Multiple-Objective Linear Programming And Analytic Hierarchy Process For Optimization Problems

Recycling Operations in Country A

Recycling is an important and complex activity in Country A. To enable timely operations, the country is divided into 10 sectors and recycling operations are commenced simultaneously in each sector. The recyclable garbage is collected from public bins, loaded into trucks, and transported to recycling sites. Each site can accommodate different amounts of recyclable garbage because of its available land size at the facility. The annual capacities for five recycling sites are given in the table below (in megatonnes): Recycling Site 1 2 3 4 5 Capacity 10 7 15 12 6 Each recycling site is installed with facilities that have different recycling efficiencies which are summarised in the table below (in percentages): Recycling Site 1 2 3 4 5 Efficiency 35% 45% 25% 75% 55% The cost of collecting and transporting recyclable garbage primarily depends on the distance between the sectors and the recycling sites.

The following table summarises the distances between each sector and each recycling site (in kilometres): Recycling Site Sector 1 2 3 4 5 1 24 10 34 52 65 2 17 15 58 64 62 3 10 20 26 66 60 4 18 25 32 57 62 5 11 22 15 55 62 6 29 34 46 54 43 7 34 43 69 43 40 8 38 42 36 53 34 9 22 29 46 53 50 10 22 46 50 42 58 Using historical data, the country estimates the annual volume of the recyclable garbage for each sector in the coming year shown in the table below (in megatonnes): Estimated Recyclable Garbage 1 2 3 4 5 6 7 8 9 10 4.6 4.6 4.7 4.2 3.8 3.9 3.4 3.3 3.9 4.1

It will cost approximately $109,603 to move one megatonne of recycling garbage for one kilometre. The management would like to maximise the amount of recycled garbage and minimise the transportation cost.

a. Formulate an multiple-objective linear programming (MOLP) model for this problem in a Word file with a brief description of an equation, and implement the MOLP model in an Excel spreadsheet.

b. Determine the optimal value for each objective in the problem.

c. Suppose the management considers maximising the amount of recycled garbage to be three times as important as minimising the transportation cost. Formulate a GP model to optimise both objectives simultaneously with a brief description of an equation in a Word file, and implement the MOLP model in an Excel spreadsheet.

Company B's Warehouse Optimization

Company B has twenty petrol stations across Melbourne. It is creating strategies to consolidate warehouse operations so that there will be three warehouses that supply the stations. The company plans to sell all its extant warehouses and build new, state-of-the-art warehouses. Each warehouse can supply multiple stations; however, each station will be supplied by a single warehouse. The location of each station is summarised in the table below: Suburb X Y Suburb X Y Ascot Vale 25 13.8 Laverton North 13.5 7.2 Avondale Heights 19.7 14.2 Melbourne 28.6 8.9 Brooklyn 18.2 9.4 Seabrook 10.9 2.3 Burnside 10.7 16.2 Southbank 29.8 8.1 Caroline Springs 9.7 16.8 St Kilda 30.4 3.4 Derrimut 10.7 10.2 Sunshine West 16.6 10.2 Flemington 24.3 11.8 Tarneit 5.2 8.1 Footscray 22.4 11 Tarneit 5.1 6.6 Footscray 23.7 11.1 Werribee 0.5 0 Hoppers Crossing 6.3 4.7 Wyndham Vale 0 2 The company wants to build its warehouses in locations that minimise the distances to each of the stations it serves. Formulate a non-linear programming (NLP) model for this problem in a Word file with a brief description of an equation, and implement the NLP model in an Excel spreadsheet. What do the results suggest? (20 marks)

Case 3 David Jones is planning to buy a new van for his work as a trady. After narrowing his choices down to three models (A, B, and C) within his budget, he is having difficulty in deciding which one to purchase. David has compared each model against one another on the basis of four criteria: price, safety, economy, and comfort. His comparisons are summarised below: Price Safety X Y Z X Y Z X 1 1/3 4 X 1 1/3 2 Y 3 1 7 Y 3 1 8 Z 1/4 1/7 1 Z 1/2 1/8 1 Economy Comfort X Y Z X Y Z X 1 1/6 1/3 X 1 1/8 1/4 Y 6 1 1/3 Y 8 1 1/3 Z 3 3 1 Z 4 3 1 David wants to incorporate all of the four criteria into his final decision. However, the criteria are not equally important. The following matrix summarises his comparisons of the importance of the criteria: Criteria Price Safety Economy Comfort Price 1 1/6 1/3 1/5 Safety 6 1 4 2 Economy 3 1/4 1 1/3 Comfort 5 1/2 3 1 Use analytic hierarchy process to compute the overall score for each van. What do the results suggest? (15 marks)

Case 4 The distributor is interested in selecting the best location for a new warehouse. After a detailed study of 10 sites, the three location finalists are Smithfield, New South Wales; Eagle Farm, Quensland; and Derrimut, Victoria. The management provides the following data on the location criteria, criteria importance, and location ratings. Conduct TOPSIS in an Excel spreadsheet to determine the best location for the new plant. What do the results suggest? Criteria Weight Ratings Objective Smithfield, NSW Eagle Farm, QLD Derrimut, VIC Land cost 4 5 7 6 Min Labour cost 6 7 6 6 Min Labour availability 8 6 7 5 Max Construction cost 5 5 6 7 Min Transportation 5 6 6 7 Max Access to customers 9 8 7 7 Max Long-range goals 7 5 7 6 Max