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decision tree analysis Er. S Sood
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decision tree analysis Er. S Sood
1. OPERA TIONS RESEARCH LESSON 33: DECISION TREE ANALYSIS Learning Outcomes: decision-tree while the same type of problem calling for a • Decision tree as an effective decision-making tool sequence of decision is depictable in multi-state probabilistic decision-tree. Dear students, this is the last in a series of lectures on manage- rial decision-making. Today we are going to discuss the beauty How A Decision Tree Appears To Be of decision tree analysis. Decision Tree Analysis One of the devices for representing a diagrammatic presentation of sequential and multi-dimensional aspects of a particular decision problem for systematic analysis and evaluation is ‘decision tree’ – whereby the:- 1. Decision problem, 2. Alternative course of action, 3. States of nature and 4. The likely outcomes of alternatives, are diagrammatically or graphically depicted as if they are branches and sub-branches of ‘horizontal tree’. As the name implies, a decision tree consists of network of :- 1. Nodes, Dear friends, having come this far, let us now try to figure out 2. Branches, the various uses of the concept that we have been discussing all 3. Probability estimates the while. 4. and Pay-offs. Uses Nodes are of two types:- 1. The decision tree diagram is useful for portraying the:- 1. Decision-node (designated as a square) and 1. Inter-related, 2. Chance node (designated as a circle). 2. Sequential and Alternative course of action or strategies originate from the 3. Multi-dimensional aspects, decision node as the main branches (decision branches). At the of any major decision-problem within the system’s framework. terminal of each decision branch, there is a chance node By drawing a decision tree, the decision maker will be in a wherefrom chance event emanate in the form of sub-branches position to vislaies the entire complex of the decision problem (chance branches). The respective pay-offs and the probabilities in all its dimensions as also the actual processes and stages for associated with alternative courses and the chance events are arriving at the final choice. shown alongside the chance branches. At the terminal of the 2. It focuses attention on the critical elements in a decision chance branches are shown the expected values of the problem over the duration of its solution, apart from bringing outcome. to light the relationship between the presently available course There are basically two types of decision-tree:- of action and the network of future events. 1. Deterministic and 3. The decision tree device is especially useful in cases where an 2. Probabilistic. initial decision and its outcome affects the subsequent decisions and where the decision-maker has to make a sequence of These can further be divided into:- decisions on major decision-problem. 1. single stage and 4. Clearly, decision trees enable the decision maker to see the 2. multistage trees. various elements of his problem in content and in a systematic The basic difference between the two is that a single stage way. Very often even a rudimentary analysis will enable certain deterministic decision-tree involves making only one decision courses of action to be eliminated and permit the decision under conditions of certainity (no chance events). Whereas,in a maker to focus upon those options where this judgment and multi-stage deterministic tree a sequence or chains of decision experience are essential. are to be made. A problem involving only one decision to be 5. The obvious advantage of decision tree structure is that made under conditions of risk or uncertainty (more than one complex managerial problems and decisions of a chain-like chance event) can be represented in a single stage probabilistic nature can be systematically and explicitly defined and evaluated. © Copy Right: Rai University 11.235 157
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Multi-dimensional decision sequences can be strung on a Decision Tree For Sowing Seeds OPERA TIONS RESEARCH decision tree without conceptual difficulties. 6. The decision tree model can be applied in various fields such as introduction of a new product, marketing strategy, make vs. buy decision, pricing assets acquisition, investment decisions and as strange an area as selecting a life partner. Remarks 1. A decision tree (as with any other quantitative technique) does not make the decision totally routine in that there will always be some criterion upon which the analysis is based (such as maximizing expected value) and that criterion must be viewed with the usual caution and the necessary care taken in interpreta- tion of the results. Further-more, a decision tree is merely a structured example of a mathematical model and as such it will inevitably be simplification of reality. Can one ever be sure that the structure of the problem has been perfectly represented and that, for example, all possible courses of Key : Decision point action have been allowed for? Chance event 2. A manager may make use of decision tree for making The above is a very simply illustration which nevertheless, decision which otherwise not easy to make. Once a decision- display the basics features of a decision tree. maker accepts that position values at the nodes of a decision diagram are logical and meaningful, he can readily make use of Example 11 decision tree. A basic value of decision three lies in expressing all An executive has to make a decision. He has four alternatives:- outcomes or events in quantitative forms, which provide 1. D1, precision in decision-making. Since the various techniques have 2. D2, been developed to take into account the impact of large number of variables, the use of decision tree is even increased. 3. D3 and 4. D4. 1. Problems in Decision Tree Though decision tree has its use in effective decision-making, it When the decision has been made events may lead such that any is not as easy as it seems to be. A decision tree, while simple in of the four results may occur. The results are R1, R2, R3 and essence, can get complex in application. One of the main R4. Probabilities of occurrence of these results are as follows: difficulties in analyzing decision tree is that even with simple R1 = 0.5, R2 = 0.2, R3 = 0.2, R4 = 0.1 two or three branch forks, tree can be quite complex. It will be The matrix of pay-off between the decision and the results is just like a bush. There it another problem in the construction indicated below: of decision tree, i.e., making assumptions and settling of probabilities from different figures in decision tree. There is R1 R2 R3 R4 often inconsistency in assigning probabilities for different D1 14 9 10 5 events. Moreover, since many mangers are involved in this D2 11 10 8 7 process, often the process becomes time-consuming. Not D3 9 10 10 11 withstanding these, a decision tree offers a solution of the D4 8 10 11 13 decision situation better than any other technique. Show this decision situation in the form of a decision tree and Dear friends, time to take up examples now.Here we go. indicate the most preferred decision and corresponding expected Illustration. value. The Two alternative courses available in the said example are to Friends, try solving it first all by yourself and then tally your grow either rice or barley; the former requiring heavy rain and solution with that given below. the latter scantly for excellent yield. Whether there will be heavy Solution or scantly rain in the season is uncertain at the time of sowing A decision tree, which represents possible courses of action and the seed (in sophisticated decision tree models probabilities can states of nature are shown in the following figure. In order to be assigned to their occurrence and their pay-offs computed in analyses the tree, we start working backward from the end the chance events occur) branches. Friends, try solving it first all by yourself and then tally your solution with that given below. Solution © Copy Right: Rai University 158 11.235
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Friends, try solving it first all by yourself and then tally your OPERA TIONS RESEARCH solution with that given below. Solution The given data can easily be represented by the following decision tree diagram. Consequence of outflow (Rs.) There are two decision points in the tree indicated by 1 and 2. In order to decide between the two basis alternatives, we have to fold back (backward induction) the tree from the decision point 2, using EMV as criterion: Evaluation of Decision Points: The most preferred decision at the decision node 1 found is by calculating expected value of each decision branch and selecting State of Expected cash Decision point Nature Probability Cash outflows Outflow the path (course of action) with high value. Decision at point D2 1. Drill up to Water struck 0.2 Rs. 12,5000 Rs. 2,500 The expected monetary value of node A, B and C is calculate as 250 fit. follows: No water struck 0.8 27,500 22,000 EMV (outflows) = 24,500 EMV (A) = 0.5 x 14 + 0.2 x 9 + 0.2 x 10 + 0.1 x 5 = 11.3 2. Do not drill EMV (outflow) = Rs. 25,000 up to 250 ft. EMV (B) = 0.5 x 11 + 0.2 x 10 + 0.2 x 8 + 0.1 x 7 = 9.8 The decision at D2 is : Drill up to 250 feet. Decision at point D2 EMV (C) = 0.5 x 9 + 0.2 x 10 +0.2 x 10 + 0.1 x 11 = 9.6 1. Drill up to Water struck 0.7 Rs. 10,000 Rs. 7,000 200 ft. EMV (D) = 0.5 x 8 + 0.2 x 10 + 0.2 x 11 + 0.1 x 13 = 9.5 Not water struck 0.3 24,500 7,350 Since node A has the highest EMV, the decision at node 1 will EMV (outflow) = Rs. 14,350 2. Do not drill EMV (outflow) = Rs. 15,000 be to choose the course of action D4. up to 200 ft. The decision at D1 is : Drill up to 200 ft. Example 12 A farm owner is seriously considering of drilling farm well. In the past, only 70% of wells drilled were successful at 200 feet of Thus the optimal strategy for the farm-owner is to drill the well depth in the area. Moreover, on finding no water at 200 ft., up to 200 ft. and if no water is struck, then further drill it up to some persons drilled it further up to 250 feet but only 20% 250 ft. struck water at 250 ft. The prevailing cost of drilling is Rs. 50 Example 13 per feet. The farm owner has estimated that in case he loss not A business man has two independent investments A and B get his own well, he will have to pay Rs. 15,000 over the next 10 available to him; but he lacks the capital to undertake both of years (in PV terms) to buy water from the neighbor. The them simultaneously. He can choose to take. A first and then following decisions can be optimal: stop, or if A is successful then take B, or vice versa. The i. Do not drill any well, probability of success on A is 0.7, while for B it is 0’4. Both ii. Drill up to 200 ft. investments require an initial capital outlay of Rs. 2,000, and both return nothing if the venture is unsuccessful. Successful iii. If no water is found at 200 ft., drill further up to 250 ft. competitions of A will return Rs. 3,000 (over cost), and Draw an appropriate decision tree and determine the farm successful completion of B will return Rs. 5,000 (over cost). owner’s strategy under EMV approach. Draw the decision tree and determine the best strategy. © Copy Right: Rai University 11.235 159
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Friends, try solving it first all by yourself and then tally your However by resorting to overtime, the company will not be in a OPERA TIONS RESEARCH solution with that given below. position to meet the high magnitude of sales. It will be able to satisfy up to the level of medium magnitude only, even if high Solution magnitude of sales results. The appropriate decision tree is shown below: Solve the problem to suggest: which option should be selected? Friends, try solving it first all by yourself and then tally your solution with that given below. Solution The decision tree, which represents possible courses of action and out comes is shown in the figure on next page: There are three decision points in the above decision tree indicated by D1, D2 and D3. Evaluation of Decision Points: Decision point Outcome Probability Conditional Expected Values Values D3 (i) Accept A Success 0.7 Rs. 3000 Rs. 2100 Failure 0.3 -Rs. 2000 -Rs. 600 Evaluation of the tree begins with the terminal nodes (rolling Rs. 1500 (ii) Stop 0 back principle) to locate the optimum strategy or course of D2 (i) Accept B Success 0.4 Rs. 5000 Rs. 2000 action. Failure 0.6 -Rs. 2000 -Rs. 1200 Rs. 800 Evaluation of Decision Points: Stop 0 D1 (i) Accept A Success 0.7 Rs. 3000 + 800 Rs. 2660 Failure 0.3 -Rs. 2000 -Rs. 600 Decision point Event Proba Resulting Profit Expected Profit -bility (in Rs. Lakhs) (in Rs. Lakhs) (ii) Accept B Success 0.4 Rs. 5000 + 1500 Rs. 2600 D2 (i) Avail (a) High sales 0.45 7 0.45 x 7 = 3.15 Failure 0.6 -Rs. 2000 -Rs. 1200 overtime (b) Medium sales 0.30 7 0.30 x 7 = 2.10 Rs. 1400 facilities (c) Low sales 0.20 3 0.20 x 3 = 0.60 (iii) Do Nothing 0 (d) Nil 0.05 -5 0.05 x (-5) = -0.25 Total expected profit = 5.6 Less ever time express = 1.5 Net EMV = 4.1 Hence, the best strategy is to accept A first, and if it is successful, Install new (a) Approval 0.7 0.7 x 5.20* = 3.64 then accept B. (ii) division from the govt. (b) Refused 0.3 0.3 x 0 = 0 Example 14 approval Net EMV 3.64 A company is contemplating whether to produce a new *Approval (a) High Sales 0.45 15 0.45 x 15 = 0 product. If it decides to produce the product it must either of new (b) Medium sales 0.30 7 0.30 x 7 = 2.10 division (c) Low sales 0.20 3 0.20 x 3 = 0.60 install a new division which needs a cash outlay of four lakh (d) Nil 0.05 -5 0.05 x (-5) = -0.25 rupees, or work overtime with overtime expenses of Rs. 1.5 Total expected profit = 9.20 Less cost of new division = 4.00 lakhs. If the company decides to install a new division, it needs New return = 5.20 the approval of Government, and the company feels that there ∴ decision at D is : go for overtime 2 facilities is a 70% chance of getting the approval. D1 (i) Produce the product 4.10 A market survey has revealed the following facts regarding the (ii) Do not produce the product 0 magnitude of sales for the new product. Hence the option should be to produce a new product and Magnitude of Probability Resulting Profit work overtime with expected monetary value of Rs. 4.10 lakhs. Sales (in Rs. Lakhs) High .45 15 Dear friends, with this we have reached the end of today’s Medium .30 7 lecture. Low .20 3 Take care. Nil .05 -5 (Loss) Bye. © Copy Right: Rai University 160 11.235
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