To have a static policy, order-up-to-levels and replenishment decisions are declared as robust int+ Order_up_to[Period] in 0..maxint; and robust int Replenish[Period] in 0..1; which give a less nervous policy. Now the best replenishment **policy is to have replenishment every**
period, Periods 1-5, with an order-up-to-level [14,21,23,20,18] respectively. The expected total cost of this scheme is $439.70.

The solution robust policy, assuming a tradeoff constant λ = 1.**0, is determined using the Stochastic**
OPL objective function minimize mv(cost,λ)**. The optimal replenishment plan is now: in Period**
1, the order-up-to-level is 35; no replenishment is planned **for Period 2; in Period 3, depending on** demand, order-up-to-level takes a value between 37 and 51; no replenishment in Period 4; and finally, in Period 5, there are replenishments in 16 scenarios. The expected total cost of following the solution robust policy is $353.06.

In Fig.8, we see that as one would expect, an increase in λ**, which actually points to a decrease**
in the objective value uncertainty, causes an increase in the expected total production and inventory costs.

![0_image_0.png](0_image_0.png)

## 10 Related Work In Decision Making Under Uncertainty

Stochastic constraint programs are closely related to Markov decision problems (MDPs). An MDP model consists of a set of states, a set of actions, a state transition function which gives the probability of moving between two states as a result of a given action, and **a reward function. A solution to an** MDP is a policy, which specifies the best action to take in each **possible state. MDPs have been** very influential in AI of late for dealing with situations involving reasoning under uncertainty [21]. Stochastic constraint programs can model problems which lack the Markov property that the next