Wednesday, July 17, 2019
Dynamic Programming Essay
changing schedule is a mathematical technique dealing with the optimization of multi leg closing processes. In this technique, decisions regarding a accredited paradox be typic onlyy optimized in stages rather than simultaneously. This generally signifies that the original decision trouble is divided into small sub-problem (stages) which spate past be handled more efficiently from the computational stance point.Basic Elements of combat-ready programTo apply Dynamic Programming, we have to pay special attention to the collar basic elements of the DP Model. They are 1. comment of the stages.2. description of the alternate(a)s at distri stillively stage.3. Definition of the states for apiece stage.Definition of the states varies depending on the situation being sit arounded. Nevertheless, as we drapeigate each application, we get out find it helpful to cope the following questions 1. What relationships bind the stages together?2. What information is undeniable to make feasible decisions at the present-day(prenominal) stage without reexamining the decisions made at previous stages?Application of the Dynamic Programming in the Business WorldWe exit try to present three application models and eventually a worked out implementation of Dynamic Programming showing the superiority of DP over the familiar or straight forward method of solution.1. hold up Force ModelIn some body structure projects, hiring and firing are exercised to maintain a do work force that meets the needs of the project. Given that the activities of hiring and firing both(prenominal) incur additional costs. In such cases, done with(predicate) the implementation of DP Model, we dope get the optimal result regarding how thelabor force should be well-kept throughout the life of the project.For exampleA twirl contractor estimates that the size of the work force ask over the next 5 weeks is to be 5, 7, 8, 4 and 6 workers respectively. Excess labor kept on the force w ill cost $300 per week and new hiring in any week will incur a fixed cost of $four hundred plus $200 per worker per week.The elements of this DP model are1. Stage iSuch problem privy optimally be work through DP Model.Equipment Replacement ModelThe longer a machine stays in serve, the higher is its precaution cost, and the lower its productivity. When a machine reaches a certain age, it may be more economical to regenerate it. The problem thus turns into determining the most economical age of a machine. Suppose that we are poring over the machine backup man problem over a span of n geezerhood. At the start of each year, we decide whether to keep the machine in returns 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 mustiness 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 increases by 10% a year. The salvage value of a 1-year-old tractor is $30,000 and decreases by 10% a year. The current annual operate 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 start 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 enliven rate offered by NCC Bank is 8% deepen 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 working capital at the end of 4 years.Such problem can also optimally be solved easily by applying DP Model. A high society is selecting the advertising for its productand the frequency of advertising by each m aterial are shown in the following skirtFrequency per week Expected Sales (In Tk. 1,000) tv set 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 optimal combination of advertising frequency and sales.SolutionStateslet X1= The frequency of advertising 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 Now we can derive the optimal valuesX1=1X2=2X3=1Expected Sales= 96,000Us ual or Straight forward method of solution muckle indicates alternative plans at each stage & Arrows lay out the decision.The features of the above exhaustive enumeration scheme are 1. All the decisions of any combination must specify before a combination can be evaluated. Here during solution, we have to make 64 alternative plans first. 2. An optimum policy cannot be determined until all combinations have been evaluated. This method is inefficient because some of the combination may not be feasible. 3. In new(prenominal) cases the number of combination may be in like manner large to allow exhaustive listing.The Dynamic Programming approach avoids the above mentioned difficulties by first pause up the problem into smaller sub-problems which are called stages in DP. A stage here signifies a great deal of the problem for which a separate decision can be made.