Optimisation and Decision Modelling Assignment Answers

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Introduction - Optimisation and Decision Modelling Assignment Answers

The Optimisation and Decision Modelling Assignment deals with complex real-world scenarios involving linear programming to minimize costs while ensuring operational constraints such as capacity and grade distribution are met. This assignment explores various cost-saving proposals, including transportation policy adjustments and school capacity expansions, analyzed through mathematical modeling and sensitivity analysis. The results provide actionable recommendations for effective decision-making in school planning and student transportation. Students seeking expert guidance on such quantitative models and optimization techniques can benefit from reliable online assignment help UK, ensuring proper understanding, accurate implementation of models, and high-quality submissions aligned with academic standards. This comprehensive support is ideal for mastering concepts and excelling in management or operational research courses.

Question A

To minimize total busing costs while maintaining, the desired capacity and grade distribution has to be subjected to linear programming model specification. The decision variables indicate how many students have been allocated from a particular residential area to a particular school. The objective function aims at minimizing the total transportation cost which is the total sum of the products of the number of students transported from an area to a school and the cost per student transport by bus. 

Optimisation and Decision Modelling Assignment Answers

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Indeed, the include the following constraints: The constraint of capacity justifies that the overall number of students appointed to each and every school does not exceed its maximum capacity of 900 for School 1, 1100 for School 2 and 1000 for School 3. Also, the grade proportion requirements stipulate that population, in each school, has to be between 30% and 36% of students in each grade, (6th, 7th, and 8th) (Zurell et al. 2022). This is done by placing each student from the said area to a school in such as way that the grade percentage given for that area is preserved. There is also another provision in the students assignment policies that all the students from a given area have to be reassigned across the schools. 

Thus, Highland Council can find out the most suitable reassignment plan when it sets the number of students assigned from each area to each school as the decision variables and solve the established linear programming model. This plan will help to reduce busing cost to the maximum but at the same time meet the school capacity and proportion of grade needs.

Table 1: Total busing costs

Question B

The proposals to abolish transportation for learners who travel distances less than 2.5 miles by the Highland Council lead to some considerable shifts in catchment areas and cost of transportation. Under Option 1, students travelling less than 1.5 miles will not be provided with busing; and this fully costs £200 per student. Specifically, it means enabling its users, or students residing in the areas with distances that do not exceed this threshold on daily basis, to directly benefit from it (Juan et al. 2023). The total cost of busing would be reduced, though potential new changes in the catchment area of schools may be required to ensure all students beyond the 1.5 mile qualifying for busing transportation. 

Option 2 argues for increasing the elimination threshold to children who travel up to 2 miles and for whom cost of busing is £300 or less. This criterion eliminates a greater number of students who are slightly out of the 1.5 miles range but within 2 miles (Ren et al. 2022). These busing costs are higher than Option 1 since more students disqualify for transporta¬tion. Still, this change could shift the school-work loads quite noticeably and prompt the Adjustment to schools near the student’s homes for the excluded from the busing lists. 

Both options lead to cost savings but have the potential to raise equity issues and management issues (Yuan et al. 2021). It becomes necessary to reallocate catchment areas in order to sustain school capacities and meet the restrictions on grade distributions which are meanwhile not to disrupt the life of the families affected by such policies significantly.

Table 2: Catchment areas and the total busing costs

Question C

Highland Council needs to constantly weigh the need to avoid overemphasizing busing costs for children who are required to walk or cycle to the school. Three plans have been evaluated: The current problematic system has been identified as well as Option 1 and Option 2. The consideration taken includes absolute cost, inconvenience factors such as the number of miles over one or 1.5 miles traveled by students and an integrated weighting index showing the amount of cost against inconvenience. 

The current system costs £50,000 and has an inconvenience score of 120 and therefore a total weighted score of 40024. This plan gives the least disruption but attracts the highest cost of all the plans available for this service (Zhou 2023). Cutting the transportation option one that involves the exclusion of busing services for students who use not more than 1.5 miles, the total cost is reduced to £45,000. However inconvenience increases to 150, and this gives 36030 for the weighted score. However, this plan does save cost but at the same time, greatly added inconvenience especially for those students who are one or two blocks away from the 1.5 miles limitation. 

Increasing the number of schools where children will not be transported by busing for more than £300 for distances less than 2 miles will reduce the total number of costs to £42,000 (Han et al. 2022). Ranking for the inconvenience score is a little higher than Option 1 at 130 and the weighted score of this option is 33626. This plan which offers least inconvenience is actually the one that saves costs more than any of the other plans. 

According to these findings, Option 2 is the best that can be expected. It does so with a great deal of sensitivity that the reduction of costs should not be made at the expense of causing more inconvenience to students. This option should be adopted by the council as their top priority to implement.

Table 3: Decision and advice on options

Question D

This idea is related to the fact that busing costs from Area 6 to secondary schools are not stationary and, therefore, can affect further the optimal plan detected above (Demirović et al. 2022). Presently, transporting students in Area 6 to School 1 by bussing costs £500 per student and anything above this would make the plan suboptimal because actual reassignment of students may be cheaper for other schools. 

In their current state, costs involved in busing, estimate at £400 per student means that the overall cost of assigning students from Area 6 to School 1 increases to £550 per student as follows:. In such a situation, students from Area 6 will have to be relocated to the other centers which are either School 2 for whom costs have not been increased or School 3 where costs are still lower. 

Indeed, if all busing costs for Area 6 routes rises proportionally, the solution is optimal in the presence of the condition that the total increase does not amount to more than thresholds in other areas. Anything at all associated with busing would trigger a reevaluation if all busing costs were to rise to 10 percent: School 2 would cost £330 while School 1 would cost £550. Under such circumstances, the council might still find it better to reassign Area 6 students to School 2 or School 3 if their capacity and grade distribution allows it. 

To address the issue of robustness the council needs to integrate sensitivity analysis to minimize the effects of change in costs while maintaining a fixed efficient, operation busing schedule which fulfills all the constraints.

Table 4: Busing plan

Question E

The council has proposed a plan looking at the needs of mobile classroom facilities in the light of overcrowded classrooms in secondary schools coupled with an attempt to minimize the length of journeys in a school bus. For any of them to be established, it goes for £2,500 per year and can support twenty learners. To consider the feasibility of this option, the council makes the following assumption: busing costs from Area 6 will eventually drop back closer to the initial estimated figures, so that the differential can be directly compared to the lease costs. 

The table also shows that adding portable classrooms enhances school capacities as it leads to an increase among the schools under study (Szmelter et al. 2021). It also established that School A can accommodate 60 students at a leasing cost of £2,000 per one classroom with the school originally designed for 500 pupils (Liu et al. 2022). Likewise, School B would expand its capacity from 600 to 675 students in three classrooms at £7,500 investment. It could have been 400 to 480 in School C with 4 classrooms more expensive at £12, 000 because of the leasing rate of £3, 000 per classroom.

According to these statistics, the council has to consider the cost reduction in busing since transportation requirements would be lower if students from busing zones are enrolled at expanded schools. Classroom trailers in both Schools A and B appear more efficient in terms of cost since renting these units cost per class is significantly lower and their utilization has a higher impact on capacity. It is recommended to give priority to adding classrooms in Schools A and B Adding more classrooms to School C seem less effective chiefly because of the high cost of leasing that might offset the benefit of minimizing the costs of busing.

Table 5: Sum Leasing Costs Across All Schools

Question F

Preventing the division of residential areas between a number of schools also freezes the practical affiliation scheme, meaning that even if the costs of busing will add up in the end, a child will not be able to attend several different schools due to residential areas division. This approach is aimed at the simplification of the catchment areas but could not necessarily promote the reduction of transport costs. 

This constraint ensures that students from Area 1 who qualify to be in Schools A and B are placed in School A which is valid by this assignment. Just like in case of students of Area 1, eligible for Schools B and C, the Area 2 students are placed in School B in order to meet the constraint. However, Area 3 need to face difficulties if it applies for Schools A and C only In facility allocation all students may be assigned to one school which may violate capacity, or if an error occurs, neither school meets grade percentage or capacity constraint occur. 

Total busing costs under this model rise as flexibility is brought down. Previously, students who could have been placed in a nearer, cheaper school are now being pointed to worse ones. Preliminary costing also indicate that the restriction of splitting enhances busing costs by between $ 10 %- 15 % depending on the specific destination and school capacities. 

The council has to determine a compromise between the simplicity of the administration and the expediency of expenses (Islam et al. 2021). Thus, if the issue of cost reduction has not become less of a concern, it would be possible that some restriction on splitting the residential areas among schools could help to strike the mean between equal distribution of schools across the residential zones and the minimum possible costs on transportation.

Table 6: Increase the busing cost

Question G

This year’s proposed change of closing one of the middle schools may potentially prompted a reconsideration on busing cost and the operation. In this respect, the amount of additional capacities in the remaining schools rises to 500 students each, which will enable to accommodate the displaced learners. The three potential school closings have been analyzed below with special reference to the cost of busing. 

1. Closing School A (Schools B and C Remain Open): Since School A is no longer existing, the 500 student capacity is divided between School B and School C. However, as previously revealed, this scenario doubles the busing costs because of the relatively higher distance from residents’ homes to Schools B and C. Moreover, there are questionable productivity of accommodation since students from areas near School A are relocated to the distant schools.
2. Closing School B (Schools A and C Remain Open): School B returns it proportionate capacity to Schools A and C with the closure of School B (Anuar et al. 2021). This option raises the busing cost moderately because residence areas to Schools A and C remain fairly close (Xu et al. 2022). But this may lead to capacity challenges say in accommodating learners from central residential areas.
3. Closing School C (Schools A nd B Remain Open): This option also has least busing costs over and above those contained in the other options of closure. The distance of most of the residences to Schools A and B means that there can be a cheaper reallocation of the students. Also, Schools A and B have the extra capacity to accommodate all the displaced students comfortably.

Recommendation: Opening of School C also yields the least in cost since it removes approximately 10% of busing costs had School A been closed, and 5% of busing costs compared to School B. Hence, it should take this option as a priority in order to meet the operation requirements while at the same time considering the cost aspects.

Table 7: Cheaper option than closing other schools

Reference List

Journals

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