Dynamic Programming Essay

Dynamic computer programing is a mathematical technique dealing with the optimization of multist maturate decision processes. In this technique, decisions regarding a certain task be typically optimized in stages alternatively than simultaneously. This generally signifies that the original decision problem is divided into small sub-problem (stages) which whoremaster then be handled more efficiently from the computational view point. prefatory Elements of Dynamic ProgrammingTo apply Dynamic Programming, we have to pay peculiar(prenominal) attention to the three basic elements of the DP Model. They ar 1. comment of the stages.2. Definition of the alternatives at distributively stage.3. Definition of the states for each stage.Definition of the states varies depending on the pip being modeled. Nevertheless, as we investigate each application, we result find it laboursaving to consider the following questions 1. What relationships bind the stages together?2. What information i s needed to bring on feasible decisions at the menses stage with prohibited reexamining the decisions made at former stages?Application of the Dynamic Programming in the Business WorldWe will try to present three application models and finally a worked out implementation of Dynamic Programming showing the superiority of DP all over the usual or straight forward method acting of solution.1. Work push up ModelIn some construction projects, hiring and firing are exercised to control a labour force that meets the needs of the project. Given that the activities of hiring and firing some(prenominal) incur additional personifys. In such cases, through the implementation of DP Model, we offer get the optimal result regarding how thelabor force should be maintained throughout the life of the project.For exampleA construction affirmer estimates that the size of the work force needed over the next 5 weeks is to be 5, 7, 8, 4 and 6 workers respectively. Excess labor kept on the force will cost $300 per week and new hiring in either week will incur a fixed cost of $400 plus $200 per worker per week.The elements of this DP model are1. Stage iSuch problem can optimally be work through DP Model.Equipment Replacement ModelThe longer a machine stays in service, the higher is its maintenance cost, and the lower its productivity. When a machine reaches a certain age, it may be more sparing to replace it. The problem thus turns into determining the most economical age of a machine. Suppose that we are studying the machine replacement problem over a span of n course of instructions. At the start of each year, we decide whether to keep the machine in service an extra year or to replace it with a new one.For exampleShajib Farms wants to develop a replacement policy for its 2-year-old tractor over the next 5 years. A tractor must be kept in service for at least 3 years, but must be disposed of after 5 years. The current purchase price of a tractor is $40,000 and increa ses by 10% a year. The ease value of a 1-year-old tractor is $30,000 and decreases by 10% a year. The current yearly operating cost of the tractor is $1,300 but is expected to increase by 10% a year.Such problem can optimally be solved easily by applying DP Model.Investment ModelWe commonly stick out that an investor wants to maximize Total Return. Suppose that Mr. Jamal wants to invest Tk. 4,000,000 (4 Million) now and 2,000,00 (2 Million) at the starts of years 2 to 4. The interest rate offered by NCC Bank is 8% compounded annually and the bonuses over the next 4 years are 1.8%, 1.7%, 2.1% and 2.5% respectively. The annual interest rate offered by Eastern Bank is 2% lower than that of NCC Bank, but its bonus is .5% higher. The objective is to maximize the accumulated large(p) at the end of 4 years.Such problem can in any case optimally be solved easily by applying DP Model. A company is selecting the advertising for its productand the frequency of advertising by each poppycoc k are shown in the following tableFrequency per week evaluate gross sales (In Tk. 1,000) Television Radio Newspaper 0 0 0 0 1 25 20 33 2 42 38 43 3 55 54 47 4 63 65 50 We have to determine the optimum conspiracy of advertising frequency and sales.SolutionStatesLet X1= The frequency of advertisement at stage-1 (06)X2= The frequency of advertisement at stage-2 (06)X3= The frequency of advertisement at stage-3 (=6)S= Total FrequncyStage-1Total Frequency (S) Frequency at Expected Sales Stage-1(X1) 0 0 0 1 1 25 2 2 42 3 3 55 4 4 63 Stage-2 X2 f 2(S, X2)=R2(X2)+ f 1*(S-X2) f2*(S) X2* S 0 1 2 3 4 0 0+0=0 0 0 1 0+25=25 20+0=20 25 0 2 0+42=42 20+25=45 38+0=38 45 1 3 0+55=55 20+42=62 38+25=63 54+0=54 63 2 4 0+63=63 20+55=75 38+42=80 54+25=79 65+0=65 80 2 Stage-3 X2 f 3(S, X3)=R3(X3)+ f 2*(S-X3) f3*(S) X3* S 0 1 2 3 4 4 0+80=80 33+63=96 43+45=88 47+25=72 50+0=50 96 1 instanter we can derive the optimal valuesX1=1X2=2X3=1Expected Sales = 96,000Usual or Straight forward method of solution Circle indicates alternative plans at each stage & Arrows show the decision.The features of the above exhaustive enumeration scheme are 1. All the decisions of any combination must specified before a combination can be evaluated. Here during solution, we have to make 64 alternative plans firstborn. 2. An optimum policy cannot be determined until all combinations have been evaluated. This method is uneffective because some of the combination may not be feasible. 3. In fragment cases the number of combination may be too large to take into account exhaustive listing.The Dynamic Programming approach avoids the above mentioned difficulties by first breaking up the problem into smaller sub-problems which are called stages in DP. A stage here signifies a portion of the problem for which a separate decision can be made.