Suppose that in the inventory problem, the storage cost...

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

Cost Function Optimization

Cost function optimization in inventory management requires formulating a function that represents total costs and then minimizing it, often through calculus. This approach identifies the point where the marginal increase in one cost component equals the marginal decrease in another, yielding the optimal order quantity.

Trade-off Between Ordering and Storage Costs

This concept involves balancing the fixed cost of placing orders with the variable cost of storing inventory. When storage costs are based on the maximum inventory level rather than the average, the trade-off analysis is adjusted, leading to a different optimal order quantity.

Inventory Control

Inventory control involves managing stock levels to balance supply and demand while minimizing associated costs. It includes deciding how much inventory to order and when to order it, taking into account various cost factors such as ordering costs and storage costs.

Economic Order Quantity (EOQ) Model

The EOQ model is a fundamental inventory management tool used to determine the optimal order quantity that minimizes total inventory costs. This model typically balances the costs of ordering and holding inventory and is adapted based on how storage costs are calculated.

Transcript

00:01 For problem number 15, if a company or a factory has to build a warehouse large enough to hold the maximum inventory, then the cost of storage would be the same no matter how full or empty the warehouse is.

00:18 So given that q is the number of units per batch and m is the annual demand of units of units per batch, so required units per year and f is the fixed setup cost per patch and k is a fixed storage storage cost per year and lastly given that g is the cost to manufacture one unit.

01:34 Okay, so given these terms we can say that that batch, only one batch, manufacturing cost, is equal to f, which is the fixed set up cost for patch, plus g, which is the cost of one unit, multiplied by q, which is all the units.

02:14 In a patch.

02:17 And we can also say that the total annual manufacturing cost would be equal to number of patches multiplied by batches, sorry, multiplied by the batch cost.

02:58 So this is a batch cost that we will use here.

03:03 And the number of patches, we can say that the total annual manufacturing and cost number of patches is m the total annual demand of units over q which is units per patch now we have the number of patches and we can multiply that by f plus g q which is the batch which gives us mf over q plus mf over q plus g okay this is the total annual manufacturing cost and using the concept of maximum inventory size we get that the total production cost which we call t of q its function in q is equal to f of m fm which is this over q plus gm plus the storage the storage cost which is k multiplied by q so getting the t dash of q we will get negative fm over q squared plus zero plus zero plus k...

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