A firm produces fire alarms and sells them to small firms in South England. In the last year the production took place in 7 plants reported in the following map. From these plants the components were shipped to several small firms. 

The plants are located at the distances reported in the following table:

  Distances (miles)  

Given imminent budget cuts, the company would like to know if it is more appropriate to run production using a smaller number of plants and afterwards transporting the finished goods to the remaining plants, which will become only distribution centres for the small firms.

The plants were managed with different operations practices. Therefore, the number of workers, the production capacity and the operational costs are different, as well as the demand forecasted. The data are reported in the following table:

 Number of WorkersCapacity  (1000 units)Demand (1000 units)Cost (£1000 )
Plant 1252812410
Plant 2502815500
Plant 3233013600
Plant 4482310400
Plant 5463317350
Plant 6242711380
Plant 7363214420

During a meeting with the local authorities it has emerged that only plants located at a distance bigger than 20 miles can be kept open simultaneously; this will permit to balance the distribution of the plants in the territory and consequently the fairness for the population of having equal opportunities. In addition, plants 1 3 5 and 7 are managed by an affiliate company that has already proposed that only 2 of these can be kept open. Following an agreement with the Union, the management has to keep an overall minimum of 120 workers employed. The management wants to keep the overall demand satisfied from all the plants. The company will base the decision for each plant on the costs recorded last year.

Task 1 

You have been called in to undertake an analysis and provide advice by addressing the following question: Which plants should the company keep open in order to minimise the cost?

  • Formulate an appropriate linear programming model to answer the question          

 [20 marks]

  • Obtain the optimal solution and produce appropriate output(s) for explaining how the solution has been obtained
    • marks]
  • Provide comments on the solution                                                            

  [5 marks]

In the plants kept opened the company will be producing at their maximum capacity. The additional components (the ones in excess of their own demand) will be transported to the other plants that are kept closed (these will be used only as distribution centres). To decide the routes, the company will consider the transportation cost from the open plants to the closed ones, proportional to the distances between plants. 

Task 2 

Compile the report for the management of the company explaining: What is the minimum cost of shipping the goods from the plants that the company will keep open, to the ones that the company will keep closed?

  • Formulate an appropriate linear programming model to answer the question           

[25 marks]

  • Obtain the optimal solution and produce appropriate outputs for explaining how the solution has been obtained

                                                                                                                                           [15 marks]

  • Comment on the solution and outline a set of relevant conclusion             
    • marks]

Task 3 

Finally, use the scientific literature to support your decision of adopting a transportation problem. Find at least two different applications where the transportation model has been used and briefly describe the approach that the authors have used. You must use appropriate sources and you must include a reference list and need to follow the APA 7th edition.  

[15 marks]