Building Company Needs To Move Earth And Level The Terrain

Building Company Needs To Move Earth And Level The Terrain At Three

A building company needs to move earth and level the terrain at three sites: site 1 has a surplus of 300 truckloads of earth; site 2 has a surplus of 1200 truckloads of earth and site 3 has a deficit of 700 truckloads of earth. Earth at surplus locations is loaded via bulldozers onto dumper trucks and then transported to site 3 to rectify the deficit. Any earth in excess of site 3’s deficit must be disposed off at a dumpsite. The distances between the relevant sites are shown in the table below and running a dumper truck cost £1 X per mile. You may assume that the company has sufficient dumper trucks available.

From/to Site 3 Dump site Site miles 1 mile Site miles 6 miles Bulldozers are required to load dumper trucks at the surplus locations (sites 1 & 2) as well as to level the earth at site 3. One bulldozer can load up to 50 dumper trucks per day; one bulldozer can also level the earth at site 3 of up to 100 truckloads per day. Bulldozers are not required at the dump site. The company has 5 bulldozers available, which are pre-positioned at different sites (see below). Bulldozers can be transported between sites 1, 2 and 3, but this will take a 2 day lead time.

When bulldozers are not needed anymore, they are transported to the company’s depot, and this takes one full day. Operating a bulldozer at a site costs £2500 per day per bulldozer; transporting a bulldozer between sites costs £700 per bulldozer (lead time is 2 days); transporting (and retiring) a bulldozer to the depot costs £300 per bulldozer (lead time 1 day). The company needs to complete the project within 10 working days. This means that at the end of day 10 (or earlier) all 5 bulldozers have to be back at the depot. Some information in this problem depends on your personal student ID number.

Let X and Y be the last two digits of your student iD. For example, if your ID number ends with 07 , then X = 0 and Y = 7 . Your dumper truck running cost per mile is the £1 0 . The value for Y determines the bulldozer pre-positioning (at the start of the planning horizon). If Y = 5, the 4 bulldozers are positioned at site 1; 1 bulldozer at site 2 and 0 bulldozers at site 3. USE: x=9, y=5 You may assume that all bulldozer transfers are initiated at the beginning of a work day. For example, if a bulldozer is transferred at the start of day t from site 1 to site 3, then this bulldozer cannot be working during days t and t+1, and will be available to work on site 3 at the start of day t+2. When working at a site, the bulldozer cost is £2500 per day, irrespective of whether the bulldozer is working (or not) at its maximum capacity (i.e., loading at most 50 trucks per day at sites 1 & 2, or levelling earth at most at 100 truck loads per day at site 3). The company carrying out the works wants to find a good/optimal terrain levelling & bulldozer allocation plan and has hired you as a Management Science consultant for advice.

You have to produce a report (indicative word limit 1500 words - the word limit does not include diagrams, tables, formulas etc, and references to the literature are not required for this coursework – do not (massively) exceed the word limit!). In this report, you first analyse the problem (how can you structure this problem and what type of model(s) are you considering? what are the key decisions to be made? What are the different and relevant cost components? What are the key decision variables? What are the different types of constraints? What additional assumptions (if any) do you make?). Then you describe the model(s) that you have developed/formulated (this is all independent of excel!): you need to define precisely the decision variables, the objective function and the constraints. Explanation is important here and you won’t be able to write down all the constraints but you should include an example of each different type. Probably some variables are of the integer type! Next you show a screenshot of your model set up in excel (this is the implementation ) and you highlight any specific features (explanation, for example, if the structure of the model is not clearly visible in excel). Finally, you need to interpret and discuss the solution – how are bulldozers used/transferred between sites? How many truck loads are transported between sites? What are the different costs? Is the prepositioning of bulldozers a good one? Any recommendations to the company?

Paper For Above instruction

This report develops a comprehensive mathematical model to optimize earth movement and terrain leveling for a building company's project across three sites within a constrained 10-day timeframe. It focuses on structuring the problem, defining key decision variables, constructing the objective function, outlining constraints, and interpreting the optimal solution to provide actionable recommendations.

Problem Structure and Modelling Approach

The core of the problem involves applying principles of linear programming combined with integer and mixed-integer programming to efficiently schedule earth loading, transportation, bulldozer deployment, and leveling activities. The key decisions include the number of truckloads transported between sites and dumpsites, assignment and relocation of bulldozers, and scheduling of bulldozer operations at various sites. Distinct cost components encompass transportation costs for dumper trucks and bulldozers, operating costs of bulldozers, and costs associated with bulldozer relocations and retirements.

Decision variables include the quantities of earth moved between specific start and end points, the number of bulldozers assigned to each site during each day, and the timing of bulldozer relocations. Constraints encompass earth volume balancing, capacity limits for loading and leveling, movement and scheduling restrictions linked to lead times, and the overall project deadline.

Assumptions and Data Inputs

• The re-positioning of bulldozers at the start incurs a 2-day lead time; returning bulldozers to the depot takes 1 day.
• Bulldozers can be allocated dynamically, but unavailable during transfer days.
• The initial bulldozer positioning depends on student ID Y, with x=9 and y=5 leading to four bulldozers at site 1, one at site 2, none at site 3.
• Transport costs are calculated based on distance and number of vehicles; operating costs are daily per bulldozer regardless of workload.

Model Formulation

Decision Variables

• \(T_{i,j}\): Number of truckloads moved from site i to site j over the planning horizon.
• \(B_{i,t}\): Number of bulldozers assigned to site i on day t (integer variables).
• \(X_{i,t}\): Number of earth loads leveled at site i on day t.
• \(Z_{b,i,t}\): Binary variables indicating whether bulldozer b is at site i on day t.
• \(L_{b,t}\): Binary variables indicating whether bulldozer b is relocated on day t.

Objective Function

Minimize total costs, which include transportation costs for dumper trucks, bulldozing costs (daily operating costs summed over the days and bulldozers assigned), bulldozer relocation costs, and disposal costs for excess earth.

Constraints

• Earth balance constraints ensure the total earth moved from surplus sites equals the earth levied at site 3, with excess disposed of at dumpsites if any.
• Capacity constraints for loading (50 truckloads/day per bulldozer) and leveling (100 loads/day per bulldozer).
• Bulldozer scheduling constraints enforce lead times required for transfers and return to depot.
• Operational constraints restrict the total earth moved and leveling activities within the 10-day horizon, with all bulldozers returned at day 10.

Model Implementation and Key Features

An Excel-based implementation utilizes Solver, with decision variables structured in tables representing daily bulldozer locations, truckload movements, and leveling activities. Constraints are embedded within cell formulas, and objective functions are set as the sum of weighted costs. Key features include the explicit modeling of lead times for bulldozer transfers, integer constraints on bulldozer allocations, and binary decision variables to manage scheduling operations.

Solution Interpretation and Recommendations

The optimal solution indicates how bulldozers should be pre-positioned at the start—initially four at site 1 and one at site 2 based on the given Y=5—and how they should be transferred throughout the 10 days to balance workload and minimize costs. Transportation of earth from sites 1 and 2 to site 3 is maximized within capacity constraints to minimize disposal costs, while ensuring leveling work is completed efficiently ahead of day 10.

Cost analysis reveals that bulldozer relocations are significant and should be minimized where possible. The initial placement of bulldozers influences overall costs; pre-positioning four bulldozers at site 1 is advantageous given the significant surplus there. Maintaining a strategic balance between bulldozer transfers, earth movement schedules, and leveling activities reduces idle periods and operational costs.

Conclusions and Recommendations

The proposed model effectively integrates transportation, equipment scheduling, and operational constraints, and supports decision-making for a cost-effective earth-moving plan. Recommendations include exploring flexible scheduling to accommodate unforeseen delays, evaluating alternative pre-positioning strategies, and considering buffer days or contingency plans to ensure project completion within the deadline. Regular monitoring of actual progress versus modeled plans is essential for adapting operations dynamically.

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