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Summary  Factory Physics

2 Inventory Control

What is the key insight of EOQ
There is a tradeoff between lot size and inventory

What is the difference between the base stock and the Q r inventory models
in the q,r model the ordering costs are nonnegligible

what is the eoq formula?
sqrt(2DA/h)

How did Harris (1913) defined the sum of the labor and material costs to ready the shop to produce a product?
Setup cost

2.2 The Economic Order Quantity Model

Consider figure 2.1. To which assumptions can the vertical lines at intervals nQ/D (with n = 1, 2, 3 ... n) be attributed? (Why are they vertical?)
Assumption 1 & 2. The vertical line represents instantaneous production and immediate delivery, and more so, a perfect overlap and similar timing between them.

How is assumption 4 represented in figure 2.1?
Assumption 4: represented in the oblique lines. Since demand is constant over time, the inventory level decreases constant over time as well, resulting in an oblique line.

Based on the 6 assumptions, figure 2.1 should actually be incorrect. How? What extra assumptions are made here?
According to assumption 6, products can be analyzed individually. This means that the oblique lines in figure 2.1 should actually not be smooth. Instead, inventory should be decreasing incrementally.
For modeling purposes, time and product are represented as continuous quantities. 
What is the EOQ in formula form?

What is the implied order cost in formula form?

A manager calculates an optimal order quantity Q* = 23,4 for product X, with an optimal order interval T* = 2,6 weeks. Yet he decides to order every 2 weeks in order quantity Q'. What is Q', what are the effects on the costs and what could be the reason for the manager to use this different strategy?
D = Q* / T* = 9 products per week. Q' = T' ⋅ D = 2 ⋅ 9 = 18 products.
Effect on costs (see 2.2.4): Y(Q')/Y* = 0,5 (Q'/Q* + Q*/Q') = 1,035. This means a 3,5% increase in costs.
Reason: to order items at intervals given by powers of 2 to facilitate the sharing of resources in a multiproduct environment (e.g. delivery trucks)

In 2.2.3 a key insight is presented as: 'there is a tradeoff between lot size and inventory'. Does this mean lot size goes down as inventory goes up?
No, a larger lot size leads to a larger inventory. A larger lot size leads to a lower order frequency. This means that a higher order frequency leads to lower lot sizes and lower inventory levels. There are decreasing returns however, as shown in figure 2.3.

Give the expression for the total (inventory, setup, and production) cost per year with relation to the EOQ model
Y(Q) = ((hQ)/2)+((AD)/Q) + cD = ((icQ)/2)+((AD)/Q) + cD

For which lot size Q is the total annual cost Y(Q) minimized?
For the value of Q for which the holding cost and setup cost are exactly balanced (i.e. the hQ/D and AD/Q cost curves cross).

With which amount will the optimal order quantity increase or decrease?
Increase: with the square root of the setup cost or the demand rate.
Decrease: with the square root of the holding cost.

With respect to the sensitivity (2.2.4) of the EQO model; When we choose a lot size Q 3,75 times as large as optimal, what will the error be in the cost function?
Y(Q')/Y* = (1/2) * ((Q'/Q*) + (Q*/Q')) = (1/2) * ((3,75/1) + (1/3,75)) =2,01. That is, a 275 percent error in lot size results in a 101 percent error in cost.

2.4 Statistical Inventory Models

Wilson (1934) breaks the inventory control problem into two distinct parts:
1: determining the order quantity
2: determining the reorder point
What parts do the news vendor, base stock and Q,r model solve?News vendor: 1, order quantity
Base stock: 2, reorder point
Q,r: 1 & 2, order quantity & reorder point 
In the news vendor model, what is the critical fractile formula?

Give 3 examples of types of stockout costs.
Some examples:
1: Loss of profit through a loss of sales.
2: Loss of future sales as customers go elsewhere.
3: Loss of customer goodwill.
4: Extra costs associated with urgent, often small quantity, replenishment orders.
5: Cost of production stoppages caused by stockout of WIP or raw materials.
(Can you see now why determining stockout costs is often very hard in practice?) 
A flag shop wants to sell a special flag for a onetime event. The manager estimated demand for the flag to be 83. He can only order in quantities of 25. He determines the stockout cost to be $10 as a consequence of lost profit. He is not sure what discount to use in the event of having stock left over after the event. However, the manager is positive this is the only cost in case of overage. When is it best to order 75 flags, when is it best to order 100? In other words: what is the critical cost of overage and how much is the discount then?
When the manager decides to order 75 instead of 100 flags, then obviously the cost of having 8 flags less than demand should be smaller than having 17 more than demand. This means that 8 Cs <= 17 Co.
When the manager decides to order 100 flags, then the opposite is true: 17 Co <= 8 Cs.
Since Cs = $10, we can calculate the critical overage cost, Co = $4,71.
Since we know that the cost of stockout is equal to $10 of lost profit, we know that the critical discount is Cs + Co = $14,71.
This means the manager should use a discount on the selling price greater than $14,71 if he buys 75. He should use a discount smaller than $14,71 if he buys 100. 
How is the inventory position defined?
Inventory position = onhand inventory + orders  backorders

What are the definitions of onhand inventory , orders and backorders?
Onhand inventory: physical inventory
Orders: resupply requests that are not yet committed to customers
Backorders: customer demands that have occurred but not yet filled 
What is the difference between net inventory and onhand inventory?
Net inventory = onhand inventory  backorders
Onhand inventory does not include backorders but is the physical inventory you can actually 'see'. 
When does 1: net inventory, 2: onhand inventory, 3: inventory position, go negative?
1: If backorders > onhand inventory.
2: Never. Onhand inventory fluctuates, but can never go negative. (Onhand inventory => 0)
3: Never. Inventory position is held constant at r+1. 
If we would use a stockout cost approach instead of a backorder cost approach for the basestock model, we could rewrite 2.29 as Y(r)=holding cost+stockout cost. What would 2.30 then look like? (With k = stockout costs per unit and D = yearly demand)
Y(r)=h I(r) + k D (1  S(r)) = h (r + 1  theta) + k D (1  phi((r + 1 + theta)/sigma))

John argues: "Using a basestock model, if the service level is 90%, so if S(r)=0,9, then obviously B(r) must be 0.1, since if out of every 10 orders, 9 can be filled from stock, 1 must be backordered." Why is John wrong?
B(r) is not a percentage but a number. It represents the average amount of backorders at any given time. Consider this example: a shop has an S(r)=0,90 and sells 1 million products a year, then 100.000 products are backordered in this year. However, at the end of the year, the number of backorders B(r) will of course not be 100.000, they are (most likely) spread out over the year. Neither will it be 0.1.

A shop buys and resells a certain product with an estimated daily demand of 500 products. The shop owner wants to use a base stock model to determine the optimal reorder level. Why is this a bad idea? Is there another model available the shop owner could use?
It is highly unlikely that a supplier will supply that many products in single quantities. The shop most likely has to purchase products in batches. In that case, the Q,r model would be a more representative model to use.

John also has a computer store. What is a reasonable distribution of the demand and why?
The Poisson distribution. Customers originate from a large population, and decide independently to computers a John's store.

John's friend Paul sells different types of bicycles. In order to control the inventory, Paul asks John to come up with a suited inventory model for each type of bicycle separately. John proposes the newsvendor model. Why is this model not completely correct?
Substitution effects are not taken into account: people can decide to buy a different type of bicycle in case their favorite type of bicycle is no longer available.

The news vendor problem, and its intuitive critical fractile solution, can be extended to a variety of applications that have more than one period. Name three features for such a common problem situation.
1. A firm faces periodic (e.g., monthly) demands that are independent and have the sam distribution G(x)
2. All orders are backordered (i.e., met eventually)
3. There is no setup cost associated with producing an order.

In relation to the news vendor model: If the demand is normally distributed, then increasing variability (i.e., standard deviation) of demand increases the optimal order (production) quantity if:
A: 1. Cs/(Cs + C0) > 0,5
2. Cs/(Cs + C0) < 0,8
3. Cs/(Cs + C0) > 0,8
4. Cs/(Cs + C0) < 0,5
and decreases it if
B: 1. Cs/(Cs + C0) > 0,5
2. Cs/(Cs + C0) < 0,8
3. Cs/(Cs + C0) > 0,8
4. Cs/(Cs + C0) < 0,5
Encircle the proper answer
A: 1. Cs/(Cs + C0) > 0,5
B: 4. Cs/(Cs + C0) < 0,5