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Semester I Examinations 2007 / 2008
Exam Code(s) 4CS, 3BI1,3BC1, 4BC2, 4BC3, 4BF1, 1EM1, 1OA Exam(s) B.Comm Degree B.Sc. Degree BIS Degree Industrial Engineering Erasmus & Visiting
Module Code(s) IE309, IE Module(s) Operations Research Operations Research I
Paper No. I
External Examiner(s) Prof. Jiju Antony Internal Examiner(s) *Ms. M. Dempsey Dr. D. O’Sullivan
Instructions: Answer any 3 questions. Show all your work clearly and explain your work. All questions will be marked equally.
Duration 2 hrs No. of Pages Cover + 5 Department(s) Industrial Engineering
Course Co-ordinator(s) Mary Dempsey
Requirements : Graph Paper Normal
Q1 IKHEHA Furniture Company makes three kinds of office furniture: chairs, desks and tables. Each product requires some labour in the parts fabrication department, the assembly department, and the shipping department. The furniture is sold through a regional distributor, which has estimated the maximum potential sales for each product in the coming quarter. Finally the accounting department has provided some data showing the profit contributions on each product. The decision problem is to determine the product mix – that is to maximise IKHAHA’s profit for the quarter by choosing production quantities for the chairs, desks and tables.
The formulation of this problem should satisfy the five requirements for standard LP
- There are limited resources
- There is an explicit objective function
- The equations are linear
- The resources are homogenous (everything is in one unit of measure, labour hours)
- The decision variables are divisible and nonnegative (we can make a fractional part of a chair, desk of table)
i) Illustrate this scenario ii) Formulate the problem in Mathematical Terms as a Linear Programming Problem iii) Plot the constraint equations and the objective function iv) Determine the area of feasibility v) Find the optimum point using the simplex method vi) If it were deemed undesirable to make a fractional part of a chair, desk or table what technique would you use
vii) Explain “the constraint is binding”
Q3 Galway Paper Company manufactures paper at three factories one in Oranmore, one in Barna the other in Moycullen. The paper is then shipped by road to two depots one in Galway city and the other in Athlone. The paper is broken from bulk into customised quantities, and then shipped to five warehouses in response to replenishment orders. Each of the factories has a known monthly production capacity, and the warehouses have placed their demands for next month. The following tables summarise the data that have been collected for this planning problem. Knowing the costs (in Euro) of transporting goods from factories to Distribution Centres and from Distribution Centres to warehouses, Galway Paper Company is interested in scheduling its material flow at the minimum possible cost.
Distribution Centre Factory
Galway City Athlone Capacity
Oranmore 1.28 1.36 2500 Moycullen 1.33 1.38 2500 Barna 1.68 1.55 2500
To
From
Warehouse 1
Warehouse 2
Warehouse 3
Warehouse 4
Warehouse 5
Galway City
Athlone 0.57 0.30 0.40 0.38 0. 1200 1300 1400 1500 1600
i) Translate this transhipment problem into a transportation problem. ii) Formulate the transportation problem as a linear programming problem. iii) Explain using a simple example how one can restrict a route in transportation modelling. iv) What is degeneracy in transportation modelling?
Q4 a) Coach Duffy is the nominated coach of the Irish relay swim team. His team is in training to compete in the Olympics in Beijing in 2008. The coach needs to organise the relay team. The relay team requires each of the four swimmers to swim a different stroke: butterfly, breaststroke, backstroke and freestyle. During training Coach Duffy has studied his top four swimmers in each of the four strokes, and he has tracked their times (in seconds) as shown in the following table. With this information, Coach Duffy is ready to assign swimmers to strokes in the relay race but there are many possibilities.
Stroke
Swimmers
Butterfly Breaststroke Backstroke Freestyle
Jack 38 75 44 27 Paddy 34 76 43 25 Mary 41 71 41 26 Ann 33 80 45 30
i) Using the Hungarian Solution method, evaluate an optimal solution for this assignment problem ii) Is this the only optimal solution? iii) Explain the use of a Dummy Row in this problem
Q4 b) Given the following network, with distances shown in miles, find the shortest route from node 1 to each of the other nodes.
