Lot Size Suppose that in the inventory problem, the storage...

Lot Size Suppose that in the inventory problem, the storage cost depends on the maximum inventory size, rather than the average. This would be more realistic if, for example, the company had to build a warehouse large enough to hold the maximum inventory, and the cost of storage was the same no matter how full or empty the warehouse was. Show that in this case the number of units that should be ordered or manufactured to minimize the total cost is $$q=\sqrt{\frac{f M}{k}}$$

Key Concepts

Model Assumptions

Key modeling assumptions include a deterministic demand rate, fixed ordering costs per lot, and storage costs that depend solely on the maximum inventory level rather than the average. These assumptions facilitate a tractable analysis where the total cost function becomes a function of the order size, and the explicit dependence on maximum inventory allows for an alternative but realistic cost minimization approach.

Optimization Techniques

Calculus-based optimization is applied to derive the optimal order quantity. By expressing the total cost as a function of the order quantity and then taking its derivative with respect to that quantity, setting the derivative equal to zero identifies the minimum point. This process leads to the square-root expression that provides the formula for the optimal lot size.

Economic Order Quantity (EOQ)

Economic Order Quantity is a fundamental concept in inventory management that determines the optimal order size to minimize the total inventory costs, including ordering and holding costs. In this modified scenario, the EOQ formula is adjusted to reflect storage costs based on the maximum inventory level, leading to a decision rule that effectively balances the trade-off between setup and capacity costs.

Inventory Cost Components

The analysis involves two main cost components: the fixed ordering (or production) cost incurred per lot and the storage cost that depends on the peak inventory level needed to accommodate the largest possible stock. This dual-component cost structure is crucial for developing a cost function that must be minimized to determine the best lot size.

Transcript

00:01 So in this question, we are going to derive our own solution to a slightly different order size problem.

00:10 So in this case, the storage costs don't actually depend on how full the warehouse is or how many things you actually have in storage.

00:20 It just depends on how big your warehouse needs to be.

00:24 In reality, i'd suspect it's probably some combination of the two.

00:29 There is a cost for just having a big warehouse and then there is a cost for having it filled up and needing to provide air conditioning, security or whatever.

00:42 But in this case, only the maximum storage size matters.

00:49 So remember how we derived the original economic order size formula.

00:54 We first figured out total costs as a function of the order size.

00:59 And then minimized with respect to this order size parameter.

01:05 So let's figure out total cost.

01:07 It has two parts, production and storage.

01:17 Now the production part is going to be the exact same as in the book.

01:22 So it's going to be f times m divided by q.

01:26 This is the startup portion.

01:29 So m divided by q is the number of batches we create.

01:34 And f is this fixed setup cost every time you're doing a batch, or a fixed order cost every time you're placing an order.

01:42 And at m over q, the total demand divided by the production per batch is the number of batches you need to produce.

01:53 So this is the setup costs, and then we have the c times m, which is just the cost of producing one unit times the total number of units we're producing.

02:07 And note that this doesn't actually depend on q, so it's not going to change our answer in the end.

02:14 Then we have storage costs.

02:16 So since the storage costs only depend on the maximum number of stuff we need to store, and we're told it's proportional, the storage costs are k times q.

02:33 So basically, this formula assumes that it's twice as expensive to store twice as much stuff.

02:40 Now, again, in reality, there's probably going to be cutoffs...

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