GROUP MEMBERS AMARASENA R.G.C. (061004D) DE MEL W.R. (061013E) DOLAPIHILLA I.N.K. (061017U) KUMARAJITH R.M.E. (061031G)

Introduced by Richard Bell G.B. Dantzig In year 1952

Dynamic Programming Dynamic Programming is an algorithm design method that can be used when the solution to a problem may be viewed as the result of a sequence of decisions.

Typical steps of Dynamic Programming  Characterize the structure of an optimal solution.  Recursively define the value of an optimal solution.  Compute the value of an optimal solution in a bottom-up fashion.  Compute an optimal solution from computed/stored information.

Application areas  Airline  Hotel  Apparel  Manufacturing  Healthcare  Broadcast  Energy  Rail  Tour operators  Cargo  Restaurants  Golf  Car rental

Applications  Network Problems  Man power scheduling  Inventory management  Resource allocation Problems  Optimal Stopping Problems

Applications cont.  Bioinformatics – computerized analysis of biological data the use of computers to extract and analyze biological data, especially in studying the nucleotide sequences of DNA and other nucleic acids  Computer science: theory, graphics, AI, systems.

Network Problem  Many applications of dynamic programming reduce to finding the shortest (or longest) path that joins two points in a given network.  For larger networks dynamic programming is much more efficient for determining a shortest path than the explicit enumeration of all paths.

e c f d g h c c Example  Several paths  Various distances Minimum distance?

Collection and delivery problems Robotics Board drilling

Inventory Problems  Dynamic programming can be used to solve an inventory problem with the following characteristics: 1. Time is broken up into periods, the present period being period 1, the next period 2, and the final period T. At the beginning of period 1, the demand during each period is known. 2. At the beginning of each period, the firm must determine how many units should be produced. Production capacity during each period is limited.

3. Each period’s demand must be met on time from inventory or current production. During any period in which production takes place, a fixed cost of production as well as a variable per-unit cost is incurred. 4. The firm has limited storage capacity. This is reflected by a limit on end-of-period inventory. A per-unit holding cost is incurred on each period’s ending inventory. 5. The firms goal is to minimize the total cost of meeting on time the demands for periods 1,2, …., T.

Resource Allocation Problems Resource-allocation problems, in which limited resources must be allocated among several activities, are often solved by dynamic programming.

To use dynamic programming to do resource allocation, three assumptions must be made:  The amount of a resource assigned to an activity may be any non negative number.  The benefit obtained from each activity is proportional to the amount of the resource assigned to the activity.  The benefit obtained from more than one activity is the sum of the benefits obtained from the individual activities.

Example  Company having several plants  Several projects  Limited investment Maximize revenue?

Optimal Stopping Problems A special class of problems involving a discrete choice are those in which there is a single decision to put an end to an ongoing problem.

 A student must decide when to give up trying to solve a homework problem.  A firm must decide when to exit an industry.  A firm decides when to stop working on the development of a product and to launch it

Applications in detail Managing warehouse space  Several items  Total volume  Value of unit volume Maximize revenue?

Replacing machines To calculate the maximum time for using a machine. Machine price Operating and maintaining cost Selling price

Applications cont.  Number of teams can be allocated to different areas in such a way that the total effectiveness is high  Number of days can be allocated for specific tasks.

 Dynamic Programming in High Density Barcodes Symbol Technology has developed a new design for barcodes, that has a capacity of several hundred bytes.

 In chess playing software Can use dynamic programming in developing chess playing software

 Dynamic programming in Civil engineering The application of dynamic programming to slope stability analysis - Stresses acting can be analysed

 Dynamic programming in the fields of structural engineering and water resources engineering. Used in continuous beams geometric layout of truss water allocation capacity expansion reservoir operation

Advantages of Dynamic programming  Better control  Less complex  Focused approach  Realistic approach

Limitations  Cost  Time  Thus dynamic programming can only be efficient when there are not too many partial results to compute!

Thank you