5 Multiple Constraints

5 Multiple Constraints - 1 A Simple Mathematical...

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1 A Simple Mathematical Programming Problem Consider a single constraint separable optimization problem of the form: max z = g i ( x i ) i = 1 n st h i ( i = 1 n x i ) b x i 0 and integers for all i . This problem can be solved by dynamic programming even when the functions g i and h i do not conform to the standard convexity assumptions of mathematical programming. We decompose the problem into a sequence of single decision variable sub-problems using the constraint to define the state variable domains and the sub-problem interconnections. This is accomplished by specifying a sequence in which the variables are to be determined; that is, imposing an artificial sequence in which the variables are to be determined. Lets assume that we will work these in the index variable name sequence, x 1 as the first sub-problem, x 2 as the second, etc. Then we have transformed the problem into one depicted by the sequential diagram (three variable problem illustration): Time 2 Time 3 s 3 s 2 s 4 XX X 1 2 3 = b Time 1 > 0 s 1 The state variables s i represent the amount of the resource b which is available for allocation in the specific and future decision periods. The last state variable, s 4 in this example, is the slack variable or how must of the initial b units are left over at the end of the allocation process and, hence, this must be a nonnegative quantity. This nonnegativity property propagates all of the way back to the initial sub-problem placing restrictions on the decision variables. Here we are assuming that the terms h i represent usage of the resource units and do not increase them, that is, h i ( x i ) 0:
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2 s n + 1 = s n h n ( x n ) 0 implies 0 h n ( x n ) s n s i + 1 = s i h i ( x i ) 0 implies 0 h i ( x i ) s i , i n , " ,1 s 1 = b We can now formulate the dynamic programming recursion equations for this problem. These are: f 1 ( s 1 ) = max{ g 1 ( x 1 ) + f 2 ( s 1 h 1 ( x 1 )) : x 1 {0,1,2, " } with h 1 ( x 1 ) s 1 }, s 1 { b } f 2 ( s 2 ) = max{ g 2 ( x 2 ) + f 3 ( s 2 h 2 ( x 2 )) : x 2 " } with h 2 ( x 2 ) s 2 }, s 2 {0, " , b } # f n ( s n ) = max{ g n ( x n ): x n " } with h n ( x n ) s n }, s n " , b } From the formulation, again we must evaluate these recursion functions in the reversion order of their definition. Thus, we solve for f n , then f n-1 , " , and finally f 1 . Example Problem . We consider the problem: max z = 13 x 1 x 1 2 + 30.2 x 2 5 x 2 2 + 10 x 3 2.5 x 3 2 st x 1 + x 2 + x 3 5 x i 0 and integers for all i . This results in the recursion functions, here we use the notation very similar to the MOR/DP language to facilitate the problem solution as well as its formulation: f 3 ( s 3 ) = max{10 x 3 2.5 x 3 2 : x 3 in 0. .5 with x 3 s 3 }, s 3 in 0. .5 ; f 2 ( s 2 ) = max{30.2 x 2 5 x 2 2 + f 3 ( s 2 x 2 x 2 in 0. .5 with x 2 s 2 }, s 2 in 0. .5 ; f 1 ( s 1 ) = max{13 x 1 x 1 2 + f 2 ( s 1 x 1 x 1 in 0. .5 with x 1 s 1 }, s 1 in 5 ; The tabular solution to this problem is given first and then the output of MOR/DP.
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5 Multiple Constraints - 1 A Simple Mathematical...

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