*Assumptions. *All the assumptions of model I except shortage occurs here. Backlogs due to shortage to be met with penalty.

One inventory cycle is given in the Fig. 6.3.

FIG. 6.3 Purchasing model with shortage

During time period t_{1} inventory exhaust and during time period t_{2} shortages developed.

Here Q _{1} = Actual inventory in hand.

Q_{2} = Shortage/Stock out

Q = Q _{1} + Q_{2}, t = t _{1} + t_{2} = cycle time.

Total cost = Holding cost + Ordering cost + Shortage cost

The optimum values are given asTotal optimum cost =

**Example 3.** The demand for a certain item is 50 units per year. Unsatisfied demand causes a shortage cost of $ 0.45 per unit per short period. The ordering cost for purchase is $ 20 per order and the holding cost is 15% of average inventory valuation per year. Item cost is $ 5 per unit.

Find the EOQ, the shortage inventory and the minimum cost.

**Solution.** D = 50 units/year

c_{2} = $ 0.45/unit/shortage period.

c_{5} = $ 20

c_{1} = 5 x 0.15 =$ 0.75/unit/year.

= 84.33 - 31.62 = 52.71