Decision trees solution key
0 likes165 views
This document presents a decision problem faced by a manufacturer. The manufacturer produces items that have a probability of being defective. These items are formed into batches of 150. The manufacturer can either screen each item in a batch to check for defects at a cost of $10 per item screened, or use the items directly without screening and incur a cost of $100 per defective item that makes it through. Based on the given probabilities of good vs. bad quality batches, the expected costs per batch are calculated for each option. A decision tree is constructed to model the problem, taking into account the option to first test a single randomly selected item from the batch before deciding whether to screen the entire batch or not. The optimal strategy is determined to be
Decision trees solution key
- 1. EMSE 269 - Elements of Problem Solving and Decision Making Instructor: Dr. J. R. van Dorp 1 EXTRA PROBLEM 6: SOLVING DECISION TREES Read the following decision problem and answer the questions below. A manufacturer produces items that have a probability of . p being defective These items are formed into . Past experience indicates that batches of 150 some are of and others are of (batches) good quality (i.e. p=0.05) bad quality (i.e. p=0.25). Furthermore, of the batches produced are of 80% good quality and These items are then used in an 20% of the batches are of bad quality. assembly, and ultimately their quality is determined before the final assembly leaves the plant. The manufacturer can either in a batch and screen each item replace defective items at a total average cost of $10 per item or use the items directly without screening. the latter action If is chosen, the cost of rework is ultimately $100 per defective item. For these data, the costs per batch can be calculated as follows: p = 0.05 p = 0.25 Screen $1500 $1500 Do not Screen $750 $3750 Because screening requires scheduling of inspectors and equipment, the decision to screen or not screen must be made 2 days before the potential
- 2. EMSE 269 - Elements of Problem Solving and Decision Making Instructor: Dr. J. R. van Dorp 2 screening takes place. However, the manufactures may take one item taken from a batch and sent it to a laboratory, and the test results (defective or non- defective) can be reported must be before the screen/no-screen decision made. After the laboratory test, The the tested item is returned to its batch. cost of this initial inspection is $125. Also note that the probability that a random sample item is defective is 0.8 * 0.05 + 0.2 * 0.25 = 0.09, and the probability that an item in a lot is of good quality given a randomly sampled item is defective is 0.444 and the probability that an item in a lot is of good quality given a randomly sampled item is not defective is 0.835. The manufactur wants to minimize his or her cost. A. Derive the cost figures in the table above. Clearly show your calculations. p = 0.05 p = 0.25 Screen 150*$10 = $1500 150*$10 = $1500 Do not Screen 150*0.05*$100 = $750 150*0.25*$100 = $3750 B. What Law of Probability was used in deriving:
- 3. EMSE 269 - Elements of Problem Solving and Decision Making Instructor: Dr. J. R. van Dorp 3 Pr(Randomly Sampled Item is Defective) Pr(Sample Item is Defective|Batch of Good Quality) =0.05 Pr(Sample Item is Defective|Batch of Bad Quality) =0.25 Pr(Batch of Good Quality)=0.80 Pr(Batch of Bad Quality)=0.20 Pr(Randomly Sampled Item is Defective) = Pr(Sample Item is Defective|Batch of Good Quality)Pr(Batch of Good Quality)+ Pr(Sample Item is Defective|Batch of Bad Quality)Pr(Batch of Bad Quality)= 0.05*0.8 + 0.25*0.2=0.09 Hence, THE LAW OF TOTAL PROBABILITY was used.
- 4. EMSE 269 - Elements of Problem Solving and Decision Making Instructor: Dr. J. R. van Dorp 4 C. Show that: Pr(Batch of Good Quality|Sampled Item is Defective) = 0.444 Pr(Batch of Good Quality|Sampled Item is Not Defective) = 0.835 FKU œ à WMH œ "Batch of Good Quality" "Sampled Item is Defective"; = "Sampled Item is Not Defective" WMH T <ÐFKU WMHÑ œ œ | T <ÐWMHlFKUÑT <ÐFKUÑ T <ÐWMHÑ !Þ!&†!Þ)! !Þ!* œ !Þ%%% T <ÐFKU WMHÑ œ | T <ÐWMHlFKU ÑT <ÐFKUÑ T <ÐWMHÑ œ œ !Þ)$& Ð"!Þ!&ц!Þ)! Ð"!Þ!*Ñ
- 5. EMSE 269 - Elements of Problem Solving and Decision Making Instructor: Dr. J. R. van Dorp 5 D. Calculate and show your calculations: FKU œ à WMH œ "Batch of Bad Quality" "Sampled Item is Not Defective" Pr(Bad Quality|Sampled Item is Defective) = ? Pr(Bad Quality|Sampled Item is Not Defective) = ? T <ÐFKU WMHÑ œ " T <ÐFKU WMHÑ œ | | " !Þ%%% œ !Þ&&' T <ÐFKU WMHÑ œ " T <ÐFKU WMHÑ œ | | " !Þ)$& œ !Þ"'&
- 6. EMSE 269 - Elements of Problem Solving and Decision Making Instructor: Dr. J. R. van Dorp 6 E. Model the decision problem in a decision tree and fill in ALL the details Test Item? Screen? Passed Test Failed Test Screen? Screen? No Test Test No Screen Batch of Good Quality Batch of Bad Quality Batch of Good Quality Batch of Bad Quality Batch of Good Quality Batch of Bad Quality No Screen No Screen Screen Screen Screen (0.80) (0.20) (0.835) (0.165) (0.444) (0.566) (0.91) (0.09) $125 $1500 $1500 $1500 $750 $750 $750 $3750 $3750 $3750 $875 $875 $750 $3750 $1500 $1625 $1625 $3875 $3875 COST
- 7. EMSE 269 - Elements of Problem Solving and Decision Making Instructor: Dr. J. R. van Dorp 7 F. How many cumulative risk profiles can be drawn for the decision tree under E? Provide an Explanation. Test Item? Screen? Passed Test Failed Test Screen? Screen? No Test Test No Screen Batch of Good Quality Batch of Bad Quality Batch of Good Quality Batch of Bad Quality Batch of Good Quality Batch of Bad Quality No Screen No Screen Screen Screen Screen (0.80) (0.20) (0.835) (0.165) (0.444) (0.566) (0.91) (0.09) $125 $1500 $1500 $1500 $750 $750 $750 $3750 $3750 $3750 $875 $875 $750 $3750 $1500 $1625 $1625 $3875 $3875 COST Upper Part: 4 , , , (T,{NS,NS}) (T,{S,S}) (T,{NS,S}) (T,{S,NS}) Lower Part: 2 , Total: 6 Strategies (NT,NS) (NT, S)
- 8. EMSE 269 - Elements of Problem Solving and Decision Making Instructor: Dr. J. R. van Dorp 8 G. Solve the tree under E using EMV and clearly show your calculations in the tree under E Test Item? Screen? Passed Test Failed Test Screen? Screen? No Test Test No Screen Batch of Good Quality Batch of Bad Quality Batch of Good Quality Batch of Bad Quality Batch of Good Quality Batch of Bad Quality No Screen No Screen Screen Screen Screen (0.80) (0.20) (0.835) (0.165) (0.444) (0.566) (0.91) (0.09) $125 $1500 $1500 $1500 $750 $750 $750 $3750 $3750 $3750 $875 $875 $750 $3750 $1500 $1625 $1625 $3875 $3875 COST $1369.51 $1625 $1392.50 $1369.51 $2541.67 $1350 $1350 $1350
- 9. EMSE 269 - Elements of Problem Solving and Decision Making Instructor: Dr. J. R. van Dorp 9 H. Describe in words the optimal decision strategy. Do not Sample an Item Randomly from a Batch for Testing, and Do Not Screen the entire Batch (No Test, No Screen in the Decision Tree)
- 10. EMSE 269 - Elements of Problem Solving and Decision Making Instructor: Dr. J. R. van Dorp 10 I. Draw the Cumulative Risk Profile for each alternative of the immediate decision, taking optimal decisions from thereon. What can you conclude with respect to dominance considerations? (Hint: You should be drawing 2 Cumulative Risk Profiles). Test Item? Screen? Passed Test Failed Test Screen? Screen? No Test Test No Screen Batch of Good Quality Batch of Bad Quality Batch of Good Quality Batch of Bad Quality No Screen Screen (0.80) (0.20) (0.835) (0.165) (0.91) (0.09) $125 $1500 $750 $750 $3750 $3750 $875 $750 $3750 $1625 $3875 COST $1392.50
- 11. EMSE 269 - Elements of Problem Solving and Decision Making Instructor: Dr. J. R. van Dorp 11 Strategy: Test Random Item and If Item Passed Test then do not Screen the entire Batch and if Item Failed Test then Screen the entire Batch T < G9=> œ )(& œ !Þ*"‡!Þ)$& œ !Þ('! ( ) T < G9=> œ "'#& œ !Þ!* ( ) T < G9=> œ $)(& œ !Þ*"‡!Þ"'& œ !Þ"&! ( ) T <ÐG9=> Ÿ )(&Ñ œ !Þ('! T <ÐG9=> Ÿ "'#&Ñ œ !Þ('! !Þ!* œ !Þ)& T <ÐG9=> Ÿ $)(&Ñ œ !Þ)& !Þ"& œ "Þ!!! Strategy: Do not Test Random Item and Do not Screen the Entire Batch T < G9=> œ (&! œ !Þ)! ( ) T < G9=> œ $(&! œ !Þ#! ( ) T <ÐG9=> Ÿ (&!Ñ œ !Þ)! T <ÐG9=> Ÿ $(&!Ñ œ !Þ)! !Þ#! œ "Þ!!
- 12. EMSE 269 - Elements of Problem Solving and Decision Making Instructor: Dr. J. R. van Dorp 12 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 $1000 $2000 $3000 $4000 $750 $750 $825 $825 $1625 $1625 $3750 $3750 $3875 $3875 0.76 0.76 0.80 0.80 0.85 0.85 CUMULATIVE RISK PROFILES CROSS. HENCE, NO DETERMINISTIC DOMINANCE AND NO STOCHASTIC DOMINANCE