inventory management iii: decision making this module discusses ordering costs, time between orders,...

Inventory Management III: Decision Making

This module discusses ordering costs, time between orders, inventory holding costs, economic order quantity (EOQ), quantity discounts, and production order quantity.

Authors: Stu James and Robert Robicheaux

© 2013 Stu James and Management by the Numbers, Inc.

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• Inventory Flow Over Time

• Ordering Costs

• Time Between Orders

• Holding Costs

• Economic Order Quantity (EOQ)

• Total Annual Cost

• Quantity Discounts

• Production Order Quantity

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Topics Covered

MBTN | Management by the Numbers

This MBTN Module is designed to help managers answer the question – how much of an inventoried item should be ordered or manufactured? The following topics will be covered:

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Inventory Flow Over Time


If demand for an item is constant over time, we can visualize inventory flow as in the graph below. In this “perfect world”, the inventory level would drop to zero just as the Order Quantity (Q) arrives to replenish it; thus, inventory would never exceed value Q. The average inventory level in the system would be represented by Q / 2. The distance from P1 to P2 would be the Time Between Orders (P).

Inventory Level

Q/2

Time

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Costs that Impact Order Quantity


Given this idyllic world of constant flows, there are two basic types of costs to consider to determine the appropriate order quantity.

Ordering Costs are the costs associated with placing an order or starting a production process. In the situation of an item that was ordered from a supplier these costs would include placing, delivering, receiving and stocking the order, etc. Do not include costs that vary directly with the quantity of the order.

Holding Costs are costs of maintaining inventory, such as capital costs (interest on money tied up in inventory), storage space (rent and utilities), inventory service costs (insurance on inventory, inventory maint.), and inventory risk costs (obsolescence, spoilage, shrinkage). Normally holding costs are described as a % of the value of inventory.

InsightYou may realize that if you order smaller quantities more frequently, annual ordering costs will be higher, but holding costs will be lower.

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Calculating Ordering Costs


Question 1: Janice is running low on inventory of brake pads at her tank repair shop. Her materials manager is paid $50K / year including all benefits, and it takes him about 2 hours to call the various potential suppliers, place the order and stock it when it comes in. He currently orders 100 pads/order. The suppliers charge a $50 flat rate for delivery of the brake pads plus $1.00 / brake pad. What is the order cost for a single order presuming a 40 hour week with 2 weeks of vacation?

Answer:

Hourly Rate = Annual Salary / Hours worked per year = 50,000 / (40 * 50) = 50,000 / (2000) = $25 / hour

Cost of Order = 2 hours * hourly rate + fixed delivery charge (only) = 2 * $25 + $50 = $100

Note that the $1/brake pad is not included as it is a variable cost

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Time Between Orders


Answer:

Number of Orders per Year = 1,000 / 100 = 10 ordersTime Between Orders = 365 / 10 = 36.5 days

Question 2: If annual demand for tank brake pads is 1000 / year and the quantity purchased is 100, how many orders are placed each year and what is the time between orders in calendar days?

Definitions

Orders Per Year = Annual Demand (D) / Quantity Purchased (Q)Time Between Orders (P) = 365 / Orders per Year (calendar days) orTime Between Orders (P) = Business Days / Orders per Yr (bus. days)

This intrigued Janice. She had never really thought about how often she ordered the pads. Further, she now wondered what her approximate cost was on an annual basis for placing all these orders.

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SCalculating Annual Ordering Costs


Definitions

Annual Ordering Costs = # Orders Per Year * Cost Per Order-- or –Annual Ordering Costs = (D / Q) * S

Where: D = Annual Demand Q = Quantity Ordered S = Order or Set-up Cost

Question 3: What are Janice’s annual ordering costs if demand is 1000 units / year, they order 100 at a time and order costs are $100?

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SCalculating Annual Ordering Costs


Answer:

Annual Ordering Costs = (D / Q) * S = (1000 / 100) * 100 = (10) * 100 = $1000

That was more than Janice expected and started her wondering if it would make sense to place fewer orders each year to lower the cost.

InsightJanice was on to something. Annual ordering costs would go down if she ordered less often, but she had a feeling that some other cost would go up if she did that. That cost would be the “hidden” cost of holding more inventory.

As an example, Janice could order 1000 brake pads at the beginning of the year and her order costs would only be $100 instead of $1000. But where would she put all those brake pads?

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SCalculating Annual Holding Costs


Question 4: Janice wonders what her annual holding costs are on the brake pads based on historical quantities ordered and unit cost. The brake pads cost $5 each (includes $1/pad delivery). She currently orders 100 at a time because of a quantity discount, and she estimates her holding costs at 25%. Calculate the annual holding costs.

Definition

Annual Holding Costs = K * C * (Q / 2)

Where :K = Carrying or Holding Cost Rate (%)C = Unit CostQ = Order Quantity

The holding costs of inventory increase with the size of the order because the average inventory level (Q/2) is higher throughout the year. Here is the formula for calculating holding costs:

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SCalculating Annual Holding Costs


Answer:

Annual Holding Costs = K * C * (Q / 2) = 25% * $5 * (100 / 2) = $62.50 / year

InsightNote that the annual ordering costs calculated previously is $1000 - far more than the annual holding costs of $62.50. Perhaps Janice should order less often (to lower the ordering costs) in exchange for a higher average inventory level and higher holding costs.

The following slide shows some different combinations of holding costs and ordering cost at different order quantities. What we’re really aiming for is to lower the combined cost. Sometimes, however, there are physical limitations (or cash limitations) on the amount that can be ordered.

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Total Annual Holding and Ordering Costs


Quantity(Q)

Holding Cost (HC)

Ordering Cost (OC) Total Cost

100 $ 62.50 $ 1,000.00 $ 1,062.50 200 $ 125.00 $ 500.00 $ 625.00 300 $ 187.50 $ 333.33 $ 520.83 400 $ 250.00 $ 250.00 $ 500.00 500 $ 312.50 $ 200.00 $ 512.50 600 $ 375.00 $ 166.67 $ 541.67 700 $ 437.50 $ 142.86 $ 580.36 800 $ 500.00 $ 125.00 $ 625.00 900 $ 562.50 $ 111.11 $ 673.61 1000 $ 625.00 $ 100.00 $ 725.00

Calculated costs for various order quantities where:

Demand (D) = 1000

Cost/Order (S) = $100

Carrying Cost (K) = 25%

Unit Cost (C) = $5

Janice’s holding and ordering costs for various order quantities:

InsightNotice that the lowest total cost is at an order quantity of 400 units!

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)Economic Order Quantity (EOQ)


Rather than create massive tables of various cost combination, we can instead use the Economic Order Quantity (EOQ) formula where the combined total of Annual Ordering Costs (OC) and Annual Holding Costs (HC) is minimized. By definition, this is where OC = HC

Definition

Economic Order Quantity = or ((2 * D * S) / (K * C))^.5

Where : D = Annual Demand (Historical or Forecasted)S = Cost per Order (Set-up Cost)

K = Carrying or Holding Cost Rate (%) C = Unit Cost

2 * D * S K * C

InsightNote that the EOQ model is based on certain assumptions. If reality differs significantly from these assumptions, EOQ is less accurate.

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SEOQ Assumptions

The classical EOQ model is helpful to guide managers who purchase products needed in their companies or to plan production schedules. Certain assumptions generally apply for EOQ to be used effectively.

Now let’s see how EOQ would look graphically in Janice’s example…


• There should be a continuous and known rate of demand. This can be derived from historical records analysis as well as sales and marketing forecasts.

• Lead times for delivery should be known. Constant is not really necessary as long as changes or variations can be anticipated and inventory managed to handle anticipated lead time variability.

• Price is independent of order quantity. We show here how quantity discounts might affect purchase quantities.

• Transport costs are independent of quantity or timing of orders.

• All market demand can be satisfied. Safety stocks (extra inventory) can be used to accommodate unusually high demand.

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Graphical representation of annual costs where:

Demand (D) = 1000



Unit Cost (C) = $5

InsightBy definition, the minimum total cost (the EOQ) is where:

holding cost = ordering cost

Note that the graph corresponds with the values from the table.

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Question 5: Janice is interested in using EOQ to help her order the appropriate amounts of brake pads. Recall that the annual historical demand has been 1,000 units/year, cost per order is $100, and the unit cost is $5. Her carrying cost rate is 25%. What is the EOQ for brake pads?

Calculated costs for various order quantities where:

Demand (D) = 1,000



Unit Cost (C) = $5

Answer:

EOQ = ((2 * D * S) / (K * C))^.5

= ((2 * 1000 * 100) / (.25 * 5))^.5

= (20,000 / 1.25) ^.5

= 400 brake pads per order (same as the table!!)

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Total Annual Cost


Answer:

Total Annual Cost= (Q / 2) * K * C + (D / Q) * S = (400/2)*25%*5 + (1000/400)*100 = $250 + $250 = $500

Note: At EOQ, the HC = OC, but remember that the actual order amount must be rounded to the nearest whole number if the calculation results in a partial unit (e.g., if EOQ is 399.5 round that to 400).

Question 6: Janice now wants to know what her total annual costs would be for brake pads if she orders the EOQ of 400.

Definition

Total Annual Cost = Holding Costs + Ordering Costs at EOQ (Q) = K * C * (Q / 2) + (D / Q) * S

Demand (D) = 1000



Unit Cost (C) = $5

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SSome Helpful Concepts


The goal of an inventory management system is to minimize these total costs while still providing excellent customer service. It is helpful to review some directional concepts at this point.

Question 7: The President of the company asks Janice how changes in the assumptions impact the business. Take a moment to think about how each variable in the EOQ formula impacts the EOQ and total annual costs.

Demand (D) = 1000



Unit Cost (C) = $5

Answer:

INSIGHTConsider how EOQ and Total Costs change when the key variables DECREASE.

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SQuantity Discounts


Let’s throw a few wrinkles into this economic model. The first one is Quantity Discounts. A quantity discount is when a supplier provides a lower price (cost) if a higher quantity is purchased. Looking at an example is probably the best way to see the impact on the purchase quantity decision.

Question 8: When Janice calls up her supplier to place an order for 400 tank brake pads, the supplier informs her that there are discounts for higher purchase quantities. Specifically, at 500 units the price is reduced to $4.50 / unit and, if she orders 1,000 units, the price goes to $4/unit. Saving money always seems like a good thing. Maybe she should increase her order to receive the discount. How many should Janice order?

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SQuantity Discounts


Answer (Step 1):

EOQ ($5.00) = 400 units (from previous example)

EOQ ($4.50) = ((2 * D * S) / (K * C))^.5

= ((2 * 1000 * 100) / (25% * $4.50))^.5

= (200,000 / 1.125) ^.5

= 177,777 ^.5 = 421.637, or 422 brake pads per order

EOQ ($4.00) = ((2 * D * S) / (K * C))^.5

= ((2 * 1000 * 100) / (25% * $4.00))^.5

= (20000 / 1.00) ^.5

= 200,000 ^.5 = 447.214, or 447 brake pads per order

As we’d expect, as the price decreases, the EOQ increases. But note how little the EOQ rises with the quantity discounts. Both are below the quantity needed to qualify for those discounts. What does that suggest?

This is more difficult than you might expect. Again, our goal is to minimize total annual costs. So the first step is to calculate the EOQs for each price level.

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SQuantity Discounts


Answer (Step 2):

Annual Cost ($5) = $500 (calculated previously) (Q=400)

Annual Cost ($4.50) = K*C*(Q/2) + (D/Q)*S (Q=500) = (500/2)*25%*$4.50 + (1000/500)*100 = $481.25

Annual Cost ($4.00) = (Q/2)*K*C + (R/Q)*S (Q=1000) = (1000/2)*.25*$4.00 + (1000/1000)*100 = $550.00

This would seem to indicate Q=500 (lowest), but are we forgetting something?

Step 2 is to calculate the total annual costs for each possible price / quantity combination. Note that since the EOQs for $4.50 and for $4.00 are below the quantity necessary to receive the discount, they are not viable choices for Q and so we must calculate using the minimum value Q that meets the quantity volume requirement.

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SQuantity Discounts


Answer (Step 3):

Annual Cost ($5) = $500 (calculated previously)

Annual Cost ($4.50) = K*C*(Q/2) + (D/Q)*S – COGS Savings of $500(Q=500) = $481.25 - $500.00 = -$28.75

Annual Cost ($4.00) = (Q/2)*K*C + (R/Q)*S – COGS Savings of $1000 (Q=1000) = $550.00 - $1000.00 = -$450.00

So, the lowest annual costs would be at an order quantity of 1000 units.

Yes!! We’ve forgotten to add in the annual savings due to a lower unit cost. So step 3 is to add in the additional savings with the discount based on annual demand of 1000 units.

At $4.50, the savings would be $.50 x 1000 = $500

At $4.00, the savings would be $1.00 x 1000 = $1000

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NAppropriate Demand Assumption


Let’s consider the importance of the demand assumption. In Janice’s example, we’ve used historical demand. Is that the best value to use? Let’s consider two possible scenarios.

Scenario 1: Janice is about to call up her supplier again with the new order quantity of 1000 units priced at $4.00. But, when Mark from the marketing department stops by and sees that she’s about to place an order for tank parts, he mentions that demand for tank parts is expected to increase because of a promotion the marketing department is planning. Mark checks with his manager and finds out the forecast for this part is actually 2,000 units, double the historical demand. How will this impact EOQ?

Answer: Janice would need to recalculate the EOQs based on annual demand of 2000 units. So, one managerial decision is whether to use historical demand or a future forecast as the best estimate for demand (D).

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NAppropriate Demand Assumption


Janice picks up the phone again, now ready to place her order now based on demand of 2000 units, but the production manager wanders by muttering, “I can’t believe we have to change the pads in the tank assembly from asbestos to ceramic, this is going to cause a delay in our production lines”. Janice know that demand for brake pads is generated from a combination of spare parts and in the production process. So, another managerial consideration is the likelihood of obsolesce. Here, Janet may need to recalculate only based on spare parts demand.

InsightWhile EOQ is a very helpful tool in the inventory planning process, remember that it is based on assumptions: that our estimate of future demand is accurate, and demand is relatively constant, that the carrying cost and order costs are accurate. Unit cost can also be subject to different seasonal prices or production cost assumptions if manufactured instead of purchased. So, don’t forget common sense!

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(ELS

)Economic Lot Size (ELS)


The Economic Lot Size (ELS) is a special case of EOQ for manufacturing processes where instead of order costs it is typically considered set-up costs for the manufacturing process.

Definition

Economic Lot Size = or ((2 * D * S) / (K * C))^.5

Where : D = Annual Demand (Historical or Forecasted)S = Set-up Cost

K = Carrying or Holding Cost Rate (%) C = Unit Cost

2 * D * S K * C

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Please see MBTN Inventory Management modules 1, 2 and 4 that cover other important concepts related to this module.

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Inventory Management - Further Reference


inventory management iii: decision making this module discusses ordering costs, time between orders,...

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