The model considers two main cost components: ordering costs and
holding costs. Ordering costs are assumed to be constant per order, regardless of the
order quantity. Holding costs, which include storage, insurance, and potential
obsolescence, are also assumed to be constant per unit of inventory held per unit of
time.
● Single Item: The EOQ model is designed for a single inventory item. It doesn't take
into account situations where you might be managing multiple products with different
demand patterns or costs.
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● No Backorders or Discounts: The model assumes that no stockouts or backorders
occur, and there are no quantity discounts offered by suppliers for larger orders.
● Instantaneous Replenishment: The EOQ model assumes that the entire order
quantity is delivered at once, rather than in multiple shipments.
While these assumptions can limit the EOQ model's real-world applicability, it remains a
valuable tool for understanding the trade-off between ordering and holding costs and for
establishing a baseline for inventory management. Businesses can then adjust the EOQ as
needed to account for factors like seasonal variations or potential discounts.
Problem 1
ABC Company sells widgets and orders them from a supplier. The company's annual
demand for widgets is 10,000 units. Each widget costs $10 to purchase, and the ordering
cost per purchase order is $50. The carrying cost per unit per year is $2. ABC Company
operates 250 days a year. Calculate the Economic Order Quantity (EOQ) for widgets.
D 10,000 widgets/year
Co $ 50 Cost per order
Cc $2 Cost per unit per
year
EOQ 707.11 widgets/order
sqrt(2*D*Co/Cc) Formula
Number of 14.14 orders/year
orders
Ordering 707.11 Annual ordering
cost cost
Hosling cost 707.11 Annual holding cost
Total cost 1,414,21 Total overhead
inventory cost
Material cost $ 100,000
Problem 2
Company XYZ sells a particular product with an annual demand of 50,000. The ordering cost
per purchase order is $200, and the carrying cost per dollar of inventory per year is $0.10.
Calculate the optimal order size (EOQ), total inventory cost, and the number of orders per
year.
D products/year
Co Cost per order
2
Cc Cost per unit per
year
EOQ products/order
sqrt(2*D*Co/Cc) Formula
Number of orders/year
orders
Ordering Annual ordering
cost cost
Hosling cost Annual holding cost
Total cost Total overhead
inventory cost
Material cost
Problem 3
ABC Inc. operates a warehouse where it stores its inventory. The annual demand for a
certain item is 80,000. The setup cost for ordering this item is $500 per order, and the
carrying cost per dollar of inventory per year is $0.05. Calculate the Economic Order
Quantity (EOQ), total inventory cost, and the number of orders per year.
D products/year
Co Cost per order
Cc Cost per unit per
year
EOQ products/order
sqrt(2*D*Co/Cc) Formula
Number of orders/year
orders
Ordering Annual ordering
cost cost
Hosling cost Annual holding cost
Total cost Total overhead
inventory cost
Material cost
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POQ or EPQ model
A variation of the basic EOQ model is the production quantity model, also referred to as the
gradual usage and non-instantaneous receipt model.
In this EOQ model the assumption that orders are received all at once is relaxed.
The order quantity is received gradually over time, and the inven-tory level is depleted at the
same time it is being replenished.
This situation is commonly found when the inventory user is also the producer, as in a
manufacturing operation where a part is produced to use in a larger assembly. This situation
also can occur when orders are delivered continuously over time or when a retailer is also
the producer.
The noninstantaneous receipt model is shown graphically in Figure. The inventory level is
gradually replenished as an order is received. In the basic EOQ model, average inventory
was half the maximum inventory level, or Q/2, but in this model variation, the maximum
inventory level is not simply Q; it is an amount somewhat lower than Q, adjusted for the fact
the order quantity is depleted during the order receipt period.
In order to determine the average inventory level, we define the following parameters unique
to this model:
p = daily rate at which the order is received over time, also known as the production rate
d = the daily rate at which inventory is demanded
The demand rate cannot exceed the production rate, since we are still assuming that no
shortages
are possible, and, if d = p, there is no order size, since items are used as fast as they are
produced.
For this model the production rate must exceed the demand rate, or p >= d.
Observing Figure we see that the time required to finish receiving an order is the order
quantity divided by the rate at which the order is received, or Q/p. For example, if the order
size is 100 units and the production rate, p, is 20 units per day, the order will be received
over five days.
The amount of inventory that will be depleted or used up during this time period is
determined by multiplying by the demand rate: (Q/p)d. For example, if it takes five days to
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receive the order and during this time inventory is depleted at the rate of two units per day,
then 10 units are used. As a result, the maximum amount of inventory on hand is the order
size minus the amount depleted during the receipt period, computed as
Maximum inventory level = Q - (Q/d)*P = Q (1 - d/p)
Since this is the maximum inventory level, the average inventory level is determined by
dividing this amount by 2:
Average inventory level = [Q (1 - d/p)]/2
Total carrying cost = Cc * average inventory level = Cc [Q (1 - d/p)]/2
In this case the ordering cost, Co, is often the setup cost for production and is sometimes
shown as Cs
Thus, the total annual inventory overhead cost is determined according to the following
formula:
TC = Co D/Q + Cc [Q (1 - d/p)]/2
Solving this for optimal value for Q we get
Problem 1
Assume that the ePaint Store has its own manufacturing facility in which it produces Ironcoat
paint. The ordering cost, Co, is the cost of setting up the production process to make paint.
Co = $150. Recall that Cc = $0.75 per gallon and D = 10,000 gallons per year. The
manufacturing facility operates the same days the store is open (i.e., 311 days) and
produces 150 gallons of paint per day. Determine the optimal order size, total inventory cost,
the length of time to receive an order, the number of orders per year, and the maximum
inventory level.
Co 150 $/order
Cc 0.75 $/gallon
Annual demand D 10,000 gallons/year
Working days 311 days/year
Daily demand d 32.2 gallons/day
Daily production capacity p 150 gallons/day
Optimum quantity Q 2256.85 gallons/order
Number of production runs D/Q 4.43 Runs per year
Lead time or production run length Q/p 15.05 days/order
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Total Ordering Cost 664.64 $/year
Co (D/Q)
Total Holding cost 664.64 $/year
Q(1- d/p)/2
Total Cost 1,329.29 $/year
Maximum inventory level 1,772.38 gallons
Problem 2
I-75 Discount Carpets manufactures Cascade carpet, which it sells in its adjoining showroom
store near the interstate. Esti-mated annual demand is 20,000 yards of carpet with an annual
carrying cost of $2.75 per yard. The manufacturing facility operates the same 360 days the
store is open and produces 400 yards of carpet per day. The cost of setting up the
manufacturing process for a production run is $720. Determine the optimal order size, total
inventory cost, length of time to receive an order, and maximum inventory level.
Co $/order
Cc $/yard
Annual demand D yards/year
Working days days/year
Daily demand d yards/day
Daily production capacity p yards/day
Optimum quantity Q yards/order
Number of production runs D/Q Runs per year
Lead time or production run length Q/p days/order
Total Ordering Cost $/year
Co (D/Q)
Total Holding cost $/year
Q(1- d/p)/2
Total Cost $/year
Maximum inventory level yards
Problem 3
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The Ambrosia Bakery makes cakes for freezing and subsequent sale. The bakery, which
operates five days a week, 52 weeks a year, can produce cakes at the rate of 116 cakes
per day. The bakery sets up the cake production operation and produces until a
predetermined number (Q) have been produced. When not producing cakes, the bakery
uses its personnel and facilities for producing other bakery items. The setup cost for a
production run of cakes is $700. The cost of holding frozen cakes in storage is $9 per cake
per year. The annual demand for frozen cakes, which is constant over time, is 6000 cakes.
Determine the Optimal production run quantity (Q), Total annual inventory costs, Optimal
number of production runs per year, Optimal cycle time (time between run starts), Run
length in working days
Co $/setup
Cc $/cake/year
Annual demand D cakes/year
Working days days/year
Daily demand d cakes/day
Daily production capacity p cakes/day
Optimum quantity Q cakes/order
Number of production runs D/Q Runs per year
Lead time or production run length Q/p days/order
Total Ordering Cost $/year
Co (D/Q)
Total Holding cost $/year
Q(1- d/p)/2
Total Cost $/year
Maximum inventory level cakes
EOQ with bulk discounts
You are the inventory manager for a company that sells widgets. Each widget costs $10 at a
base price. You place orders throughout the year to meet an annual demand of 12,000
widgets. The cost of placing an order (setup cost) is $20. You also incur a holding cost of
10% of the unit price per year for each widget you hold in inventory.
Discounts Offered: Your supplier offers quantity discounts based on the order size:
Order less than 300 widgets: No discount (price = $10/unit)
Order 300 to 500 widgets: 2% discount (price = $9.80/unit)
Order more than 500 widgets: 5% discount (price = $9.50/unit)
In table below, use values of Q between 200 and 800 in steps of 50. What do you observe?
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no of ordering Average Carrying Carrying Material Material
Q orders cost Stock cost/unit cost price cost Total Cost
200 60.00 1200.00 100 1 100 10 120,000.00 121,300.00
250 48.00 960.00 125 1 125 10 120,000.00 121,085.00
300 40.00 800.00 150 0.98 147 9.8 117,600.00 118,547.00
350 34.29 685.71 175 0.98 171.5 9.8 117,600.00 118,457.21
400 30.00 600.00 200 0.98 196 9.8 117,600.00 118,396.00
450 26.67 533.33 225 0.98 220.5 9.8 117,600.00 118,353.83
500 24.00 480.00 250 0.95 237.5 9.5 114,000.00 114,717.50
550 21.82 436.36 275 0.95 261.25 9.5 114,000.00 114,697.61
600 20.00 400.00 300 0.95 285 9.5 114,000.00 114,685.00
650 18.46 369.23 325 0.95 308.75 9.5 114,000.00 114,677.98
700 17.14 342.86 350 0.95 332.5 9.5 114,000.00 114,675.36
750 16.00 320.00 375 0.95 356.25 9.5 114,000.00 114,676.25
800 15.00 300.00 400 0.95 380 9.5 114,000.00 114,680.00
How we work out this problem
EOQ computation for each price range
Quantity range Price D Co Cc EOQ
Less than 300 10 12000 20.00 1 692.82
300 to 500 9.8 12000 20.00 0.98 699.85
More than 500 9.5 12000 20.00 0.95 710.82
Quantity Ordering Holding costs Material Costs Total Costs Range
costs
692.82 NEGLECT Less than 300
300 800 150 120,000 120,950 boundry
699.85 NEGLECT 0 to 500
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500 480 245 117,600 118,325 boundry
710.82 337.64 337.64 114,000 114,675.28 More than 500
We will place an order for 710 units each order
Number of orders = 12000/710 = 16.901
Problem 2
You manage inventory for bolts. Annual demand is 15,000 units. Ordering cost is $30, and
holding cost is 12% of the unit price ($5/unit) per year. The supplier offers these discounts:
Order less than 200 units: No discount
Order 200 to 400 units: 3% discount
Order more than 400 units: 5% discount
However, there's a minimum order quantity of 150 units due to supplier restrictions.
Problem: Determine the optimal order quantity considering the discounts and minimum order
quantity.
Quantity range Price D Co Cc EOQ
Quantity Ordering Holding costs Material Costs Total Costs Range
costs
Quantity ordered =
Number of orders =
Problem 3
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You manage inventory for bolts. Annual demand is 15,000 units. Ordering cost is $30, and
holding cost is 12% of the unit price ($5/unit) per year. The supplier offers these discounts:
Order less than 200 units: No discount
Order 200 to 400 units: 3% discount
Order more than 400 units: 5% discount
However, there's a minimum order quantity of 150 units due to supplier restrictions.
Problem: Determine the optimal order quantity considering the discounts and minimum order
quantity.
Quantity range Price D Co Cc EOQ
Quantity Ordering Holding costs Material Costs Total Costs Range
costs
Quantity ordered =
Number of orders =
Wagner-Whitin Altorithm
The Wagner-Whitin algorithm, developed in 1958, is a powerful tool for inventory
management specifically focused on lot-sizing problems. Unlike the Economic Order
Quantity (EOQ) model, it tackles a more complex scenario: determining the optimal order
quantity for each period within a planning horizon, considering variable demand patterns.
Here's how the Wagner-Whitin algorithm tackles lot sizing:
● Dynamic Approach: While EOQ assumes constant demand, the Wagner-Whitin
algorithm is dynamic. It takes into account the known demand for each period
throughout the planning horizon. This allows for a more optimized approach to
ordering, potentially reducing overall inventory costs.
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● Minimizing Total Costs: Similar to EOQ, the Wagner-Whitin algorithm aims to
minimize total inventory costs. It considers ordering costs (setup costs) and holding
costs associated with each period's demand. By dynamically calculating lot sizes, it
attempts to find the most cost-effective balance between these factors for each period.
● Forward Recursive Algorithm: The Wagner-Whitin algorithm employs a dynamic
programming technique called a forward recursive algorithm. It starts by solving a
single-period problem and then builds upon that solution, considering future periods
and their demands. This allows it to find the optimal ordering strategy for the entire
planning horizon.
Benefits of using Wagner-Whitin Algorithm:
● Optimal Lot Sizing: When demand varies across periods, the Wagner-Whitin
algorithm provides a more accurate solution for lot sizing compared to EOQ,
potentially leading to significant cost savings.
● Flexibility: It can handle situations with backordering (not having enough stock to
fulfill demand) or lost sales (running out of stock completely).
Limitations of Wagner-Whitin Algorithm:
● Computational Complexity: The Wagner-Whitin algorithm can be computationally
intensive, especially for problems with a large number of periods or complex cost
structures. As a result, it might not be practical for very large-scale inventory
management problems.
● Assumption of Known Demand: The algorithm requires accurate forecasts of
demand for each period within the planning horizon. Inaccurate forecasts can lead to
suboptimal solutions.
In conclusion, the Wagner-Whitin algorithm is a valuable tool for businesses dealing with
variable demand and complex inventory management needs. While it requires more
computational power than EOQ, it offers the potential for significant cost savings by
optimizing lot sizes across different periods.
Problem 1
Problem Statement: We have a company that produces a certain product. The demand for
this product varies over time, and the company needs to decide how much to produce in
each period to meet demand while minimizing inventory holding and setup costs. The
company can produce in batches, and there is a setup cost associated with each batch
produced.
Given Data:
● Demand for each period:
○ Period 1: 100 units
○ Period 2: 150 units
○ Period 3: 200 units
○ Period 4: 100 units
● Setup cost: $1000 per batch
● Inventory holding cost: $10 per unit per period
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Let's break down the computations for each step of the Wagner-Whitin algorithm in tabular
format.
Step 1: Create the Cost Matrix
Periods/Demand 100 150 200 100
100 250 520
1 12200
0 0 0
100 250 520
2 12200
0 0 0
100 250 520
3 12200
0 0 0
100 250 520
4 12200
0 0 0
Step 2: Apply Wagner-Whitin Algorithm
Step 2.1: Period 1
In this step, we compute the total cost for each batch size in period 1.
Batch Size Setup Cost Holding Cost Total Cost
100 0 1000 1000
150 0 1500 2500
200 0 2000 5200
Step 2.2: Period 2
In this step, we compute the total cost for each batch size in period 2, considering the optimal
batch sizes from period 1.
Batch Size Setup Cost Holding Cost Total Cost
100 1000 1500 2500
150 1000 2250 3250
200 1000 3000 5200
Step 2.3: Period 3
In this step, we compute the total cost for each batch size in period 3, considering the optimal
batch sizes from periods 1 and 2.
Batch Size Setup Cost Holding Cost Total Cost
100 1000 2000 3000
150 1000 3000 4000
200 1000 4000 6200
Step 2.4: Period 4
In this step, we compute the total cost for each batch size in period 4, considering the optimal
batch sizes from periods 1, 2, and 3.
Batch Size Setup Cost Holding Cost Total Cost
100 1000 1000 2000
150 1000 1500 2500
200 1000 2000 5200
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Step 3: Final Solution
The optimal batch size for each period is 100 units, resulting in a minimum total cost of
$2000.
This breakdown shows the computations made in each step of the Wagner-Whitin algorithm,
helping to understand how the optimal production plan is determined.
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