OSC EOQ Model

demand rate

(R) Fixed, continuous demand rate that must be satisfied over time

replenishment

• All units of an order arrive at the same time (instantaneous replenishment).

Set up/ordering cost

(K)

• Incurred for each order placed with a supplier.

• Measured in cost/setup. Components include:

• Hours of labor to setup a machine or operation.

• Materials spent in setup operations.

• Fixed ordering costs (administrative work, shipping costs, time spent ordering).

Holding cost

(h)

• Cost of holding inventory per unit of time (e.g., 1/year, 0.24/,month).

• Measured in units of time per item. Components include:

• Warehousing, insurance, obsolescence, and perishability.

• Largest typical component: Opportunity cost of capital (money tied up in inventory).

many orders (low batch size Q)

• The Mechanics: You purchase or produce small amounts of inventory at a time. To meet your total annual demand, you must place orders constantly.

• The Advantage (Low Holding Costs): Because batches are small, your warehouse stays relatively empty. Your average inventory on hand is low, which minimizes holding costs such as rent for storage space, insurance, refrigeration, and the risk of obsolescence or spoilage.

• The Penalty (High Setup Costs): Every time an order is placed or a production run is started, you incur a fixed setup cost. This includes administrative processing, flat-rate shipping fees, delivery unloading, or the labor and downtime required to calibrate a manufacturing machine for a new product run. Frequent ordering multiplies these fixed costs.

• Results in low average inventory levels, but frequent ordering (higher setup costs).

Few Orders (High Batch Size Q)

• The Mechanics: You purchase or produce massive quantities all at once. Because each batch covers demand for a long time, you rarely need to place new orders.

• The Advantage (Low Setup Costs): By ordering infrequently, you avoid paying repeated administrative, shipping, or machine setup fees.

• The Penalty (High Holding Costs): To clarify your prompt, the higher holding costs are caused directly by the large batch size, not the infrequency of the orders. When a massive shipment arrives, it sits in a warehouse for months before it is fully sold or used. This creates high average inventory levels, tying up your capital and maximizing storage and insurance costs.

• Results in high average inventory levels, but infrequent ordering (higher holding costs).

Average Inventory Formula

The formula Average Inventory = Q/2 is the mathematical foundation for calculating how much inventory you actually hold on a day-to-day basis.

To understand why we divide the order quantity ($Q$) by two, it helps to visualize a business's inventory levels as a "sawtooth" pattern over time. Here is the step-by-step logic behind the formula:

• The Peak (Inventory = $Q$): The moment a delivery truck drops off your new order (or a production run finishes), your warehouse is at its maximum capacity. If you ordered 1,000 units, your inventory level spikes immediately to 1,000.

• The Depletion (Steady Sales): Over the coming weeks, you steadily sell or use those items. Your inventory level gradually drops from 1,000 down to 900, then 500, then 100.

• The Bottom (Inventory = 0): Just as you sell your very last unit (hitting 0), your next order of 1,000 units arrives, and the cycle repeats.

Because your inventory is constantly and evenly draining from its maximum ($Q$) down to nothing ($0$), you do not actually hold $Q$ units in your warehouse for the entire year. The mathematical average of a straight line declining from $Q$ to $0$ is exactly the midpoint.

It is important to note that the $Q/2$ formula assumes constant, perfectly predictable demand and instant replenishment.

$$(Q + 0) / 2 = Q / 2$$

total inventory cost

Total Inventory Cost (C(Q)) is the sum of annual/monthly ordering costs and holding costs:

Economic Order Quantity Formula

(EOQ)

The Economic Order Quantity (EOQ or $Q^*$) formula calculates the exact mathematical "sweet spot" where your two conflicting inventory costs—setup costs and holding costs—are perfectly balanced. At this exact quantity, your total annual inventory costs hit their absolute minimum.

Here is how the variables in the formula

$$Q^* = \sqrt{\frac{2KR}{h}}$$

work together to find that balance:

• $K$ (Setup/Ordering Cost): The fixed penalty you pay every time you place an order or set up a machine, regardless of how many items you order.

  
  

• $R$ (Demand Rate): The total number of units you need to sell or use over a specific period (usually a year).

  
  

• $h$ (Holding Cost): The cost to store exactly one unit of inventory for that entire period (rent, insurance, depreciation, and tied-up capital).

  
  

The mechanics of the formula:

• The Numerator ($2KR$) pushes batch size UP: If your total demand ($R$) is massive, or your fixed fee to place an order ($K$) is extremely expensive, the top half of the fraction grows. The formula compensates by telling you to order more items at once so you don't have to pay that high setup fee as often. (The "$2$" is a mathematical constant derived from the calculus used to find the minimum of the total cost curve, relating directly back to the $Q/2$ average inventory formula).

  
  

• The Denominator ($h$) pushes batch size DOWN: If an item is bulky, perishable, or highly valuable, it becomes incredibly expensive to store ($h$). Dividing by a large $h$ shrinks your $Q^*$. The formula is forcing you to order smaller batches to keep your warehouse lean and avoid high carrying fees.

  
  

• The Square Root balances the curve: Because average inventory is based on $Q$ and order frequency is based on dividing by $Q$, your total costs form a U-shaped curve on a graph. Taking the square root mathematically isolates $Q$ at the exact bottom of that U-curve—the precise point where the downward slope of ordering costs perfectly crosses the upward slope of holding costs.

• Flat Cost Curve: Total cost curves around Q* are very flat. You can mis-specify Q* by a reasonable margin without significantly altering total ordering + holding costs.

• Insensitivity: The model is highly robust to minor misspecifications of model parameters and assumptions.

• Practical Benefits: Makes it easier to adjust lot sizes for quantity discounts, order in convenient units (like dozens), and consolidate orders for different products with limited cost penalties.

number of orders per period

r/q

order cycle

time between orders

q/r

production cycle

• A sequence of operations that repeats immediately and identically, sharing the same setups and production runs.

batch

• A collection of flow units produced together.

Batch size

(Q)

• The number of flow units produced in one production cycle before the process repeats.

Batch size formula

This formula is used to calculate the required batch size needed to achieve a specific target capacity or flow rate, taking into account setup times and processing times.

Here is what each component means and how the formula works:

The Components

• Batch size ($B$): The total number of units produced in one production cycle before switching over.

  
  

• Capacity: The target production rate you want the resource to achieve (e.g., matching the flow rate of a downstream bottleneck like assembly).

  
  

• Setup time ($S$): The total time spent changing over or prepping the machine for the batch.

  
  

• Processing time ($p$): The time it takes to process a single unit.

  
  

How to Read the Formula

$$\text{Batch size} = \frac{\text{Capacity} \times \text{Setup time}}{1 - \text{Capacity} \times \text{Processing time}}$$

1. The Numerator ($\text{Capacity} \times \text{Setup time}$): This calculates the total time lost to setups per unit of flow time. If you want a higher capacity or have a longer setup time, you need a larger batch to spread that setup penalty out.

   
   

2. The Denominator ($1 - \text{Capacity} \times \text{Processing time}$): This represents the fraction of total time available for setups after subtracting the time strictly spent actively producing units.

   
   

Why Use It?

As shown in your lecture examples, if an upstream machine (like a milling machine) needs to match a downstream bottleneck (like assembly operating at $0.33$ units/minute), plugging those numbers into this formula tells you the exact "just right" batch size. Running a batch size smaller than this result will cause the machine to spend too much time on setups, dropping its capacity below your target; running it much larger will bloat your inventory.

Capacity increases as

Capacity increases as the batch size gets larger.

• The Concept: When a machine has to change over from making Product A to Product B, it incurs a setup time (e.g., 60 minutes of downtime). During that setup, zero units are produced.

• Why it happens: If you run tiny batches, your machine is constantly stopping for setups, meaning a massive percentage of its total working hours is wasted on downtime. By making the batch size larger, you "amortize" (spread out) that single 60-minute setup penalty over hundreds or thousands of units.

• The Result: The machine spends less relative time sitting idle and more time actively producing, which increases its overall output capacity per hour. (Note: Capacity maxes out at a theoretical ceiling of 1/p when batch size approaches infinity, meaning setup time becomes negligible)

inventory increases as the

the batch size gets larger.

• The Concept: This ties directly back to the Average Inventory = Q/2 formula.

• Why it happens: When you produce in massive batches, you dump a huge quantity of products onto the floor all at once. Because demand happens gradually over time, those goods cannot be sold instantly; they must sit in a warehouse waiting to be shipped or used in assembly.

• The Result: Larger batches create high peak inventory spikes and raise your average inventory levels. According to Little's Law (I = R x T), holding more inventory also forces your flow times to stretch out, meaning products spend more time sitting stagnant.

utilization may increase as

as the batch size gets larger.

• The Concept: Utilization is the percentage of time a resource actually spends working versus sitting idle (Utilization = Time Producing / [Time Producing + Idle Time]).

• Why it says "may increase":

  • If your machine is currently bottlenecked by frequent setups, increasing your batch size eliminates that downtime, driving up its active production time and boosting its utilization.

  • The Catch: Utilization only goes up if the machine is actually constrained by capacity. If your process is demand-constrained (meaning you already produce enough to satisfy customer orders), running larger batches will just cause excess inventory to pile up uselessly rather than increasing productive utilization

economies of scale

What are Economies of Scale?

Economies of scale occur when a company's average cost per unit decreases as its production volume or batch size increases.

In simple terms: the bigger you make your batches or production runs, the cheaper it becomes to produce each individual unit.

Why Do Economies of Scale Happen?

It all comes down to fixed costs versus variable costs:

• Fixed Costs (Setup/Ordering Costs): Expenses that stay exactly the same no matter how many items you make—such as machine changeover time, administrative ordering fees, or factory prep work.

  
  

• Variable Costs: Expenses that scale directly with volume, like raw materials or direct labor per unit.

  
  

When you scale up your volume, your fixed costs don't increase; they just get amortized (spread out) over a much larger number of units.

The Connection to Batching and Inventory

Economies of scale are the primary driver behind why companies choose to batch their production rather than making items one-by-one:

1. Spreading Setup Penalties: If it takes 2 hours and $500 in labor to calibrate a machine for a product run, running a tiny batch of 10 units means each unit absorbs a massive $50 setup penalty. Running a batch of 1,000 units slashes that setup penalty to just $0.50 per unit.

   
   

2. Lower Per-Unit Procurement Costs: As seen in your EOQ and scale economics tables, higher annual demand rates allow companies to leverage bulk ordering, reducing the per-unit impact of fixed purchasing overhead.

   
   

A Quick Real-World Example

Imagine you run a bakery and want to bake specialty cookies:

• The Setup: Prepping the kitchen, mixing sanitizing solutions, and configuring the automated depositor takes 1 hour of fixed labor ($50), regardless of whether you bake 10 cookies or 1,000 cookies.

  
  

• Small Batch (10 cookies): The setup cost alone is $5.00 per cookie.

  
  

• Large Batch (1,000 cookies): The setup cost drops to $0.05 per cookie.

  
  

The Catch: While economies of scale encourage you to make massive batches to lower your production costs, remember the Operations Paradox—pushing batch sizes too high to chase local cost efficiency will bloat your inventory, tie up your working capital, and slow down your overall flow times.

setup time

(s)

• All non-producing time associated with preparing a resource for production (includes machine changeovers, scheduled maintenance, or forced idle time).

Processing time

• The time needed to produce one complete flow unit of output.

impact of batch size on capacity

This trade-off is one of the foundational dilemmas in operations management. It explains why running a facility efficiently is a constant balancing act between time (capacity) and money (inventory).

• The Mechanics: Whenever you switch a machine or production line from making Product A to Product B, you waste time on setup (changeover)—calibrating machines, cleaning equipment, or reloading software.

• Amortizing Setup Time: If your setup time is 1 hour, and you run a batch of 10 units, that setup time wastes 6 minutes per unit. But if you run a batch of 1,000 units, that same 1-hour setup time is spread out ("amortized") across all 1,000 units, dropping the setup penalty to a fraction of a second per unit.

• Reaching the Ceiling (1/p): As your batch size ($Q$) approaches infinity, setup time becomes completely irrelevant. Your process capacity hits its absolute maximum theoretical ceiling, which is determined purely by the raw processing speed ($p$) of the machine.

Because total available time is fixed (e.g., an 8-hour shift = 480 minutes), every minute spent setting up a machine directly steals time from active processing:

Smaller Batches: Frequent setups increase total setup time, shrinking the remaining time available for actual running. Effective capacity drops.

Larger Batches: Fewer setups minimize setup downtime, converting more of the shift into productive run time. Effective capacity expands toward theoretical design capacity

Impact of batch size on inventory

• Large Batches ($Q$): Because you produce or order in massive quantities to save on setup time, those goods sit around waiting to be used or sold. As covered by the Q/2 formula, this bloats your average inventory levels.

• Longer Flow Times (Little’s Law): Little’s Law states that Inventory = Throughput Rate × Flow Time (I = R x T). If your inventory ($I$) goes up because you are hoarding massive batches, but your sales rate ($R$) stays the same, your Flow Time ($T$) must increase. Products spend more time sitting stagnant on shelves.

• Small Batches: Drastically lower your inventory and speed up flow times, but if your setup times are long and you use small batches, your machines will spend more time being set up than actually producing. This starves your process of capacity.

How batch sizes impact set up times?

Making larger batches saves setup time because the total time spent preparing a machine or process is amortized (spread out) over a larger number of produced units.

In production, total setup time depends on the number of batches, not the number of items produced.

The Mathematics of Batch Setup Savings

Setup time per unit decreases as batch size increases:

\text{Setup Time per Unit} = \frac{\text{Total Setup Time}}{\text{Batch Size}}

Scenario Comparison

Assume a machine requires 60 minutes of setup (cleaning, calibration, tool changes) before running:

• Small Batches (10 units per batch): \frac{60 \text{ minutes}}{10 \text{ units}} = \mathbf{6 \text{ minutes of setup time per unit}}

• Large Batches (1,000 units per batch): \frac{60 \text{ minutes}}{1,000 \text{ units}} = \mathbf{0.06 \text{ minutes (3.6 seconds) of setup time per unit}}

By increasing the batch size, the non-productive setup overhead per unit drops significantly.

Key Operational Benefits

• Increased Machine Utilization: Less time spent idling for changovers means more capacity is available for actual production (higher uptime).

• Higher Labor Efficiency: Workers spend a greater percentage of their shift actively producing parts rather than performing tear-downs and rebuilds.

• Lower Direct Production Costs: Spreading fixed setup labor costs across more units lowers the individual cost per unit (economies of scale).

The Trade-Off: Work-in-Progress (WIP) and Storage Costs

While large batches minimize setup time per unit, they introduce secondary operational costs:

Advantage of Large Batches	Disadvantage / Hidden Cost
Lowers setup time per unit	Increases inventory holding costs
Maximizes equipment throughput	Lengthens production lead times
Reduces frequent line disruptions	Creates higher risk of defect cascades

⚠️ Manufacturing Trade-off: Extreme reliance on large batch sizes can obscure quality issues and lock up capital in excess inventory. Lean manufacturing principles (such as SMED - Single-Minute Exchange of Die) focus on reducing the setup time itself so that companies can run smaller batches without incurring heavy setup time penalties.

Operations dilemma between capacity and inventory

The Capacity Goal (Favors Large Batches): Maximize machine uptime, spread out setup costs, and output as much volume as possible. The downside: You trap massive amounts of working capital in inventory and lengthen flow times.

1. The Inventory Goal (Favors Small Batches): Minimize holding costs, keep warehouses lean, and ensure fast, responsive flow times. The downside: You risk severe capacity bottlenecks if machines are constantly shut down for changeovers.

Finding the exact balance point between these two extremes is what operational optimization tools—like the Economic Order Quantity (EOQ) and lean manufacturing systems—are designed to solve.

Bottleneck constraint

• The Bottleneck Constraint: In any multi-step process (like making a product that goes through mixing, baking, packaging, and assembly), the slowest step sets the pace for the entire operation. No matter how fast mixing or packaging can run, the whole factory can only output as many units as the bottleneck step allows.

• Non-Bottleneck "Free" Capacity: Steps with extra capacity are considered "free" because they sit idle part of the time. Speeding them up won't increase overall factory output; it will only create work-in-process (WIP) inventory piling up right before the bottleneck.

• Target Batch Size Calculations: When you have a strict market demand target ($R$), you can't just pick random batch sizes. You have to calculate a batch size that ensures your bottleneck resource spends enough time actually producing (rather than wasting time on changeovers) so it can keep pace with $R$ without starving downstream steps like assembly.

multi product dilemma

• The Multi-Product Trap: Imagine a factory that used to make only one flavor of soup. They only needed one setup per production cycle. Now, market demand shifts and they want to make ten different flavors of soup.

• Expanding Setup Times: Because you are switching flavors constantly, you trigger a setup/changeover every time you switch lines. The total setup time per cycle explodes because you are stopping the machines ten times instead of once.

• The Chain Reaction:

  1. More product variety —> More setups per cycle.

  2. More setup time ——> You lose valuable capacity.

  3. To claw back that lost capacity, operations managers are forced to increase the batch size ($Q$) for each flavor so they run each one longer before switching.

  4. The Catch: Larger batch sizes mean massive inventory spikes, higher holding costs, and slower flow times across the board.

The Ultimate Takeaway: High product variety is great for marketing and sales, but it creates a brutal operations headache. It forces larger batch sizes to protect capacity, which directly contradicts the goal of keeping inventory lean.

SMED

explain how world-class companies break free from the trap of having to choose between high inventory and lost capacity.

1. SMED (Single-Minute Exchange of Die)

SMED is a lean manufacturing framework designed to drastically slash equipment changeover (setup) times down to "single-digit minutes" (under 10 minutes).

To understand how it works, you have to split setup tasks into two categories:

• Internal Setup Tasks: Things that can only be done while the machine is completely shut down (e.g., unbolting an old mold and bolting in a new one). This directly cuts into your process capacity.

• External Setup Tasks: Things that can be done while the machine is still running (e.g., pre-fetching tools, gathering raw materials, pre-heating components, or cleaning parts ahead of time).

Why SMED solves the Operations Dilemma: Remember the trade-off: long setup times force you to run large batches, which bloat your inventory. SMED attacks the root cause. By shifting as many tasks as possible from internal to external before the machine even stops, you shrink total downtime. With setup times nearly eliminated, you can run small batches frequently without sacrificing a single ounce of capacity and without piling up excess inventory.

Operations paradox

he Operations Paradox (Local Efficiency vs. Global Performance)

This is one of the most important mindset shifts in operations management. It highlights the danger of trying to optimize individual parts of a business at the expense of the whole system.

• Local Efficiency: Looking at one specific machine, worker, or department and trying to keep their utilization at 100% (e.g., running massive batches on a stamping machine so it never sits idle).

• Global Performance: Looking at the entire system from end to end—measuring total lead times, inventory holding costs, cash flow, and how quickly you can actually deliver value to the customer.

The Trap: Maximizing local efficiency almost always destroys global performance. If you force a machine to run massive batches to keep its local utilization looking "efficient," you flood the factory floor with Work-In-Process (WIP) inventory. That excess inventory clogs up aisles, extends flow times, hides quality defects, and makes the company sluggish and unresponsive to real customer demand. True operational excellence means sacrificing local efficiency at non-bottleneck steps so that the global system flows smoothly.