Advantages Of Dynamic Programming

4.2. Dynamic programming
4.2.1. Introduction. The dynamic programming algorithm defines the relationship between problem states and states by splitting the problem, so that the problem can be solved by recursion (or divide). Dynamic programming algorithm is similar to divideand- conquer method, it breaks up the problem into several sub problems (phase) and then solve the sub stage one by one. The solution of previous problem can be reference information to the next sub problem. When solving the problem of sub problem, various possible solutions for the local solutions are listed, only the one who may be the optimal one saved by decision. Via figuring out each sub problem successively, the solution for the initial problem can be achieve.
4.2.2.

As long as the sub problem is calculated, its result will be record to this table. This is the basic idea of dynamic programming. The specified condition various to different programming algorithms, but they have a same filling format.
4.2.3. Pro and Con. The advantage of dynamic programming is that it can obtain both local and total optimal solution. Also, practical knowledge can be used to gain the higher efficiency of dynamic programming. However, there is no unifiedstandard model for dynamic programming, multiple condition may appear during the solving process. Also, dimension disasters may occur when solving numerical methods.
Consider about the aspects about modeling the VRP, it could be separated into three part: vehicle flow, commodity flow and set partitioning. Based on these aspect, three kinds of exact algorithms will be introduced: Vehicle-indexed formulations, Commodity flow formulations and Set partitioning formulations. Vehicle-indexed formulations part including two-index vehicle flow formulation, in which each arc of the graph can be shown by the indexes. the three-index vehicle flow formulation, in which each

4.3.1. Two-index vehicle flow formulation. Two-index vehicle flow formulations first proposed by Laporte, Nobert and Desrochers [Laporte et al. 1985], which extent the TSP problem. In Two-index vehicle flow formulations, define xij as a binary variable which represent the times arcij belong to the optimal solution.
(V F) Min
X
[i;j2E] cijxij (12) s:t: X j2V nf0g x0j = 2m (13)
X
ik xkj = 2 (k 2 V n f0g) (14)
X
i2S;j =2S or j2S;i=2S xij  2b(S) (S 2 V n f0g) (15) xij = 0 or 1 (i; j 2 V n f0g) (16) x0j = 0; 1 or 1 (j 2 V n f0g) (17)
In this formulation, the m can be constant or variable. The constraints 13 and 14 described the degree of each node. The b(S) which commonly defined as P i2S qi
Q

in
Constraints 15 is the least number of needed vehicles for all customers(S). These constraints not only make the connection between subset customers and depot, but also prevent the excess of the capacity. Finally, integer requirements on the variables specified in constraints 16 and 17.
Three-index vehicle flow formulation: this formulation is based on research result of
Fisher and Jaikumar [Fisher and Jaikumar 1981] and Fischetti et al. [Fischetti et