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# Operations Modeling IOE 202

UM

GPA 3.76

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This 16 page Class Notes was uploaded by Loy King on Thursday October 29, 2015. The Class Notes belongs to IOE 202 at University of Michigan taught by Marina Epelman in Fall. Since its upload, it has received 11 views. For similar materials see /class/231589/ioe-202-university-of-michigan in Industrial Engineering at University of Michigan.

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Date Created: 10/29/15

IOE 202 lecture 13 outline gt Announcements gt Last time gt Making decisions and sequences of decisions under uncertainty IOE 202 Operations Modeling Fall 2008 Page 1 Last time gt Different types of queueing systems characterized by gt Characteristics of arrivals and customer behavior gt Service discipline gt Service characteristics gt Steady state analysis of queueing systems with exponentiay distributed interarrival and service times gt MIVIl system when p g lt 1 gt MIVIs system when p lt 1 gt MIVIss system IOE 202 Operations Modeling Fall 2008 Page 2 Decision analysis problem Deciding on summer plansl V Bill is an undergraduate student It is now early in the Fall semester but Bill is already considering his options for summer employment V In the end of August Bill interviewed for a summer internship at a major consulting company The interview went well and Bill thinks he has a good chance about 60 of getting an offer of an internship with a 14000 summer salary V Also Bill has the opportunity to return to the summer job he held last year The job will pay 12000 for the summer However the offer to return to this job will only remain open until the end of October while the internship offers are made in mid November Therefore Bill has to accept or decline the job before he knows if he has an offer of an internship V lf Bill gets offered an internship he needs to respond by the end of November 1Adapted from Bertsimas and Freund Data Models and Decisions IOE 202 Operations Modeling Fall 2008 Page 3 Deciding on summer plans continued gt If Bill were to turn down the job he could either accept the potential internship offer were it indeed to materialize or he could look for a different summer job via the University placement service gt Interviews in the placement office begin in December Bill collected some data on the salaries for summer jobs obtained through the placement office and he thinks that the percentages in the table are a good approximation of his own chances of getting similar offers Salary of students offered this salary 21600 5 16800 25 12000 40 6000 25 0 5 gt Bill39s main criterion for differentiating between different options is salary gt What should Bill39s decision be IOE 202 Operations Modeling Fall 2008 Page 4 What are the decisions Bill is facing IOE 202 Operations Modeling Fall 2008 Page 5 Analyzing and structuring Bill39s decision problem V Here Bill is facing a sequence of decisions under uncertainty The decisions Bill must make must be considered chronologically V V The decisions Bill will make in the future will depend on what decisions he made in the past and what additional information he gained since then ie how some of the uncertainties got resolved V A convenient way of representing Bill s decision making together with the uncertain events that affect it is via a decision tree IOE 202 Operations Modeling Fall 2008 Page 6 Structure of a decision tree gt Decisionsactions are reprsented by boxs a k a dec orks or dec s on nodes gt Each line emanating from a decision fork corresponds to a possible decisionaction that can be made at this point they are called branches gt Uncertain events are reprsented by circls or chance forks or chance nodes gt Branches emanating from chance nodes dscribe all possible outcomes of the associated uncertain event the probability of each outcome is written on the corrsponding branch gt Final branchs of the tree are assigned numerical valus or payoffs associated With this sequence of decisions and events based on the decision criterion in our case salary lOE 202 Ovevztmns Modelmg Fall 2qu Pzge 7 Bill s decision tree 312000 314000 Ammo Autprmnzrmhtp 21000 z ctmhzrmhtp u go replacement o ciz 1 e 500 Ryan Internship o fznzd 25 396 wan w Internship 5 5 12000 3925 memo ns o Inmvmhtp noro zwzd 321300 hm to go no placement o ciz s 31 eaoo 25 39 5 4312000 25 N quot5 35000 lOE 202 Ovevztmns Modelmg Fall 2qu Pzge a Using the tree to make a decision gt Note that the path followed to reach each payoff is determined both by the decisions made and by random events outside of the decisionmaker s control gt Bayes Decision rule Using the best available estimates of probabilities of the respective uncertain outcomes calculate the expected value of the payoff for each of the possible sequence of decisions Choose the decisions with the maximum expected payoff IOE 202 Operations Modeling Fall 2008 Page 9 Backward induction procedure for Bayes39 Decision rule gt Start the calculations on the right hand side and move left ie backwards in time gt Whenever you reach a chance node calculate the expected payoffs at that point in the process gt Whenever you reach a decision node choose the decision with the largest expected payoff at that point in the process IOE 202 Operations Modeling Fall 2008 Page 10 Bill39s optimal strategy gt In October contact my previous boss and tell him I will not be back next summer gt In November if offered an internship accept it gt If not offered an internship I will pursue the services offered by the job placement service in December IOE 202 Operations Modeling Fall 2008 Page 11 Another decision problem Scheduling an outdoor event gt You are in charge of organizing and outdoor event as part of the Ann Arbor summer festival scheduled to take place on June 15th The earnings from the event will depend heavily on the weather if it rains on 0615 the show will loose 20000 otherwise ie if it is sunny the show will earn 15000 Historically the likelihood of it raining on any given day in mid June is 27 Today is May 31 You have the option of canceling the event by the end of today but if you do so you will then loose your 1000 deposit on the facilities V What should you do Construct your decision tree and base your answer on Bayes decision rule IOE 202 Operations Modeling Fall 2008 Page 12 Decision tree for scheduling an outdoor event IOE 202 Operations Modeling Fall 2008 Page 13 Scheduling an outdoor event continued gt Suppose that you can also cancel the show on June 14th but if you do so you must pay a fee of 10000 The advantage of waiting until 0614 is that you can listed to the next day weather forecast on the local news station before making you final decision V According to station records the station s next day forecast in mid June is sunny 90 of the time V When the weather forecast was sunny the next day turned out to actually be sunny 80 of the time when the weather forecast was rain it actually rained 90 of the time Extend the above decision tree to obtain your new strategy IOE 202 Operations Modeling Fall 2008 Page 14 IOE 202 lecture 5 outline gt Announcements gt Last time gt Restricting the variables to be integer gt Other modeling possibilities with integer variables IOE 202 Operations Modeling Fall 2008 Page 1 Last time Linear programming models gt Blending model gt Shipping model gt Postal employee scheduling IOE 202 Operations Modeling Fall 2008 Page 2 Postal employee scheduling A post office requires different numbers of employees on different days of the week The number of employees required is as follows Wed 15 Sun 11 Sat 16 Thu 19 Fri 14 Tue 13 Mon 17 Day Min required Union rules state that each employee must work 5 consecutive days and then receive 2 days off For example an employee might work Wednesday through Sunday and be off Monday and Tuesday The post office wants to minimize the number of employees it needs to hire while meeting the daily requirements Decisions to make The employee schedule in this problem is completely determined by how many employees start their shifts on each of the seven days of the week IOE 202 Operations Modeling Fall 2008 Page 3 Modeling Postal Employee schedule Schedules Variables X1 number of employees starting on Mondays X2 number of employees starting on Tuesdays X7 number of employees starting on Sundays IOE 202 Operations Modeling Fall 2008 Page 4 LP model for Postal Employee schedule m i n x1 X2 X3 S t X1 X1 X2 X1 X2 X3 X1 X2 X3 X1 X2 X3 X2 X3 X3 X1 7 X2 7 X3 7 X4 X4 X4 X4 X4 X4 X47 Let s solve with Excel Solver IOE 202 Operations Modeling Fall 2008 X5 X5 X5 X5 X5 X5 X6 X6 X6 X6 X6 X6 7 Page 5 X7 X7 X7 X7 X7 X7 X7 Number hired Z 17 Monday 2 13 Tuesday 2 15 Wednesday 2 19 Thursday 2 14 Friday 2 16 Saturday 2 11 Sunday 2 o Integer values needed The optimal solution of the Post Office staffing problem given by Excel 1 1 X1 2 1g X2 2 5g with the objective value of 22 Since hiring part time employees or fractional people is not allowed we need to ensure that only integer values of the decision va ria bles are considered 1 X3OX47 7X5OX6 1 3 5 37X7 7 To do so in Excel In the Solver dialog box add a group of constraints on the decision variables to be integer New optimal solution X1267X267X320X4277X5207X647X7O7 with the objective value of 23 Could we have predicted the direction of change in this value IOE 202 Operations Modeling Fall 2008 Page 6 St Postal employee scheduling modified X1 X2 X3 X4 X5 X6 X7 Total number hired X1 X4 X5 X6 X7 Z 17 Monday X1 X2 X5 X6 X7 Z 13 Tuesday X1 X2 X3 X6 X7 Z 15 Wednesday X1 X2 X3 X4 X7 Z 19 Thursday X1 X2 X3 X4 X5 Z 14 Friday X2 X3 X4 X5 X6 Z 16 Saturday X3 X4 X5 X6 X7 Z 11 Sunday X1 X2 X3 X4 X5 X6 X7 2 O integer Note that the above is still a linear model but with integer varia bles problems of this type are referred to as Linear Integer Programs or IPs Solver tip Under Options set Tolerance to O to get the optimal solution IOE 202 Operations Modeling Fall 2008 Page 7 V V V V IOE 202 Operations Modeling Fall 2008 Use of integer and binary variables Integer variables are used in models in which values of some or all variables represent non divisible quantities number of employees hired number of airplanes manufactured etc Binary variables ie variables that are allowed to take on only values of O and 1 are useful when we need to decide whether or not to undertake an activity Contrast this with divisible or general integer variables which model the quantity or level of the activity Through various tricks of the trade many complex situations involving both yes no and quantitative decisions can be represented and solved as Integer Programming models Page 8 Selecting courses to satisfy requirements To get an IOE minor a student must take at least two Math courses at least three IOE courses and at least two EECS courses Course Math214 OE310 EECSZBO OE265 OE373 OE366 EEC8283 Math IOE EECS gt Math214 is a prerequisite for OE310 gt OE265 is a prerequisite for OE366 gt Credit is not given for both EECS28O and EECS283 What is the least number of courses the student can take to satisfy the major requirements Note Here we need to make a yes no decision about taking each course IOE 202 Operations Modeling Fall 2008 Page 9 Formulation of a mathematical model of course selection Decision variables represent decisions by variables Objective function express the performance criterion in terms of the decision variables should it be minimized of maximized IOE 202 Operations Modeling Fall 2008 Page 10 Formulation of a model of course selection cont Constraints express all explicit and implicit constraints and restrictions on the values of the decision variables Optimal solution IOE 202 Operations Modeling Fall 2008 Page 11 Combining binary and divisible variables To clean up a polluted river the state is going to build some pollution control stations Three sites are under consideration Two different pollutants need to be controlled and the state legislature requires that at least 80000 tons of pollutant 1 and at least 50000 tons of pollutant 2 be removed The relevant data is indicated in the table below the last two columns indicate how much of each pollutant each station removes by processing 1 ton of water Cost of building Cost of treating Pollutant 1 Pollutant 2 1 ton of water removed removed Site 1 120000 20 030 030 Site 2 60000 30 050 020 Site 3 40000 40 010 050 What is the cheapest way of satisfying the legislation requirements IOE 202 Operations Modeling Fall 2008 Page 12 Operational decisions in the pollution problem gt What decisions do you need to make gt What performance measure are you using to compare different decisions gt What constraints restrictions must your decisions satisfy gt What assumptions are being made IOE 202 Operations Modeling Fall 2008 Page 13 Observations about the pollution problem gt We are not required to build all three stations We get to decide which ones to build These decisions are represented by binary variables gt We get to decide how much water to process at each station These decisions are represented by divisible variables gt We cannot process water at non existent stations Need to link the values of the divisible and binary variables via constraints IOE 202 Operations Modeling Fall 2008 Page 14 Formulation of a mathematical model for the pollution problem Decision variables represent decisions by variables Objective function express the performance criterion in terms of the decision variables should it be minimized of maximized IOE 202 Operations Modeling Fall 2008 Page 15 Formulation of a model for the pollution problem cont Constraints express a explicit and implicit constraints and restrictions on the values of the decision variables Optimal solution IOE 202 Operations Modeling Fall 2008 Page 16 An investment model Presently you have 1000 to invest Cash flows associated with 5 available investments are shown in the table you can put no more than 500 in any investment In addition to these investments you can invest as much money as you want into 12 month CDs which pay 6 interest How should you invest to maximize your cash at hand at the end of year 3 Investment Now Yearl Year2 Year3 A 100 140 B 100 115 C 100 128 D 100 115 E 100 132 Added twist For each investment you put money in you need to pay a brokerage fee of 50 at the time of investment Note each year you can only invest cash available on hand IOE 202 Operations Modeling Fall 2008 Page 17 Formulation of a mathematical model for the investment problem Decision variables Objective function Maximize cash on hand at the end of Year 3 IOE 202 Operations Modeling Fall 2008 Page 18

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