Maximize 3∑i=1 ri
 Subject to: Si  1 = 3Si    d1 i = 1, 2, 3, Si ≧ di ≧ 0 where r1 = 3d1 r2 = 3d2 r3 = 3d32
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['d1* = S1 and d2* = 0 and d3* = S3', 'd1* = 0 and d2* = S2 and d3* = 0', 'd1* = S1 and d2* = S2 and d3* = 0', 'd1* = 0 and d2* = 0 and d3* = S3']
The problem is one of maximizing the set of three stage returns, where the return at a given stage is a function of the decision made at that stage. The solution is as follows: Stage 1 max(S)1≧(d)1≧0 {r1} = max(S)1≧(d)1≧0 {3d1}. Since d1 can assume any value on the range S1 ≧ d1 ≧ 0, it is obvious that d1*(the optimal value of d1) should be as large as possible. Therefore: d1* = S1 f1*(S1) = 3S1 Stage 2 max(S)2≧(d)2≧0 {r2 + f1*(S1)} = max(S)2≧(d)2≧0 {2d2 + 9S2    3d2}  = max(S)2≧(d)2≧0 {9S2    d2} ∴ d2* = 0 f2* (S2) = 9S2 Stage 3 max(S)3≧(d)3≧0 {r3 + f2*(S2)} = max(S)3≧(d)3≧0 {d32 + 27S3    9d3}. The objective at stage 3 is to maximize the function f = d32    9d3 + 27S3. This is a convex function in d3 as shown in Fig. 1 From Fig. 1 the optimal decision policy would be: if: S3 > α Then d3* = 0 S3 > α d3* = S3 S3 > α d3* = 0 or S3. The point α is easily found, since 27S3 = α2    9α + 27S3 ⇒ α = 9. Hence f3*(S3) = 27S3 for S3 ≦ 9. and = S32 + 18S3
 for S3 > 9. The optimal decision policy is now available for any in put state S3. Solution values are given in Table 1 for selected inputs. Table  1S,d1*d2*d3*Optimal Returnf3*(S3)3270081.065400162981(54)00(9)2431272012360