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1. Python: Created full algorithm for solving periofic review’s stochastic inventory problem in real life

Situation

Weekly demand for refrigerators at an appliance store has a Poisson distribution with 4. The holding and stockout cost for refrigerators at the store are $40 and $125 per week, respectively. Replenishment orders for refrigerators incur a fixed cost of $150.

Find the optimal parameters (s, S), and the corresponding optimal cost.

Solution Approach

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1. Let y* minimize g(y); then s* ≤ y* ≤ S*

   • We only need to look for s≤y* and S≥y*

2. Let s = max{y < y*|g(y, S) ≤ g(y)}; then s is the optimal reorder point for S

   • Given S, the biggest s smaller than y* such that the cost per period of the corresponding (s, S) policy does not exceed the single period cost at that reorder point is optimal for S

3. If S’ and S are two order-up-to levels, then g(s*(S),S) < g(s*(S’),S’) if and only if g(s*(S’),S) < g(s*(S’),S')

   • Suppose we use the optimal reorder point for S’ (s*(S’)) together with some other S; if this is better than using this reorder point with S’ itself, then the best solution with S is better than the best solution with S'

4. If (s*, S*) are optimal with cost g* = g(s*, S*) then S* ≤ max{y ≥ y*|g(y) ≤ g*}

   • Given a proposed (s, S), if there is a y’ ≥ y* with single-period cost g(y’) less than or equal to g(s, S) and S > y’, then (s, S) cannot be optimal

5. We can now state an algorithm for finding (s*, S*)

Algorithm

While g(S) ≤ g': 
    If g(s', S) < g',
        Set S'=S
            While g(s,S')≤g(s+1):
                Sets=s+1
            End while
        Set s'=s;g'=g(s',S')
    End if
    Set S = S + 1
End while
Return (s', S')

Result

• Demo
• Code