DYNAMIC PROGRAMMING

By -

Dynamic programming is a useful mathematical technique for making a sequence of interrelated decisions. It provides a systematic procedure for determining the optimal combination of decisions.

In contrast to linear programming, there does not exist a standard mathematical formulation of “the” dynamic programming problem. Rather, dynamic programming is a general type of approach to problem solving, and the particular equations used must be developed to fit each situation. Therefore, a certain degree of ingenuity and insight into the general structure of dynamic programming problems is required to recognize when and how a problem can be solved by dynamic programming procedures. These abilities can best be developed by an exposure to a wide variety of dynamic programming applications and a study of the characteristics that are common to all these situations. A large number of illustrative examples are presented for this purpose. (Some of these examples are small enough that they also could be solved fairly quickly by exhaustive enumeration, but dynamic programming provides a vastly more efficient way of solving larger versions of these examples.)

Comments

Post a Comment

Popular posts from this blog

DUALITY THEORY:THE ESSENCE OF DUALITY THEORY

By -
Image
THE ESSENCE OF DUALITY THEORY Given our standard form for the primal problem at the left (perhaps after conversion from another form), its dual problem has the form shown to the right. Thus, with the primal problem in maximizatio n form, the dual problem is in minimization form instead. Furthermore, the dual problem uses exactly the same pa r amete r s as the primal problem, but in different locations, as summarized below. 1. The coefficients in the objective function of the primal problem are the right-hand sides of the functional constraints in the dual problem. 2. The right-hand sides of the functional constraints in the primal problem are the coef- ficients in the objective function of the dual problem. 3. The coefficients of a variable in the functional constraints of the primal problem are the coefficients in a functional constraint of the dual problem. To highlight the comparison, now look at these same two problems in matrix notation (as introduced at the beginn
Read more

NETWORK OPTIMIZATION MODELS:THE MINIMUM SPANNING TREE PROBLEM

By -
Image
THE MINIMUM SPANNING TREE PROBLEM The minimum spanning tree problem bears some similarities to the main version of the shortest-path problem presented in the preceding section. In both cases, an undi r ecte d and connected network is being considered, where the given information includes some mea- sure of the positive length (distance, cost, time, etc.) associated with each link. Both problems also involve choosing a set of links that have the shortest total length among all sets of links that satisfy a certain property. For the shortest-path problem, this property is that the chosen links must provide a path between the origin and the destination. For the minimum spanning tree problem, the required property is that the chosen links must provide a path between ea c h pair of nodes. The minimum spanning tree problem can be summarized as follows: 1. You are given the nodes of a network but not the links. Instead, you are given the po- tential links and the positive length for each
Read more

NETWORK OPTIMIZATION MODELS:THE SHORTEST-PATH PROBLEM

By -
Image
THE SHORTEST-PATH PROBLEM Although several other versions of the shortest-path problem (including some for directed networks) are mentioned at the end of the section, we shall focus on the following simple version. Consider an undi r ecte d and connected network with two special nodes called the origin and the destination. Associated with each of the links (undirected arcs) is a non- negative distance. The objective is to find the shortest path (the path with the minimum total distance) from the origin to the destination. A relatively straightforward algorithm is available for this problem. The essence of this procedure is that it fans out from the origin, successively identifying the shortest path to each of the nodes of the network in the ascending order of their (shortest) distances from the origin, thereby solving the problem when the destination node is reached. We shall first outline the method and then illustrate it by solving the shortest-path problem encountered by the See
Read more