Chapter 15

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# Chapter 15 - PowerPoint PPT Presentation

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

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:
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 Policywith Backordering
• Retailers may not stock all demand. Orders placed during shortages are backordered.
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:
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:
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?
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.
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:
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).
• 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.
• 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