A system used when demand occurs in only a single point in time.

**Target service level**

For a single-period inventory system, the service level at which the expected cost of a shortage equals the expected cost of having excess units.

**Target stocking point**

For a single-period inventory system, the stocking point at which the expected cost of a shortage equals the expected cost of having excess units.

**CHAPTER 11** • Managing Inventory throughout the Supply Chain **343**

The goal of a **single-period inventory system** is to establish a stocking level that strikes the *best balance* between expected shortage costs and expected excess costs. Developing a single-period system for an item is a two-step process:

1. Determine a **target service level** (*SLT*) that strikes the best balance between shortage costs and excess costs.

2. Use the target service level to determine the **target stocking point** (*TS*) for the item.

We describe each of these steps in more detail in the following sections.

For the single-period inventory system, service level is simply the probability that there are enough units to meet demand. Unlike a periodic and continuous review system, there is no re-order period to consider here—either there is enough inventory or there isn’t. The target service level, then, is the service level at which the expected cost of a shortage equals the expected cost of having excess units:

Expected shortage cost = expected excess cost

or:

(1 – p)CShortage = pCExcess | (11.12) | ||

where: | |||

p = probability that there are enough units to meet demand | |||

(1 – p) = probability that there is a shortage | |||

CShortage = shortage cost | |||

CExcess = excess cost | |||

The target service level (SLT) is the p value at which Equation (11.12) holds true: | |||

(1 – SLT)CShortage = SLTCExcess | |||

SLT = | CShortage | (11.13) | |

CShortage + CExcess |

where:

*C*Shortage* *=* *shortage cost* C*Excess* *=* *excess cost

Let’s use Equation (11.13) to test our intuition. Suppose the shortage cost and the excess cost for an item are both $10. In this case, we would be indifferent to either outcome, and we would set the inventory level so that each outcome would be equally likely. Equation (11.13) confirms our logic:

SLT = | CShortage | +10 | = 0.50, or 50% | |||

= | ||||||

CShortage + CExcess | +10 + +10 | |||||

But what if the cost associated with a shortage is much higher—say, $90? In this case, we | ||||||

would want a much higher target service level because shortage costs are so much more severe | ||||||

than excess costs. Again, Equation (11.13) supports our reasoning: | ||||||

CShortage | +90 | = 0.9, or 90% | ||||

= | ||||||

CShortage + CExcess | +90 + +10 |

EXAMPLE 11.5 | ||

Don Washing is trying to determine how many gallons of lemonade to make each day. Don | ||

Determining the Target | ||

needs to consider a single-period system because whatever lemonade is left over at the end | ||

Service Level at Don’s | of the day must be thrown away due to health concerns. Every gallon he mixes costs him | |

Lemonade Stands | $2.50 but will generate $10 in revenue if sold. |

**344 ** **PART IV** • Planning and Controlling Operations and Supply Chains

In terms of the single-period inventory problem, Dan’s shortage and excess costs are defined as follows:

*C*Shortage* *=* *revenue per gallon* *–* *cost per gallon* *=* *+10.00* *–* *+2.50* *=* *+7.50

*C*Excess* *=* *cost per gallon* *=* *+2.50

From this information, Don can calculate his target service level:

SLT = | CShortage | = | +7.50 | = 0.75, or 75% |

CShortage + CExcess | +7.50 + +2.50 |

Interpreting the results, Don should make enough lemonade to meet all demand ap-proximately 75% of the time.

EXAMPLE 11.6 | |||||

Every day, Fran Chapman of Fran’s Flowers makes floral arrangements for sale at the local | |||||

Determining the Target | |||||

hospital. The arrangements cost her approximately $12 to make, but they sell for $25. Any | |||||

Service Level at Fran’s | leftover arrangements can be sold at a heavily discounted price of $5 the following day. | ||||

Flowers | Fran wants to know what her target service level should be. | ||||

Fran’s shortage and excess costs are as follows: | |||||

CShortage | = revenue per arrangement – cost per arrangement = +25 – +12 = +13 | ||||

CExcess | = cost per arrangement – salvage value = +12 – +5 = +7 | ||||

Fran’s target service level is, therefore: | |||||

SLT = | CShortage | +13 | = 0.65, or 65% | ||

= | |||||

CShortage + CExcess | +13 + +7 | ||||

Fran should make enough arrangements to meet all demand approximately 65% of | |||||

the time. |

To complete the development of a single-period inventory system, we next have to translate the target service level (a probability) into a target stocking point. To do so, we have to know some-thing about how demand is distributed. Depending on the situation, we can approximate the demand distribution from historical records, or we can use a theoretical distribution, such as the normal distribution or Poisson distribution. Furthermore, the distribution may be continuous

(i.e., demand can take on fractional values) or discrete (i.e., demand can take on only integer values). Example 11.7 shows how the process works when we can model demand by using the normal distribution, while Example 11.8 demonstrates the process for a historically based dis-crete distribution.

EXAMPLE 11.7 | In Example 11.5, Don Washing determined that the target service level for lemonade was: | |||

Determining the Target | ||||

CShortage | +7.50 | |||

Stocking Point for | = | = 0.75, or 75% | ||

Normally Distributed | CShortage + CExcess | +7.50 + +2.50 | ||

Demand | Don knows from past experience that the daily demand follows a normal distribution. | |||

Therefore, Don wants to set a target stocking point (TS) that is higher than approximately | ||||

75% of the area under the normal curve. Figure 11.11 illustrates the idea. |

**CHAPTER 11** • Managing Inventory throughout the Supply Chain **345**

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