MAT001/13CARDIFF UNIVERSITYEXAMINATION PAPERAcademic Year:2012-2013Examination Period:AutumnExamination Paper Number:MAT001Examination Paper Title:Operational Research MethodsDuration:THREE hoursDo not turn this page over until instructed to do so by the Senior InvigilatorStructure of Examination Paper:There are EIGHT pages.This examination paper is divided into TWO sections.There are NINE questions in total.There are no appendices.The mark obtainable for a question or part of a question is shown in square brackets alongside the question.Students to be provided with:The following items of stationery are to be provided at the start of the examination:TWO answer books.Instructions to Students:Answer the questions in Section A, and TWO questions from Section B.Only the best two answers from Section B will be taken into account.Answer each section in a separate answer book.The use of a translation dictionary between English or Welsh and another language, provided that it bears an appropriate departmental stamp, is permitted in this examination.Section A(Answer ALL questions)1 For the following project:ActivityPre-RequisiteDurationA-1BA3CA6DA2EA4FD,E3GC,F1HB,G2(i) Draw the network and calculate the project duration and identify the critical path. 5(ii) Identify the activities that can be lengthened in duration without lengthening the total project duration. How much can the duration of these activities be increased by? 2(iii) If we wanted to reduce the total project duration, how would we proceed? 12 Consider the following game:s1s2r1(4,2)(1,1)r2(1,1)(2,4)(i) Identify the pure Nash equilibria. 3 (ii) Obtain the mixed Nash equilibria (clearly stating the results you use) 5 3 A company must decide on 3 possible options (options 1,2 and 3) for investment. There are 2 possible outcomes that might occur: outcome A and outcome B.OptionIf A occursIf B occurs145222106533027The returns on investment are given by the following table:It is known that PA=0.2. Use the following 5 approaches to inform the decision: MaxMax 1 MaxMin 1 MinMax Regret 2 Maximum likelihood 2 Maximum expected 24.Consider the following linear program:minimise 2x1 + 2x2 + 4x3s.t. 5x1 + 7x2 + 9x3 55X1 + x2 + x3 10x1, x2, x3 0The problem is solved using the simplex dual method and the optimal tableau is:x1x2x3s1s25/2012-1/2B15/210-1-7/2-2000A02(i)Show that A=2 and B = 5/2 2(ii)Find the other optimal solution. 2(iii) Perform sensitivity analysis on:(a) the co-efficient of x1 in the objective(b) the right hand side value of the first constraint3(iv)How would you change the formulation if x2 had to be either 0 or between 5 and 10 (note you are not required to solve this problem)? 25.Consider the following road network joining towns A, B, C, D, E and F:RoadLengthA to B 7A to C 4A to F 9C to D 6C to E 8C to F 14D to E 8All roads can be traversed in either direction. Where should a hospital be located if it is to be located on the road A to C and the objective is to minimise the maximum distance anyone from any of the towns has to travel to the hospital?9 6.(i) Give the integer programming formulation of the travelling salesman problem and show that it is not unimodular. 3(ii) Explain how you would solve the travelling salesman problem using tabu search.2(iii) Explain how you would solve the travelling salesman problem using ant colony optimisation.3Section B(Answer TWO questions)7(i) Consider an inventory model with the following assumptions: Goods are produced at a fixed rate of r per unit time. Demand is constant at a rate of D per time period (Dr) Setup cost for a production run is K. There is no lead time associated with an order. Unit production cost is p. Holding cost for one unit of inventory is h per time period. No shortage of demand is allowed.Obtain the size (q) of regularly spaced orders of a fixed size that minimise the cost per period. (Note that you are expected to include all necessary arguments) 16 (ii) With the optimal order size q* what is the optimal cost? 5(iii) A boat company has demand for 150 boats per year. The company can produce boats at a rate of 1 per day. It costs 300 to set up a production run