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Intro to Optim Meth

by: Loy King

Intro to Optim Meth IOE 310

Loy King
GPA 3.76


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Class Notes
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This 9 page Class Notes was uploaded by Loy King on Thursday October 29, 2015. The Class Notes belongs to IOE 310 at University of Michigan taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/231596/ioe-310-university-of-michigan in Industrial Engineering at University of Michigan.

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Date Created: 10/29/15
131 Computer project problems Katta G Murty Lecture slides 1 A Product mix problem A company has machines of types 1 and 2 Using them and two raw materials R1 R2 the company makes and sells products 131132133 Relevent data given below In the table M1 M2 are type 1 2 machine times in minutes Resource Input units ton of product Daily P1 P2 P3 availability M1 1 2 1 430 M2 3 0 2 460 R1 tons 1 4 0 420 R2tons 1 1 1 300 Min daily demand 70 Max daily demand 240 Net pro t ton 300 200 500 Formulate problem of maximizing total daily net pro t of company subject to all the constraints as an LP Solve the LP using AMPL will get you a higher grade if you use this software or any other software Write a report explaining the output and all the useful planning information that can be derived from it 2 Problem with Local content constraint A US company is planning to set up a plant in a foreign country to manufacture a hi tech product to take advantage of the cheaper labor in that country The product needs 10 components some in multiple copies For 2 1 to 10 let m no copies of component 2 used to produce 1 unit of nal product cih delivered unit purchase cost of component 2 from cheapest supplier in host country Gig delivered unit purchase cost of component 2 from cheapest global supplier Data provided in following table 199 Component 2 mi cig cih 1 20 025 030 2 2 2 20 400 3 4 5 30 620 4 1 8000 9500 5 1 4550 52 00 6 2 7 25 725 7 12 2 40 320 8 4 3 00 450 9 4 12 00 1925 10 1 50 00 100 00 The local supplier is the cheapest supplier globally for component 6 but all other components have a foreign source as the cheapest supplier Assume that no supplier has capacity constraints The host country has a domestic content policy which stipulates that a speci ed fraction a or higher of the total value of all the components in the manufactured product must come from domestic suppliers For 2 1 to 107 de ne the decision variables xi fraction of component 2 sourced from cheapest local supplier 1 7 xi yi fraction of component 2 sourced from cheapest global supplier Assume that the annual production target is T 106 units of the product Using the above decision variables formulate the problem of minimizing the total component cost7 subject to the domestic content policy7 as an LP Solve the model for several values of a say7 07 0257 0507 and 1 Discuss how the total component cost per unit product varies as a increases 3 Cooking oil blending A small company makes a very ne cooking oil used in used in high quality restaurants for salad oil It is a blend of 5 different vegetable oils Following table gives the posted prices of raw vegetable oils over a 6 month period of next year These prices can be locked in now by means of futures contracts 200 Raw oil prices 100 liter vat Month Price of oil Corn Peanut Sa ower Sesame Sun ower Jan 179 172 203 188 172 Feb 180 140 172 203 203 Mar 149 156 203 219 172 Apr 195 187 187 172 187 May 164 172 234 188 156 Jun 211 125 218 156 141 Hardness 50 42 20 61 88 The hardness of the blend weighted average of hardness of oils in the blend must be between 30 and 58 The company s blending plant has a monthly capacity for processing at most 250 vats of corn peanut and sa ower oils combined at most 200 vats of sesame and sun ower oils combined At the start of the 6 month period assume that there are 500 vats of each kind of vegetable oil in storage The plan must provide for the same quantities of each oil in storage at the end of the 6 month period For each oil the storage capacity is 1000 vats the cost of keeping the oil in storage from one month to the next is 8vat The amount of cooking oil produced 096sum of the amounts of vegetable oils blended Given the product s popularity all the cooking oil that is produced in a month can be sold in that month itself at a price of 290vat It is required to formulate a 6 month plan that gives the amount of each vegetable oil to purchase each month the amount of cooking oil to produce each month and the amount of each vegetable oil to hold over in storage for the succeding month Formulate this as an LP Solve the LP and write a brief report explaing the output and any useful planning information you can get from it 4 Forming a committee A school committee is forming an ad hoc committee to prepare a very important report on all aspects of the school district About the audience The school committee chair is Barbara Firestone CPA Ms Firestone is also the Controller of a major automotive supplier AutoParts Inc She 201 knows a lot about schools nance and accounting but not a lot about engineering She is familiar with the types of problems industrial engineers can solve but she is not at all familiar with industrial engineering problem solving methods Ms Firestone needs to be able to explain her choices of citizens to appoint to this ad hoc committee As chair she is authorized by the bylaws of the school committee to appoint citizens to an ad hoc committee She does not need the approval of the other members of the school committee Since this is a very important report and it ts her management style she plans to seek a consensus on appointments to the ad hoc committee from school committee members before actually making the appointments About you You are a Staff Engineer with a very prestigious engineering consulting rm IOE 310 Inc You would like to be a Senior Engineer IOE 310 Inc has a reputation for rapidly promoting engineers who bring business into the company You are new to the consulting business and want to build a reputation as an outstanding engineering consultant About the scenario You learned that the school committee was forming the ad hoc committee from your cousin who is a secretary with the school district As a concerned citizen you decided to volunteer your time to help the school committee You set up an appointment and met with Ms Firestone School Committee Chair She explained her problem to you and was very pleased to have your assistance During your meeting you had an opportunity to talk a little about business and you learned that AutoParts Inc is having some minor problems with inventory control and in other areas in which you and your colleagues at IOE 310 Inc have considerable expertise The management team at AutoParts Inc is meeting next month for a quarterly budget review Ms Firestone told you that she is going to recommend funding for expert consulting services You seemed to have stumbled upon a business opportunity here What you have accomplished so far Ms Firestone gave you a list of 42 different types of expertise numbered 0 to 41 that she feels need to be represented on the ad hoc committee You surveyed 33 people numbered 0 to 32 who are willing to serve on the ad hoc committee and identi ed their areas of expertise The data is given in the table E is the number of expertise type and cit with E are the people with that expertise Your technical assignment Ms Firestone feels that it is very important to keep the size of the committee small Determine the smallest committee that contains all types of expertise Your communication assignment You de nitely have a moderately di icult report writing situation you need to write a recommendation and you can not make any simplifying assumptions Your task goal is to have your recommendation on the citizens to appoint to the ad hoc committee approved by Ms Firestone and the other members of the school committee Your social goal is to increase your credibility with Ms Firestone Her perception of your abilities as an engineer are sure to be greatly in uenced by the quality of your report 202 E Cit with E E Cit with E E Cit with E 0 2471822 1 8061932 2 7361325 3 451429 4 101827 5 11122023 6 7121422 7 1161828 8 13161728 9 412171822 10 614181932 11 313172526 12 451429 13 161827 14 11202331 15 7121422 16 16182832 17 131732 18 19262830 19 15161832 20 17222628 21 28121417 22 2691128 23 35232627 24 491415 25 0817 26 1101319 27 271224 28 081618 29 671328 30 28121724 31 926283132 32 67152331 33 5141924 34 72128 35 3113031 36 4182732 37 8172532 38 72332 39 3282930 40 5131830 41 7263132 5 Setting up Telemarketing centers A company wants to set up 3 or 4 telemarketing centers They identi ed 7 different sites as being suitable locations for setting up these centers These centers are to serve customers in eight area codes 501 918 316 417 314 816502606 Let cij Communication costs in hour to the company if customers from area codej are allocated to be served by a center set up at site 239 Cost in millionyear of operating a communi cation center at site 239 Data on these cost coef cients is given below Formulate a model to determine the best sites to set up the centers and the allocation of area codes to the centers set up so as to minimize the total annual operational and communication costs for serving all the area codes given above subject to the constraints Solve the model Explain the computer output and the optimum solution obtained 203 Site 239 d cij for area code j 501 918 316 417 314 816 502 606 1 DA 5 14 35 29 32 25 13 14 20 2 AT 8 18 18 22 18 26 23 12 15 3 LV 4 22 25 12 19 30 17 26 25 4 DE 9 24 30 19 14 12 16 18 30 5 LR 3 19 20 23 16 23 11 28 12 6 MP 45 23 21 17 21 20 23 20 10 7 SL 55 17 18 12 10 19 22 6 22 DA Dal as TX AT Atlanta GA LV Louisville KY DE Denver CO LR Little Rock AR SL St Louis MO MP Memphis TN 204 Aim of Computer Projects in Our LP Courses Katta G Murty IOE Dept U of Michigan Ann Arbor H To make sure that our students can model amp solve at least well structured problems using software systems in common use 10 To make sure that they derive all useful planning information from model such as o lnfeasibility analysis if model infeasible ways to make it feasible 0 Opt Sol amp check for its reasonableness o Slack values at Opt critical resources slack resources 0 When is opt dual sol the marginal value MV vector for RHS constants o Derive planning info by marginal analysis for example Making one unit of a new product needs 3 units resource 1 MV of 10 net pro tunit 5 units resource 2 MV 15unit 7 units resource 3 MV 20unit Then this product worth making if its net pro t potential 2 3 X 10 5 gtlt15 7 X 20 215unit 3 To give them practice in preparing good reports interpreting the output from the model An example Project Problem Optimization of crude oil re ning amp blending 3 suppliers of crude with following data Availability Price Bday B Crude 1 60000 1525 2 90000 1300 3 80000 1400 Distillation outputs at fractionater 1P Interm Product 205 Yield of I B B DN DHO DGO P Crudel 019 027 038 005 2 016 032 027 013 3 002 024 026 039 Processing cost 06B of crude Distillation capacity 110000 Bday 2nd stage processing of some 1Ps in a CC Catalytic cracker Only DHO DGO processed here N normal level H high level Capacity either 100000 B of DHOday or 50000 B of DGOday or any convex combination Yield at CC B B CN CHO P DGO at N 048 052 015 DHO at N 018 08 011 DHO at H 032 069 01 Processing cost 095B at N 15B at H Blending 1Ps into nal products DHO gt Heating oil Sells for 3975B Daily demand S 40 000B DGO gt Diesel fuel 39B No limit on demand P CHO 17 2 mix gt Heavy fuel oil 30B No limit DN CN 20 a wherea 2 17 mix gt Gasoline Quality T as a 4325B Demand S 40000 B Need determine complete daily production plan to max daily net pro t We encourage use of AMPL in our classes After obtaining opt primal amp dual sols students analyze to answer following type questions Which crude supply to increase amp how much can we pay per B amp still break even What is the bene t from expanding capacity at fractionater amp the CC What is the bene t of increasing demand for gasoline amp heating oil etc Many nice computer project problems with data can be found in following books 206 K G Murty Operations Research Deterministic Optimization Models Prentice Hall 1995 K McAloon amp C Tretkoff Optimization and OomputationalLoogic Wiley lnterscience 1996 Aim of Computer Projects in Our lP amp Combinatorial optimization Courses How to model the problem so that a good solution can be obtained within reasonable time using a combination of 1P amp heuristic approaches An example Project Problem Planning for the military training of US Army National Guard forces ANG using mobile combat simulators ANG forces reside in 368 towns amp cities spread all over the country each of these places is called an Armory ANG forces grouped into platoons each with 16 soldiers Each platoon is allotted 4 tanks Mobile trainer is a platoon level trainer it trains a whole platoon at a time using 4 tanks 21 mobile trainers to be elded Army identi ed 29 sites in whole country these sites to be called HQ sites which are suitable to serve as home bases for these mobile trainers At each home base 2 1 mobile trainers can be stationed ANG training takes place only during weekends Constraint Each unit must receive training at a training center S 100 miles from their armory Home bases for mobile trainers can serve as training centers Besides these a set of Secondary training sites STS to be selected from among the 368 armories HQ sites not selected as home bases to serve as training centers The mobile trainers will travel from their home bases to cover the STS and give training to units visiting them for training Each armory has 3 platoons to be trained all of them should be allocated to the same training center A mobile trainer can train upto 3 platoons during a weekend 207


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