Inventory Decisions with Certain Factors Chapter 15 If I order too little, I make no profit. If I order too much, I may go broke. Every product is different. Help me!—A Retailer’s Plea Inventory Decisions with Certain Factors

Elements of Inventory Decisions There are four basic inventory system costs: Ordering costs Procurement costs Inventory holding or carrying costs Inventory shortage costs Demand is usually erratic and uncertain. We assume it is smooth and predictable. That makes it easier to develop mathematical models. These can later be made more realistic. Order quantity is the main variable. With no uncertainty, we can schedule deliveries to arrive exactly when we run out.

The Economic Order Quantity (EOQ) Model The decision variable is Q = Order Quantity There are four parameters: k = Fixed cost per order A = Annual number of items demanded c = Unit cost of procuring an item h = Annual cost per dollar value of holding items in inventory An order quantity is to be found that minimizes: Total = Ordering + Holding Procurement Annual cost Cost

The Economic Order Quantity (EOQ) Model Inventory level has a cycle beginning with a new shipment’s arrival. T = Q/A = Duration of inventory cycle

The Economic Order Quantity (EOQ) Model The annual ordering cost is the number of orders times the cost per order: The annual holding cost is the cost per item held 1year times the average inventory: The annual procurement cost is the product of annual demand and unit cost: Procurement cost = Ac

The Economic Order Quantity (EOQ) Model The total annual inventory cost is: We drop Ac from the above, since that amount will not vary with Q. Ac is not a relevant cost. That provides the function to be minimized, the total annual relevant inventory cost:

The Economic Order Quantity (EOQ) Model It may be shown using calculus that the level for Q minimizing the above is the economic order quantity Problem. A software store sells 500 Alien Saboteurs annually. The supplier charges $100 per order plus $20 each. It costs $.15 per dollar value to hold inventory for a year. How many should they order, how often, and at what annual relevant inventory cost?

The Economic Order Quantity (EOQ) Model Solution: The following parameters apply: A = 500 k = 100 c = 20 h = .15 The economic order quantity is The inventory cycle duration is T = Q/A = 183/500 = .366 year or 133.6 days The total annual relevant inventory cost is:

Optimal Inventory Policy with Backordering Retailers may not stock all demand. Orders placed during shortages are backordered.

Optimal Inventory Policy with Backordering The new model adds the order level S, that quantity on hand when a shipment arrives. A shortage cost applies, based on a penalty p for being one item short for a year. New total annual relevant inventory cost: Optimal order quantity and order level:

Optimal Inventory Policy with Backordering Shortage penalty p applies over a year, but cost prorates to fractions of items or years. Example: The retailer suffers lost profit on future business equal to $.05 each day that one Alien Saboteur is on backorder. That translates into p = $.05×365 = $18.25. Solution: The order quantity is computed:

Optimal Inventory Policy with Backordering Solution: The order level is computed: The relevant cost is = $253.81 + 217.47 + 36.31 = $507.59 The above is smaller than before, even though there is a shortage penalty and shortages. Why?

Optimal Inventory Policy with Backordering There is a net savings in holding costs and a slight reduction in ordering costs. Those outweigh increased cost due to shortages. The number of backorders is Q – S. Here that quantity is 197 – 169 = 28. The annual shortage cost is only $36.31, because durations of shortage (for last of the 28) are only 28/197 = .142 year (52 days). The results suggest that: Retailers will run short, if they can get away with it! But backordering must make sense.

Optimal Inventory Policy with Backordering Nobody backorders cigarettes or gasoline. Sales for those products are lost during shortages. This model does not apply for them. The shortage penalty p is not usually known. But it may be imputed from existing policy. The service level L is used for that purpose: L = proportion of time fully stocked The imputed shortage penalty is: p = hcL 1 - L

Economic Production-Quantity Model The inventory model may be extended to finding the optimal production quantity.

Economic Production-Quantity Model The new parameter is the annual production rate B. Parameter k is the production setup cost. The variable production cost per unit is c. The total annual relevant inventory cost: The economic production quantity:

Economic Production-Quantity Model Example: Water Wheelies have annual demand of A =100,000 units and are made at the rate of B = 500,000. Production costs are k = $2,000 setup and c = $5 variable. It costs h = $.40/year to tie up a dollar. Economic production quantity is Total relevant cost is TC(8,944)

More Elaborate Models Incorporate a second one-time shortage penalty (done in Chapter 16). These models are for single products. Add additional products. Incorporate uncertainty regarding: Demand (done in Chapter 16). Lead-time for delivery of order (Chapter 16). Incorporate lost sales (done in Chapter 16). Extend to single period products (Ch. 16). NOTE: The basic EOQ model works very well even when its ideal conditions don’t apply. It is very robust.

Inventory Spreadsheet Templates Economic Order Quantity Sensitivity Analysis Backordering Production

Economic Order Quantity Model (Figure 15-3) 2. Enter the problem information in F6:F9. 1. Enter the problem name in B3. Optimal order quantity Optimal total annual relevant cost and time between orders

Sensitivity Analysis (Figure 15-6) A sensitivity analysis shows how answers vary as data changes. Here the fixed order cost, k, varies. 1. Enter the problem name in B3. 2. Enter the problem information in F6:I9. The fixed order cost has a diminishing effect on the results. For example, a 100% increase in k causes both Q* and TC(Q)* to increase by only 41%.

Graphing the Sensitivity Analysis (Figure 15-7) Graphing sensitivity analysis results makes It is easier to see relationships.

Backordering Model (Figure 15-9) 1. Enter the problem name in B3. 2. Enter the problem information in F6:F10. Optimal total annual relevant cost and time between orders Optimal order quantity and order level

Production Model (Figure 15-13) 1. Enter the problem name in B3. 2. Enter the problem information in F6:F10. Optimal time between production runs, duration of production run, and total annual relevant cost. Optimal order quantity