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OPI M 101 Management Science F A Q , Semester 2, 2010/2011 (Check the date to ensure that the F A Q is updated. I will be updating this F A Q as I get more questions.) Updated: Date:

01- 01- 2011 (start) 08-02-2011 17-02-2011

General

Q1: I would like to ask if the finals formula sheet would be identical to that of the trial paper, i.e. to say that ONLY the following formulas will be provided and NOTHING else:

-ALL the queuing formulas under different models -Bayes' Formula

A1: Yes, Only the formulas in the trial paper will be given. Q2: For the challenging problems in homework, do we need to submit it to you also? A2: No. I will go through the challenging questions in class next week. You do not need to submit them. It will be good if you try the challenging questions. Q3: Do we have to write the assignment by hand, or is a typed copy acceptable? A3: Both are fine with me. Q4: What are the necessary chapters of the textbook to read for the course? A4: See Lecture in week 1.

Q5: I've noticed that there is a solver for Mac on Vista. May I know if Windows users like myself need to get any equivalents?

A5: The answers are equivalent. The note in smuvista is for MAC users. Windows users will have no problems. Q6: May I know if it is necessary to complete the warm-up questions in the assignments? What about the supplementary problems on Vista? Also, will the supplementary problems be required for submission at any time? A6: Only the graded questions from the assignment are graded. The rest are for your practice.

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Q7: May I know where to submit our Assignment 1 on the 30th of August? Also, is there a specific time cut-off for that day? I.e. 12pm?

A7: There is no deadline in time. I will collect the assignments the following morning. Q8: Instead of photocopying all the questions from the 10th edition, do you instead have a list of questions that 9th edition users do not have which are available in the 10th and at the same time, also count as required questions? That would be more efficient for me to photocopy in my opinion.

A8: The questions are random. I do not have the list. You will have to compare with a 10th edition. I believe the numbering of the questions are random, hence, unless you type it out yourself, you will be making a copy of almost every page of the question.

Assignment 1 Q1: ( Question 4) And since it's maximizing of profits, can i approach the question such that the function Z is minimizing the cost, since profits=revenue-cost? (or isit strictly the function Z must be maximizing profits?) A1: For this question, the question is asking strictly max profit and revenue. It is stated in this question because there are more than one way to consider the objective. Q2: With regards to Homework assignment 1, practice question part B, Could you please explain why under the constraints section, you do not combine the calorie restraint into 1 line? E.g. 1500 < 520x1 + 500x2 + 860x3 + 600x4 _ 50x5 +460 x6 + 240x7 + 180x6 <2000 A2: Either way is fine. But if you planned to input into excel, you need to express it into two constraints. We will talk about this when we cover excel.

Q3: A department store has a schedule for the next season, which requires the number of employees in each day of the week be at least 22, 17, 13, 14, 15, 18, and 24 (from Sunday to Saturday). The employees are grouped into 7 shifts each with 5 consecutive working days. Shift 1 takes Sunday and Monday off, shift 2 Monday and Tuesday off, and so on. The wage per employee per day (including the off days) is $40. Employees working on Fridays earn an extra $5 for those days. Employees working on Saturdays and Sundays earn an extra $10 for those days, and are compensated additional $10 for transportation charges. Management wants to meet the schedule requirement at the minimal wage cost. Formulate this as a linear programming problem. (12 marks ) Does this mean that employees who get to earn the additional pay for Saturdays and Sundays have to work on EITHER Saturday OR Sunday only to be eligible, or do they have to work on BOTH days before they get paid the additional $10? A3: If you work on Sat or Sun, you get paid $10 extra. If you work on both days, you earn extra $20.

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LP/ G raphical Solution Q1: My question is if Strict Inequalities can be used in the formulation of a LP model due to its continuous nature. For example: 2X + 1Y < 3. If we are able to use strict inequalities, how do we identify the number infinitesmially smaller than 3? A1: Even though the phrasing of the question suggest a strict inequality, we still use a <= inequality. The reason is that with a strict inequality, we do not have extreme point. This hinders us from solving the problem. Q2: In today's lecture, I asked you whether the slack variable tends to relate to profit maxmisation solutions and the surplus variable lends itself to cost reduction solutions. I had a bit of difficulty differentiating the slack and surplus variable. Is the slack variable so called because the firm would be 'slacking' if it did not fully utilise all its resources? And the surplus so called because the firm does not want to convert any excess raw materials into surplus products? A2: Slack is the general term used (in mathematics). From a business perspective, in a < constraint, it represent unused resource, in a > constraint, it represent surplus/excess. In the excess case, if the firm has to produce at least 50 products to meet certain standard, it should do the bare minimum of 50 products. However, if the optimal solution requires 60 products, it is still implementable but it has exceeded the pre-requite by 10 products. Q3: Is it crucial to define our decision variables correctly. For instance C can represent # of

clocks sold. While C can also represent # of clocks produced. Is there a definite answer for this definition or is it flexible?

A3: There are more than one way to define the decision variables. Make sure that your definition make sense. Q4: When plotting the graph

· I am still unsure of how accurate does the graph needs to be. · Must we include all the equations of constraint/objective function or is it

enough to merely draw the graph · Must we find co-ordinates for all extreme points and the Z(optimum solution)

for each extreme point? · When finding optimum solution base on the gradient approach, we must

clearly indicate the shift of the objective function? · Finally when there is a change in obj function, must we indicate on the graph?

A reasonable sketch of the graph is sufficient. You will still need to solve the simultaneous equations for the extreme points. It will be good to indicate the shift of the objective function using an arrow.

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Q5: Lastly, when we vary the RHS of the eqn, we basically have to look at 2 things namely the

shadow price and the range of the allowable increase/decrease to determine if it is feasible to vary the RHS value right?

A5: Yes. you are right.

Q6: For the assignments, do the answers we give have to be whole numbers? (for example 2 footballs, as opposed to 2.3 footballs)

A6: Leave your solutions in decimal. We are dealing with LP not integer programming. This will be clear in the next lecture. Q7: the concept of a negative shadow price. If a decrease in value will result in a decrease in negative shadow price, does that mean that the resulting effect on the profit/Z value will be Positive increase? A7: Yes, you are right. Q8: For the midterm on Monday, you mentioned in class that we would be able to use our Graphic Calculators. If one of the questions requires us to solve the model using the graphical method, besides drawing the graph, do I still have to write out the workings? Like writing out the simultaneous equations to find out the coordinates of the points of intersection of the graphs / optimal solutions? Or can I just draw the graph, then write "using the GC, the coordinates of the points of intersections are (x,y)"? A8: You can write "using the GC, the coordinates of the points of intersections are (x,y)" Q9: one of my shadow prices is negative. I would like to ask where i went wrong. A9: Yes, it is likely that the shadow price be negative. However, the interpretation is the same. In this case, every increase in RHS will cause a decrease in Obj value. Q10: (x1+x2)/(x1+x2+x3+x4) >= 0.60 as per stated in qtn 5. Excel keeps prompting that it is non-linear and hence cannot be assumed linear. I've checked my values and other formulas several times and still cannot resolve the error. By removing the constraint from solver it can be computed, but the resulting ratio is less than 0.6 A10: (x1+x2)/(x1+x2+x3+x4) >= 0.60 is nonlinear. Recall in lecture that I mention about making all inequalities to linear. For this case, move the denominator to the RHS. You will have a linear constraint. (See Lecture 3 (I think) on the example to make hotdog mixture.) Q11: When i analyzed e sensitivity report for question 5(b), I encountered a 1E+30 answer for the allowable decrease of x3 and allowable increase for x4. May I ask what that means, and also what answer I should write in my asssignment. A11: It means x3 is unbounded below and x4 is unbounded above. So x3 is less than some value and x4 is greater than some value.

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Q12: Do we need to find the sensitivity range? Or can we just derive the answer from the graph? A12: You do not need the whole sensitivity range. But to solve for it using the gradient, you will be solving for either the lower bound or upper bound. You cannot read it from the graph because the graph is not drawn to scale. Q13: while doing the supplement exercise, i realize that in your ans for sensitivity analysis, u stated something called the dual value. Is it the same as shadow price? Can I say like for example, i will choose to increase labour because the shadow price is $2, hence for every increase in one unit of labour, profit will rise by $2 which is higher than othe shadow price. Is it correct to say it in this way? Is it the same as u saying the dual value is $2 hence higher than the rest? A13: Yes, it is the same as shadow price. For this course, we will stick to "shadow price". Q14: We always have to let one variable be x1 and the other be x2 when doing LP, however does it make any difference in the answer when I swap the variables for x1 and x2? For instance, if i let X1 be apples and X2 be pears, then would the graphical solution for X2 wrt x1 change if I let X1 be pears and x2 be apples? I'm asking this as I get a different solution for qn 9 of chapter 3 in the 9th edition of our textbook, where I let x1 be the no. of yards of corduroy and x2 be the no of yards of denim instead of vice versa as shown on the solutions. A14: There will be no difference in the final solution if you swap the variables, as long as you define them clearly. The only difference if you were to solve it graphically is to axis. As mentioned in class, if your y-axis is x1 ( i normally use x2), then every line on the graph has the form X1 = m X2 + c. If you stick to this understanding, the solution will be the same. Q15: i have a doubt regarding question 5bi, the question ask if what if the rack space change from 18000 to 18500, this part i understand. But i dun really get the second part, is the question asking what if there is 20000 increase in rack space from 18000 or is it asking if what if the rack space change from 18000 to 20000? A15: The question is asking rack space change to 20,000. Q16: The second question i have in mind is that what if the scenerio pose to ask is greater than the allowable increase or decrease? for example what if 5bi u were to to ask like what will happen if the reck space increase from 18000 to 19000, the change here is 1000 but the allowable increase is merely 560. so how do we solve this then? A16: Since it is outside the range, you have to resolve the problem by excel (which is easy). In exam, you can state that the sensitivity report is not sufficient to answer the question. This will be review again when I talk about the mid term review. Q17: While doing the assignment, I encountered some difficulties. How do we find the objective function line? On the class notes,slide 19 of Lecture 2, you used 800 as a value for Z. How did you obtain that value to get the objective function line?

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A17: When plotting the obj function line, as mentioned in class, the Z value is missing (part of the y-intercept). Recall that in plotting a line y=Mx + C, we need M (which we have) and C( the y-intercept which depends on Z). Once we have a line, we can move parallel to it to get all the points. Q18: so I can just plot any value as the Z value and then use it to solve the equation? A18.2: Yes, Plot as value with any Z value. And you can move parallel to that line to maximize or minimize it. Q19: For your first set of supplementary questions, for chap 2, qns 3,9,13 & 37, the questions

were solved using graphical approach but the answers you provided analyzed the extreme points and derived the z function value from the coordinates. After that, inspection of the function values at the extreme points was done and the optimal solution was derived from there.

Does that mean that we are advised not to draw the objective function in our presentation of answers? (Because that was what I did, and I admit it would look more messy) A19: Yes, you need to draw one objective function to show me. You do not need to draw several with different z. The solutions I provide are suggested solutions. As discussed in class, using extreme points is not exactly graphical solution.

Q20: In the HW Assignment, I have a doubt in one of the questions relating to employee shifts.

For employees working on saturdays and sundays, the problem says they earn an extra $5 for those days. Does that imply that they earn $5 for each of those days(meaning they earn $10 for working on both days), or just $5 for working on either one or both the days?

A20: if you work on sunday, you earn extra $5. If you work on both sat and sun, you earn $5 x 2.

Q21: For assignment 1 Q5, I got a negative shadow price for one of the constraint on the number of magazine. May I know how do I interpret that? Is it the case where increase will lead to loss of profit and decrease will lead to gain.

A21: It is interpret the same. For every increase in RHS, objective function will change by

SHADOW PRICE (can be +ve or -ve) x INCREASE in RHS

Q22: And lastly , for Q6, the excel results that I obtained include decimals for the number of cask of wines for Red and Blush. Do I have to round down by imposing a integer constraint?

A22: No. You do not round down or up the values. We will address this in the lecture on Integer programming.

Q23: for qn 2b, i realize that the slack variables in the solution provided is minus s1 (greater or equals to). Then why is it in some case we use plus s1 (less than or quals to). What is the intuition or reasoning behind this?

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In both cases, does slack means the same thing? that is is it equals to unused or excess resources for both case? A23: This is a ">=" constraint. In most of our examples, our constraints are "<=" constraints.

Q24: The Q1 in assignment 1 is the same as Q1b in Mid term of Sem 2. However, the objective functions for both are different. Why is this so? Assignment 1 solution - min 285x1 + 290x2 + 290x3 + 290x4 + 290x5 + 285x6 + 280x7 Mid term paper solution - min 205x1 + 200x2 + 205x3 + 210x4 + 210x5 + 210x6 + 210x7 Are the 2 Questions different? Or are both answers correct?

A24: The first equation represent cost including off-days. The second represent cost excluding off-days. The first equation is right. I was lenient in marking last term for this question, hence I accepted both formulations.

Q25: I have a question about the midterm. For the graphical method, is the optimal point always the intersection between the constraints?

A25: Yes, you can always find the optimal point at the intersection. That may not be the only point, but it exist. See slide 25 on lecture for LP. Q26: When defining variables, or when writing constraints, is it necessary to state the units? Would it be ok to omit them? E.g. 5X1 + 4 X2 >= 50 (Cents)

A26: You do not need to state the units.

Q27: I realized there are some questions in the textbook that require us to know the conversion of pounds to ounces, square feet, foot etc. May I ask if we need to know these conversions for mid term or exam?

A27: I will not be testing on that.

Q28: Just to be sure, when the qns tells you that demand for a certain pdt X1 is roughly 1000, do we always put the dd constraint as >= or <=? Should we cap it at 1000, or go on to produce more than 1000? A28: The question will state explicitly. eg. demand is at most 1000 or demand is less than 1000. Q28.1: Or do we look at the context of the question (e.g. if they restock every week, then it makes more sense to put constraint as <=1000) A28.1: Yes, it depends on the context.

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Q29: For  questions  on  graphical  solutions,  is  it  necessary  to  draw  multiple  objective  function  lines  and  show  the  shift?  or  just  drawing  one  will  do?    A29:  Drawing  one  will  be  sufficient.          Q30:    Some  questions  in  the  textbook  and  assignments  use  the  terms  'profit'  and  'revenue'  interchangeably.  Is  this  allowed?  because  based  on  this  definition  I  may  obtain  a  different  objective  function  and  a  different  optimal  solution,  especially  if  budget/cost  is  one  of  the  constraints.    A30:  You  should    be  careful  with  the  definition  in  the  problem.    Revenue  typically  refers  to  sales  in  our  problems.    On  the  other  hand,  profit  should  consider  the  cost  of  the  raw  materials.      

Integer Programming Q1: I was revising the lecture notes when I came across this part about taking an arbitrary huge number in the form of: X1 <= M1y1, where M1 is the arbitrarily huge value. My understanding is that it has something to do with the fact that if y1=0, then X1=0 as well but why cannot it be simply X1<=Y1. Even if y1= 1, X1 will be <= Y1 as well. A1: your X1 need not be only less than 1. X1 is typically production amount. So if y1 is 1, you produce X1. without M1, you restrict production of X1 to be only 1. Q2: if we get decimal places for our answers in Excel, do we need to do anything (round off)? A2: No, you do not round up the value. The reason will be revealed in the integer prog lecture. Q3: For the integer programming section, when we do a formulation for the constraint for integer

front of y1? Say for the midterm paper, we had the answer of X1 + X2 <= 500,000Y1

than the constraint (X1+X2 <= 100,000). A3: Any large value of M will be enough. Consider the following constraints: X1 + X2 <= 500,000Y1 X1+X2 <= 100,000 If Y1 =1, then first constraint becomes redundant. Hence, X1+X2 <= 100,000. If Y1=0, then first constraint implies X1+X2=0, since X1,X2>=0. Then second constraint becomes redundant.

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Q4: SP: 10th edition Chapter 5, 13, can I take 18000y1 as an expense budget constraint? Why x1-10y1<=0? A4: No. You do not need to put 18000 as part of budget. To be more precise, we should in fact include it in budget. X1<=10 y1 is actually the x<=My constraint.

cost when we are told that we should maximize sales?

cost. However, the fixed cost is stated as an annual cost of 18,000. Should we include the fixed cost of 18,000 inside the constraint?

A5: This is a fix cost problem. Yes, the question state max sales, but we should consider the fixed cost involved. For questions which I set, I will state this more explicitly. The fix cost is captured in the next constraint, x1 <= 10 y1. This is the form of the M-method.

Q6: function (8000X1 + 6000X2) take into account the expenses so that is it (8000-1000)X1+(6000-2000)X2? If feel this is so because the 8000 and 6000 are the returns for the condo and the land, respectively,

X2 >= 0? That was the solution in the lecture notes question as land could take on fractional values.

A6: The first question is solving for the returns. So we do not factor in the cost of the condo. See line 2 of question. Yes, You are right. x1>=0 and integer, while x2 do not need to be integer. This is a typo.

Q7: How will we know when to use binary or integer in the midterm? I was looking at the Integer programming slides when the model about the construction of a facility (Swimming pool, tennis court, etc.) came up, anpossible to build 4 athletic fields? That way the solution can be an integer and not only binary.

A7: The slide in the lecture contains the summary of all the main points of the question. It will be stated in the question if it is possible to build 4 athletic fields. If it is stated, then you are right, we need integer (not binary) constraints.

Q8: for question 2 in the assignment, the vehicles are considered in terms of integers right? And, when we solve the problem, are we required to submit the answer report from excel as well?

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A8: The vehicles have to be non-integers. Otherwise, it will take a while for you to solve the problem. It is optional to submit the answer report for this question.

As mentioned in class, integer problems are difficult to solve. Hence, we want to avoid it as much as possible. In this problem, the vehicles are of reasonable large values, hence continuous variable is a good assumption.

Q9: Midterm- From Q4a, Should there also be a constraint for something that suggests either design "1&2" or "3"?

A9: y1+y2=1 is enough. Q10: Midterm - From Q4a, Should x1, x2 => 0 (and also integer?) A10: No. we do not need the integer constraint. We try to avoid integer constraint as much as possible because it is difficult to solve. Q11: In general, how do we know if the variables are supposed to be integers or continuous? As

in problem 2 I had considered vehicles to be indivisible, hence integers, however during the exam, how will we know when to use integers and when not to?

A11: By default, it will be continuous. Unless the question ask you to model it into a mix integer program.

Q13: Its regarding writing constraints for maximizing & minimizing LP problems. Is it the case

greater

Im quite confused especially regarding min. problems! Cos for example in lecture notes 4,slide 37-thought the constraint should be: x1a+x2a+...+x6a <= 12. Cos the DVs cannot exceed the available capacity right? But the answer in slide 38 is x1a+x2a+...+x6a = 12, and I noted down that you said in class that you can also write it as >= and that it cannot minimizing LP?

A13: . There is no restriction. In a 2-D problem, imagine the feasible region to be a square which is away from the x-axis and y-axis. In this case, the constraints is a mixture of < and >, and regardless of whether you minimize or maximize, there will be a solution (either one of the four corners). In lecture 4, slide 37-38, do not be too carried away by the term available capacity in the problem. We want to move products from the farms to the plants. How much do we move? It depends on the capacity of the plant. Hence, in this particular problem, available capacity is analogous to the max we can store at the plant. Recall that in transportation problem, we try our best to move as many products as possible from the source to the destination. Use this understanding in this problem. Q14: The questions are:

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(1) In the objective, how come it is not stated as such: Min z = 18x1aY1 + 15x1BY1 + 12x1CY1 ....... I was thinking that we should add in "Y1" (2) Why is that the constraints for available capacity are "equal" and not "less that or equal to?" A14:

1) Modeling this way makes the model nonlinear. 2) This will be clear next lecture when we cover network.

Q15:  If  the  question  states  "use  an  easily  solvable"  prog,  r  we  supposed  to  include  the  interger  constraint  or  otherwise?    A15:    Integer  constraint  is  a  tough  constraint  to  handle.    So  we  should  not  consider  integer  until  told  otherwise  or  required  by  the  problem.  

Network Q1: In your questions of determining the shortest possible route, I understand that we have to write the permanent set, branch and time. Will u require us to draw out the entire possible routes of nodes? Or should we only draw the shortest route? Or can we May I also use this opportunity to ask if directions in Xij are important. i.e when I formulate the problems into LP format, do I treat X34 and X43 the same? A1: I will need the permanent set, branch and time. You need to conclude it by the final solution. To reduce the number of variables, you need small number to large number. i.e x34 is needed instead of x43. Q2: I have a question on maximum flow. How should we present the answer? Should we just have 1 diagram and do all the workings on 1 diagram or should we have to draw multiple diagrams for the workings for each path? A2: It depends on the size of your writing and diagram. Present it in a few networks if the working is too messy. Q3: Shortest Route Problems: Can we model a question if the direction of the arrows is not

given?

A3: Yes.

Q4: Shortest Route Problems: For the modeling of such problems, slide 23 of the lecture notes ed, given that

the there can be the case of 1-2 and 1-3 both selected at the same time.

A4: The LP model only solves for a fixed start pt and a fixed end pt. The solving by hand solves for the shortest route from the start pt to all the other pts.

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Q5: Maximum Flow Problems: Can we model a question if the direction of the arrows is not

given?

A5: Yes. You only need the numbers (left and right of each node which represent the direction of the arc).

Q6: For supplementary problem, chapter 6 qn 10, the shipments are prohibited from Miami to Chicago, and you provided in your solution a constraint of X23=0. If I use the method of giving a very high coffecient to x23 in the objective coefficient, for e.g. 100x23, do I still need to put the constraint x23=0? A6: Either method is fine. (set high coefficient or set constraint x23=0.) Q7: For supplementary problem chapter 7, qn 35, I noticed that you included this part on 'branch in which total capacity is used', do we need to include that in our presentation of the answer in the exam? [The assignment 4 solution for the maximum flow question did not include that component] If we need to include the part on 'branch in which total capacity is used' in the exam, does 'branch in which total capacity is used' means that there is 0 capacity left? Then, may I check why branch 3-5 is also included in the 'branch in which total capacity is used' when it still have capacity? A7: Ignore the column 'branch in which total capacity is used' . Q8: Is the assumption that items can only flow from a lower node number to a higher node

number always true? Do we need to state this assumption when solving the problem?

A8: Yes, it is always true for this course. You do not need to state this.

Q9: When drawing out flow in the network model, we use darker lines to show the flow of the items in the model. Is it necessary to show the darker lines in the exam?

A9: You do not need to show the dark lines. The permanent set and the branches will be sufficient.

Q10: When solving shortest route problems, is it necessary to draw out the models for each step, or will presenting the solution with only [Permanent Set Branches Distance] suffice? The textbook draws out each and every model for every step of the solution.

A10: [Permanent Set Branches Distance] is sufficient.

Q11: Is there a way to formulate an LP for minimal spanning tree? Are we required to know how to formulate it? It is not presented in the textbook.

A11: We did not cover LP formulation for min spanning tree. Hence, you only need to know how to solve it.

Q12: about transportation problems, would we be asked to solve it manually or it would be just about formulating the LP? A12: You only need to know the formulating of LP.

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Q13: For Transshipment Problems, why do we need to indicate whether it is a balance problem or not? A13: If it is not balanced, you need to put in the appropriate inequality. See transportation problem. Q14: For a question on maximal flow model, are we required to show a square box with a digit zero inside if the route is not used in our flow? Or do we just need to show a square box with the respective digits for routes that have flow? A14: Showing routes with flow is sufficient. Q15: For assignment 2 qn5, do we assume that the family can only travel from a node of lower number to a node of higher number? A15: for solving by hand, you need to follow the small to large node. For solving by LP, we have to assume small to large node. Q16: For Assignment LP, lets say in the case of operators assigned to tasks, if one operator must do 2 tasks or if the are more operators than tasks and they must all work, how will the constraints be affected? A16: Treat this problem as a transportation problem with supply (operators) more than demand (task). Q17: Why is the capacity constraint for the mock paper qn 2b X71<=31 and not 48? Is the capacity constraint always the larger value? To calculate the value, we just have to add up the total flow from node 1, and also add up the total flow into the last node and then take the larger value to be the capacity constraint? A17: It is either the sum of the total flow out of node 1 or the total flow into last node. In this case, it is either 31 or 43. Choose either one. We do not need a accurate upper bound. Q18: 1) For maximal flow problems, must we calculate/show the backward flow? Is it okay to leave that out and just show the forward flow? 2) Also for the LP for maximal flow, why do we need to put e.g. x61 instead of x16, when the flow is from node 1 to node 6 (as defined in our Decision Variable: Xij) 3) For transshipment problems, do we need to define the middle nodes twice since, i.e. Xij, the middle nodes act as location i first and then destination j. is it okay just to use it as i first and then j but not specify? A18:

1) No. You need to update the back flow. 2) I am fine with either. They both mean the same. 3) You can use the same xij, but i and j will have different range. Do state the range. Eg. i

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Q19: I would like to ask in the question has prohibited routes. We can assign a very large relative transportation cost. But how much is large enough? And if we don't use this way, we add in the constraint x=0, so the other constraints we keep the same? A19: The value of the large cost depends on the parameters. If the parameters of the problem is from 1-100, then 1000 is large enough. Yes, add the constraint x=0 is possible too. Q20: Can I ask for the trial paper right, the maximum flow problem, must we do it using D1, D2 etc. or can I just write everything on the first diagram? A21: You can do on 1 diagram. But you have to be neat so that I can read your solution.

Decision Analysis

Project Management

E ND