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Semester I Examinations 2008 / 2009
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. Where you may think that additional information is needed, please clearly state so and make appropriate assumptions and/or use reasonable estimates that will enable you to proceed. Duration 2 hrs No. of Pages Cover + 5 Department(s) Industrial Engineering
Course Co-ordinator(s) Mary Dempsey
Requirements : Graph Paper Normal
Q
Farm2Juice makes Organic Drinks. These drinks are labelled Booster, KickStart, Healthwise and Energiser and use a combination of fresh fruit blends and other fresh organic ingredients. The fresh fruits used are strawberries, oranges and apples. As increased numbers of customers demand high quality organic drinks the company employs a farm to juice order winner focus. Fresh fruit is perishable therefore Farm2Juice focuses on the speedy usage of fresh fruit and the maximisation of profits.
The blend requirements of each type of drink are as follows: each Booster requires 5oz of strawberries, 2oz of oranges and 3oz of apples, each Kickstart requires 1oz of strawberries, 3oz of oranges and 2oz of apples, each Healthwise requires 9oz of strawberries, 4oz of oranges and 5oz of apples and each Energiser requires 12oz of strawberries, 1oz of oranges and 10oz of apples. There are 2600ozs of strawberries, 1200ozs of oranges and 1600ozs of apples available.
The Drinks are sold through outlets, which have an estimated, the maximum potential sales for each product in the coming day. The accounting department has provided some data showing the profit contributions on each product. The profit associated with each container of drink is as follows: Booster €12.00, KickStart €9.00, Healthwise €25.00 and Energiser €30.
The decision problem is to determine the optimal product mix – that is to maximise Farm2Juice profit for the day by choosing production quantities for the Booster, KickStart, Healthwise and Energiser
The formulation of this problem should satisfy the five requirements for standard LP
- There are limited resources and there is an explicit objective function
- The equations are linear
- The resources are homogenous (everything is in one unit of measure)
- The decision variables are divisible and nonnegative (we can make a fractional part of a Booster, KickStart, Healthwise and Energiser i) Illustrate this scenario ii) Formulate the problem in Mathematical Terms as a Linear Programming Problem iii) Find the optimum point using the simplex method iv) If it were deemed undesirable to make a fractional part of a Booster, KickStart, Healthwise and Energiser what technique would you use? v) Explain “the constraint is binding”
Q3 A & E Toys have two factories, one in China and one India. The toys are shipped to three distribution centres in Europe one in France one in Poland and the other in Finland. The containers are then broken into orders and then shipped to three warehouses in response to replenishment orders at supplier sites throughout Europe. 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. A & E Toys is interested in scheduling its material flow at the minimum possible cost.
Source Transhipment Sink 1
Distribution Centre Factory
France Poland Finland Capacity (tons) China 16 10 12 300 India 15 24 17 300
To From
Warehouse 1
Warehouse 2 Warehouse 3
France 6 8 10 Poland 7 11 11 Finland 4 5 12 Demand (tons)
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) An accountancy firm based in Dublin wants to assign three recently hired NUI, Galway graduates, Burke, Collins, and Duffy to regional sales districts in Galway, Cork and Waterford. But the firm also has an opening in Mayo and would send one of the three there if it were more economical than a move to Galway, Cork or Waterford. It will cost €1000 to relocate Burke to Mayo, €800 to relocate Collins there, and €1500 to move Duffy. What is the optimal assignment of personnel to offices? (Hint: Use a Dummy Row.)
Office
Hiree
Galway Cork Waterford Mayo
Burke 800 1100 1200 1000 Collins 500 1600 1300 800 Duffy 500 1000 2300 1500
i) Using the Hungarian Solution method, evaluate an optimal solution for this assignment problem ii) Is this the only optimal solution?
Q4 b) Every day Farm2Juice must transport containers of juice from the factory to the warehouse. This involves going through several cities. The transportation manager would like to find the route with the shortest distance. The road network is shown below with the factory labelled 1 and the warehouse labelled
