Different models are used to manage inventory for products that are continually available (like milk) or products available for limited time (like seed).The Economic Order Quantity (EOQ) model determines the least cost level of inventory to carry, as well as costs. News Vendor models are used for products only available for a single period.

EOQ and News Vendor models have proved useful for managing inventory for many years, analyzing tradeoffs among major cost components. These models are robust and easy to customize to particular industries. Their approach to costing is similar reflecting levels of inventory, as well as shipping costs or quantity discounts.

Inventory costs fall into three classes:

1) carrying costs of regular inventory and safety stock;

2) ordering or setup costs;

3) stockout costs. Inventory control systems balance the cost of carrying inventory against the costs associated with ordering or shortfalls

Firms carry extra inventory to guard against uncertain events. Known as safety stock, the purpose of this inventory is to provide protection against stockouts. Safety stock is costed just like regular inventory, it is an interest rate times the level of safety stock.

If less is sold than expected during the 10 days or if the shipment arrives early, we will still have inventory on the 10th day and no customer service problems are encountered.

Managing the uncertainty surrounding safety stock is the key to reducing inventory levels.

stockout costs involve lost sales when no inventory is on hand. Such costs fall as inventory (and customer service) levels increase. The relationship between stockout costs and inventory depends upon the accuracy of the demand forecast and the ability of the firm to recognize and react to a change in demand.

One way to evaluate an inventory management policy is to choose a service level target. From this target, the inventory policy will determine the inventory requirements and associated costs of providing that level of service. A higher service level implies that more inventory will be held as safety stock.

**Newsvendor Model**

From sweatshirts in EOQ to summer dresses in Newsvendor. The big difference is that while sweatshirts were continuous selling items, the demand for summer dresses is limited to summer months. Once the summer season is over, the unsold dresses must be heavily discounted. You are a local design firm that designs northwest-accented summer dresses, sources them from China and sells them through retailers here.

The problem is that for a particular summer dress, total demand during the summer season is hard to predict. All you can do is to make a guess, that is, develop a *probability distribution of demand*. Let us generate our demand with the throw of a regular dice; it can be any number from 1 to 6, each with probability 1/6.

On the supply side, the lead time from your Chinese supplier is long. There is no possibility of making multiple orders. You make one order before the summer season starts, sell as many as you can during the season and then whatever is left is discounted. Let us say that *per unit purchase cost c* is $80. For any units that you are able to sell *per unit revenue r* is $100. For the units you are not able to sell during the season, let us say that you can discount them and are able to sell them at a *per unit salvage value s* of $30.

The big decision is the *order quantity S* of dresses you should order from your Chinese supplier at the beginning of the summer season.

*The Trade-off*

If you order a very large quantity, there is a bigger chance that you will not be able to sell all of them. There will be *excess* units at the end of the season that you will have to discount. You will lose money on them. On the other hand, if you order a small quantity, there is a bigger chance that you will be *short*. That is, there will be some demand you will not be able to satisfy. You will not be able to make as much money as you could have.

The order quantity decision resolves this trade-off between the expected cost of having excess inventory and the expected cost of falling short. We will call the sum of these two costs as* Mismatch cost*. The optimal order quantity will minimize mismatch cost.

*Marginal Costs*

To resolve this trade-off, we start with defining marginal costs of excess and shortage.

*Marginal Cost of excess C _{e}*

**is defined as the cost of having one unit excess. You bought this unit for purchase cost of c=$80, were not able to sell it during the season and then had to discount it down to the salvage value of s=$30. The cost to you is $80-$30=$50. That is, in this setting,**

*Ce=c-s*.

*Marginal Cost of shortage C _{s}* is defined as the cost of having one unit short. Had you bought this unit for purchase cost of c=$80, you would have been able to sell it during the season for a revenue of r=$100. We say, that the cost to you for being one unit short is $100-$80=$20. That is, in this setting,

*C*

_{s}=r-c.* **Service Level*

Service level is the chance that you will be able to meet all the demand in a single period (summer season). Suppose you bought an order quantity S=3 units. Recall that demand is any number between 1 and 6 with equal probability 1/6. In this case, you will be able to meet all the demand only if demand is either 1 unit, 2 units or 3 units. That is, the probability that demand is less than or equal to S=3 units. This probability is known as cumulative probability and is given by the sum of the probabilities that demand is 1, demand is 2, and demand is 3 = (1/6)+(1/6)+(1/6)=3/6=1/2. That is, if you buy S=3 units, you will provide a service level of 50%.

Here is a quick table to provide cumulative probabilities in our case:

Demand | 1 | 2 | 3 | 4 | 5 | 6 |

Probability | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |

Cumulative
Probability |
1/6 | 1/6+1/6
=2/6 |
2/6+1/6
=3/6 |
3/6+1/6
=4/6 |
4/6+1/6
=5/6 |
5/6+1/6
=6/6=1 |

*Optimal Service Level and Optimal Order Quantity*

Single-period model tells us that, given the marginal costs of excess and shortage, C_{e} and C_{s}, the *optimal service level is given by (C _{s}/(C_{s}+C_{e}).* In our case, the optimal service level is equal to 20/(20+50)= 0.2857.

*Optimal order quantity S** is the minimum size of the order that will be able to provide the optimal service level. Going by the above table, if you buy, for example, S=1, you will be able to provide a service level of 1/6=0.1667 which is less than the optimal service level we wish to provide. If we buy 2, service level is 2/6=0.3333 and we will be able to satisfy the optimal service level requirement of 0.2857. Therefore S*=2.

Rule: compute optimal service level and find the minimum value of demand for which cumulative probability, for the first time, equals or exceeds optimal service level. That is the optimal order quantity.

*Summary of Formulas for Continuous Demand: Normal Distribution*

The demand distribution we considered above is a *discrete* distribution because demand can only take a limited number of values. In some real settings, it is easier to work with the assumption that the demand follows Normal distribution with a *given mean and standard deviation*. Normal is a continuous distribution because demand can take any value. In this case, we can use the following formulas:

Given per unit revenue r, per unit purchase cost c and per unit salvage value s:

Marginal cost of excess C_{e}=c-s; Marginal cost of shortage C_{s}=r-c.

Optimal service level = C_{s }/ (C_{s}+C_{e})

Given a normally distributed demand with given mean and standard deviation

compute z = spreadsheet function Normsinv (required service level)

Order quantity that can provide required service level = mean + z*standard deviation

Alternatively, given an order quantity S, the service level it can provide =

Spreadsheet function = Normdist (S, mean, standard dev., TRUE)

For Normal distribution, we can also compute the following:

Expected shortage = Std. Dev.*{ Normdist(z,0,1,false) -z +z Normdist(z,0,1,true)}

Expected excess = S – mean + Expected shortage

Expected mismatch cost = C_{s}*Expected shortage + C_{e}*Expected excess

Expected profit = (r-c)* mean – Expected mismatch cost

**How much to order and when to order?**

One of the major objectives of any supply chain is to cater to the demand in the most efficient manner. One of the ways of having achieving such efficiency is: cater to the demand by minimizing the inventory levels as much as possible.

Essentially, there are two fundamental decisions that help us manage inventory. They are:

- How much to order?
- When to order?

** How much to order?**

__Newsvendor Model__

We decide on how much to order depending on the cost of over-stocking and the cost of under-stocking. For example, say you are company that sells cakes. A cake costs you $1.24 to prepare and you sell the cake at $2.49. But, if you cannot sell the cake within 24 hours, then you have to sell it to another local vendor at $0.99. In this case, the

Cost of under-stocking (Cu) is: $1.25 (the profit that you lose in not being able to sell the cake)

Cost of over-stocking (Co) is: $0.25 (amount that you lose because you have to sell at a discount to a vendor)

The optimal service level (SL*) is: Cu/(Cu+Co). The optimal quantity that you need to order is the smallest quantity Q at which the service level exceeds the optimal service level (SL*).

How can I get the service levels? You can get the demand distribution and the service levels from the past data.

The above formula is also called the newsvendor formula. This is used when the product has a limited shelf life and inventory cannot be carried over.

__Economic Order Quantity Model__

The other approach to determine quantity is called the Economic Order Quantity model.

Let us make two assumptions:

- Demand will be steady (no variance)
- Lead time for delivery order is zero. The order is immediately delivered whenever an order is issued.

In such a scenario, I will always order whenever the inventory goes to zero. Immediately the order comes and my inventory reaches Q again. The rate at which inventory goes to zero is the throughput rate (R) itself.

So, the time between two orders is Q/R

Therefore, the order frequency is R/Q.

Every order has certain fixed costs associated with it. For example, even if you order 1 unit of an AC there will be some $2000 of fixed costs (assume fixed cost will be a step cost). So, therefore you want t order as many units as possible so that the fixed costs of an order are spread over large number of units.

But at the same time, if you order more number of units then you have to bear more costs for the inventory carrying costs. Let’s look at what is an optimal solution for under this trade-off situation.

Fixed cost paid per period = S*R/Q (S is the fixed cost for every order)

Cost of holding inventory (H) = cost of keeping one unit in inventory for a certain period

The sum of both fixed cost and the cost of holding the inventory has to minimum. Under such conditions the optimal quantity to order is the Economic Order Quantity (Q*) = sqrt (2RS/H). If you centralize your inventory, then it helps in inventory optimization because: if demand increases by 2 then quantity increase by only sqrt(2).

This can be rounded off to the nearest packaging standards that are required for your supplier.

Now, let’s negate one of the two assumptions we made in the EOQ Model.

Let’s say we have a lead time of 3 weeks before the order comes.

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A table for reference to understand which order quantity is better