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Chapter 15

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

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Chapter 15

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  1. 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

  2. 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.

  3. 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:

  4. 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

  5. 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

  6. 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:

  7. 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?

  8. 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:

  9. Optimal Inventory Policywith Backordering • Retailers may not stock all demand. Orders placed during shortages are backordered.

  10. Optimal Inventory Policywith 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:

  11. Optimal Inventory Policywith 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:

  12. Optimal Inventory Policywith 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?

  13. Optimal Inventory Policywith 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.

  14. Optimal Inventory Policywith 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:

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

  16. 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:

  17. 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)

  18. 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.

  19. Inventory Spreadsheet Templates • Economic Order Quantity • Sensitivity Analysis • Backordering • Production

  20. 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

  21. 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%.

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

  23. 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

  24. 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

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