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

Lot Size Suppose that in the inventory problem, the storage cost is a combination of the cost described in the text and the cost described in Exercise $15 .$ In other words, suppose there is an annual cost, $k_{1},$ for storing a single unit, plus an annual cost per unit, $k_{2},$ that must be paid for each unit up to the maxi- mum number of units stored. Show that the number of units that should be ordered or manufactured to minimize the total cost in this case is $$q=\sqrt{\frac{2 f M}{k_{1}+2 k_{2}}}$$

Key Concepts

Inventory Management

Inventory management is the discipline of planning, organizing, and controlling the flow and storage of goods to balance costs and service levels. It involves decisions about how much to order or produce, when to reorder, and how to store goods, all to ensure efficient operations while minimizing costs.

Economic Order Quantity (EOQ) Model

The EOQ model is a cornerstone of inventory management that calculates the optimal order quantity which minimizes the total costs associated with inventory. It balances the trade-off between ordering costs (costs incurred when placing orders) and holding (or storage) costs (costs associated with keeping inventory on hand), resulting in a formula that minimizes the sum of these costs over a planning period.

Inventory Holding Costs

Inventory holding costs are the expenses related to storing unsold inventory over a certain period. These costs can include both fixed costs (such as storage space and insurance) and variable costs that vary with the number of units held. Understanding and accurately estimating these costs is essential for determining efficient inventory levels and order quantities.

Ordering Costs

Ordering costs refer to the expenses incurred each time an order is placed, regardless of the order size. These include administrative costs, processing costs, and sometimes transportation or setup costs. Reducing the frequency of orders can lower these costs, but must be balanced against the potential increase in holding costs.

Cost Optimization in Inventory Models

Cost optimization in inventory models involves analyzing and balancing various cost components, such as ordering and holding costs, to find the condition where the total cost is minimized. This involves the application of mathematical techniques such as calculus to derive optimal solutions that help in effective decision-making regarding order quantities and inventory policies.

Transcript

00:01 This question is another variation on the economic order size or economic lot size formula.

00:09 So in question 15, i go into a bit of detail about different assumptions in the model.

00:15 So if you're curious about that, i refer you back to that one so that we can just get started on this question.

00:23 So the first step is, again, to figure out total cost.

00:27 Then we're going to take the derivative with respect to q, equal to zero, figure out the so yeah, let's go.

00:37 T, the total cost, has a production part, which is not going to change relative to the books.

00:45 Oh wow, relative to the books derivation or the derivation in question 15.

00:54 So we have production and we have storage costs.

00:57 So the storage costs are going to be the thing that changes.

01:01 So production is again f times m over q plus some c times m times m and then in this case we have storage costs which depend on the yearly storage so k1 times q and the maximum storage a 2 times k so a quick aside about where this one has comes from.

01:38 So the idea is we have q things in storage at some point and then a gradual even decrease and once our source at zero we order another batch of q and we have the same gradual even decrease then we order another one we have the same gradual even decrease etc etc so the total cost so the this guy is time and this guy is, let's say storage.

02:17 So the total cost of having something in storage is sort of the average number of things in storage will be one half queue.

02:35 And if the cost of storing something is just proportional to the number of stuff you've stored, the storage costs, are equal to half, or equal to that constant times your average stored amount of stuff.

02:53 So the average stored amount of stuff is half q.

02:57 The costs are equal to unit storage times half q.

03:06 And then we have the max storage cost k2 times q.

03:11 Who's origin? i explain them slightly more detail in number 15.

03:15 So now we need to minimize this guy.

03:21 So how do we minimize? well, we want to minimize with respect to q.

03:26 So we want to investigate dt, dq, and we want this equal to zero.

03:33 So first order of business is what is dt, dq? well, similar to last question, we get minus f times m over q squared plus one half, k1 plus k2 times q.

03:57 So i write it in this form with the q outside of the parentheses to sort of illustrate how similar it is to last time...

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