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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

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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
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 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 model4
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 model5
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 model6
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 model7
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 model8
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
Optimal Inventory Policywith Backordering
  • Retailers may not stock all demand. Orders placed during shortages are backordered.
optimal inventory policy with backordering10
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 policy with backordering11
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 policy with backordering12
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 policy with backordering13
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 policy with backordering14
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
Economic Production-Quantity Model
  • The inventory model may be extended to finding the optimal production quantity.
economic production quantity model16
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 model17
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
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
Inventory Spreadsheet Templates
  • Economic Order Quantity
  • Sensitivity Analysis
  • Backordering
  • Production
economic order quantity model figure 15 3

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

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 the Sensitivity Analysis(Figure 15-7)

Graphing sensitivity analysis results makes It is easier to see relationships.

backordering model figure 15 9
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
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