Backorder Inventory Model

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Backorder Inventory Model In this model, we assume that stock outs (and backordering) are allowed. In addition to previous assumptions, we assume that sales will not be lost due to a stock out. Because, we will back order any demand that can not be fulfilled B: Backordering cost per unit per year b: The amount backordered at the time the next order arrives Q – b: Remaining units after the backorder is satisfied

Total Annual Cost = Annual Setup Cost + Annual Holding Cost + Annual Backordering Cost 



Annual Setup (Ordering) Cost = (D/Q) . S

Annual Holding Cost = (Average Inventory Level) . H By using the graphical ratios, we know that:  

T1 / T = (Q – b) / Q Therefore, if we replace T1/T in the above equation we get   

Average Inventory Level = (Q – b)2 / 2Q

By using the graphical ratios, we know that: T2 / T = b / Q Therefore, if we replace T2/T in the above equation we get Average Backordering = b2 / 2Q and 

We find optimum order quantity (Q*) and optimum backordering quantity (b*) by taking the derivatives of dTC/dQ = 0 and dTC / db = 0 and then putting the values in their places. 

Quantity Discount Model

A quantity discount is simply a reduced price (P) for an item when it is purchased in LARGER quantities. A typical quantity discount schedule is as follows: Since the unit cost for the Third discount is the lowest, We might be tempted to order 2000 or more units. However, this quantity might not be the one that minimizes the Total Cost. Remember that, As the quantity goes up, the holding cost increasesHere, there is a trade off between reduced product price (P) and increased holding cost (H). Total Cost = Setup Cost + Holding Cost + Product Price (Cost) Total Cost = DS / Q + QH / 2 + PD where P is the price per unit To determine the minimum Total Cost, we perform the following process which includes 4 steps: Step 1: Assume that I: is a percentage value, and I . P represents the holding cost as a percentage of price per unit (P). For each discount alternative, calculate a value of Q* = [2DS / IP]1/2 Here, instead of using a value of H, the holding cost is equal to I . P 

That is, If the item is expensive (such as a Class A Item), Its holding cost will be higher. Since the price of item (P) is a factor in Annual Holding Cost, we can no longer assume that the holding cost is constant (such as H) when price changes. Step 2: For any discount alternative, If the calculated optimum order quantity (Q*) is too low to qualify for the discount range, Then, Adjust the order quantity upward to the lowest quantity that will qualify for the particular discount alternative. Step 3: Using the total cost (TC) equation above, compute a total cost for every order quantity (Q). Use the adjusted Q values. Step 4: Select the discount alternative which has the minimum Total Cost (TC). 

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