Stochastic Lotsizing Problem
Stochastic lotsizing models can be applied when the demand is dynamic and random. These planning approaches simultaneously consider randomness and dynamic conditions. Period demands are forecasted with expected value $\mu_t$ and standard deviation $\sigma_t$. There are several ways to cope with uncertainty under these conditions.
1. Current approach used by software vendors
The standard approach currently applied by most software vendors is a simple modification of the MRP planning approach. Compute safety stock in advance and then add the safety stock to the net demand. Next compute the lot sizes with any applicable procedure.
Comment: This procedure is flawed, as usually the safety stock is not calculated correctly. In addition, it is not taken into consideration that the lot sizes have an impact on the absorption of risks. For example, with large lot sizes, the safety stock required to guarantee a target service level may be zero or even negative. As a consequence, the target service level which should be the basis for the safety stock calculation is met only by chance.
2. Compute order-up-to levels for all periods of the planning horizon
Fix the times of ordering and the target order-up-to levels. Then wait until demand is observed and order the difference between the current inventory and the target order-up-to level. The target order-up-level can be computed exactly, based on a mathematical model.
Comment: This procedure leads to random variations of the order sizes. This may not acceptable, particularly if lot sizes transfer into capacity requirements, which will be random, too.
3. Compute order quantities for all periods of the planning horizon in advance
This approach requires the application of a stochastic lotsizing model that fixed the optimum setup periods and the optimum production quantities in advance. The demand data are given in terms of expected values $\mu_t$ and standard deviations $\sigma_t$ per period $t$. These are usually the outcome of a forecasting procedure.
The following example shows the optimum lot sizes and the inventory development when a stochastic lotsizing model is used where the constraint is in effect that at the end of every production cycle the target fillrate is $\beta=90$ %. The results presented are the exact solution of the lotsizing model.
| t | $\mu_t$ | $\sigma_t$ | Setup | Lot size | $E\{I^p\} | $E\{F\}$ | $\beta_t$ | Costs |
| 1 | 200 | 60.00 | 1 | 335.47 | 135.7157 | 0.2484 | 1.00 | |
| 2 | 50 | 15.00 | 0 | - | 87.8269 | 2.1112 | 0.99 | |
| 3 | 100 | 30.00 | 0 | - | 20.7670 | 32.9401 | 0.90 | 1244.31 |
| 4 | 300 | 90.00 | 1 | 770.62 | 456.0893 | 0.0007 | 1.00 | |
| 5 | 150 | 45.00 | 0 | - | 306.3241 | 0.2347 | 1.00 | |
| 6 | 200 | 60.00 | 0 | - | 122.9693 | 16.6453 | 0.97 | |
| 7 | 100 | 30.00 | 0 | - | 58.5920 | 35.6227 | 0.93 | |
| 8 | 50 | 15.00 | 0 | - | 36.5873 | 27.9953 | 0.90 | 1980.56 |
| 9 | 200 | 60.00 | 1 | 461.00 | 222.3548 | 5.2130 | 0.97 | |
| 10 | 150 | 45.00 | 0 | - | 102.4447 | 30.0899 | 0.90 | 1324.80 |
| Total: | 4549.67 |
Used symbols:
$E\{I^p\}$: expected inventory at the end of period $t$
$E\{F\}$: expected backorders at the end of period $t$
$\beta_t$: fillrate at the end of period $t$
Comment: This procedure fixes the lot sizes in advance and thus leads to reliable capacity requirements. Extension to capacitated lotsizing models under capacity constraints have recently been developped.