N.B. There are only 20 constraints mentioned here as some constraints refer to numerous models.
Using this data, the problem was formulated as a linear program and solved using an Excel spreadsheet, with following assumptions made:
- That only 50 units of models from the Grand Estate Series were built on the lake
- That the additional land not mentioned in the advertised size of the models in the Country Condominium and Grand Estate series was used for additional yard space (or otherwise)
Results – Planned Community at Lake Saddleback
Based on the results of our model, we recommend that LSDC build the amounts of each model illustrated in the table below, as this would generate a total profit of $126,702,703.56:
(Please see page A0 for more detailed results)
Table 1
Key
(L) = Model built on the lake
(NL) = Model not built on the lake
(P) = Premium model
(NP) = non-premium model
The most noticeable results from these recommendations are:
- That out of all of the Grand Estate models to be built on the lake, the recommended number of units of the Trump model was 26, whereas the recommended number of units for the three other models was 8, the bare minimum according to the constraints set to the planned community. We assume that this indicates that the Trump models built on the lake are very profitable for LSDC
- That the recommended number of units of the Grand Cypress Premium models is 0. We assume that this indicates that this model is unprofitable for LSDC
These factors will be explored further later on in the report.
The Excel tool ‘Solver’ has allowed us to produce an Answer Report, Limits Report and Sensitivity Report for the problem (See Pages A1 – A8). These reports allow LSDC a better understanding of the recommendations provided by our company, and are briefly explained below:
Answer Report
The variables labeled as ‘binding’ on the Answer Report (for example, the maximum number of units built of the Weeping Willow model, which provides a result of 211) are those variables which have utilised resources to their full capacity, and whose status is therefore optimal; any adjustments to these would affect maximum profit. The ‘slack’ values of the non-binding values indicate that these variables did not use all of the available land to its full capacity (for example, the maximum number of units built of the Golden Pier model, which provides a result of 40). However, even though we could have theoretically had a higher quantity of models, the constraints affecting the project have prevented us from doing so.
Limit Report
The limit report suggests that all of the models contribute equally to the objective value of $126,702,703.56 (except the Grand Cypress Premium model because the recommended number of units built for this model is zero). We assume that this apparent equality is due to the numerous constraints affecting the problem.
Sensitivity Report
The sensitivity report illustrates details about reduced costs, for example, by indicating that in order to make the unprofitable Grand Cypress Premium model worth building, the profit per unit for this model would need to increase by $335.34.
The shadow prices in the Sensitivity Report indicate the change in LSDC’s total profit that would result if the organisation were to produce one more unit of any given model, thereby allowing us to differentiate the less profitable models from the more profitable models. For example, if LSDC were to use one more unit of the Bayview Premium model, this would increase the total profit by $,2917, whereas if the organisation were to make one more unit of Vanderbilt on the lake, total profits would decrease by $1,320 as a result. The range of values for which these shadow prices are valid are illustrated by the ‘Allowable Increase’ and ‘Allowable Decrease’ figures for each constraint.
The report also shows that the range of optimality for each model via the ‘Allowable Increase” and “Allowable Decrease”; for example, we see that the number of Trump model units not built on the lake could increase by 1320 or decrease by 2507 and not affect the total profit generated (providing everything else remained the same).
Building the sports/recreation complex
LSDC also wants to evaluate the feasibility of building a 10 acre sports/recreational complex within the area used for the planned community. The implementation of this idea would result in a decrease in the total land available for housing to 290 acres, and would cost LSDC $8,000,000. However, the value of all models (except Golden Pier) would increase as a result of developing the complex, thus generating additional profits for the organisation.
We have evaluated and solved this problem using an Excel Spreadsheet by treating it as an extension of the linear programming problem of developing the complex. In order to do so, we changed the constraint for total land available for the planned community to be ≤ 290 acres. The profit figures per model were then adjusted according to the new specifications, and added together to form a gross profit for the organisation. $8,000,000 was then deducted from this gross profit to account for the cost of building the complex, resulting in a final net profit figure for the organisation.
Results- Building the Sports/Recreation Complex at Lake Saddleback
Based on the results of our model, we recommend that LSDC do build the sports/recreation complex, as it would generate a net profit of $138,374,938.71, an increase of $11,672,235.20 on developing the community without the complex. Also, the resulting decrease in the variety in models is not substantial, as illustrated in Table 2:
(Please see pages B0-B8 for more detailed results)
Table 2
Key
(L) = Model built on the lake
(NL) = Model not built on the lake
(P) = Premium model
(NP) = non-premium model
As Table 2 suggests, the decrease in the number of units built if the sports/recreational complex would only be 56, a mere 3% decrease. Also, the maximum decrease of units built of any one model is the 5% decrease that would occur in the number of Trump units (not on the lake) built.
‘What if’ Analyses
Some of the recommendations derived from the formulation of this problem have encouraged us to further explore some of the constraints, to test the effects of these changes on the planned community as a whole:
1) “What if we increased the maximum limit of the number of Grand Estate Models to be built on the lake to 100 (instead of 50)?”
The results for the planned community at Lake Saddleback have shown that out of all of the Grand Estate models to be built on the lake, the recommended number of units of the Trump model was 26, whereas the recommended number of units for the three other models was 8, the bare minimum according to the constraints set to the planned community. This suggests that the profitability of the Trump models on the lake is quite high, and it is for this reason that we wanted to test the effect of increasing the maximum limit for all the Grand Estate models on the lake.
The results for this ‘what if’ analysis are as follows:
Table 3
Table 4
As illustrated by Table 3, the number of Trump models built on the lake would increase from 26 to 76 (66%). Table 4 indicates that the resultant total profit has increased by $2,860,000 compared to the total profit generated by the original planned community problem, indicating that this model is by far the most profitable model in the Grand Estate product (if not all the products). We assume that the revised number of units for the Trump model on the lake would have increased even more (and possibly to 100 units) had it not been for the constraint stating that the minimum number of all other models on the lake is 8 units.
(Please see pages C0-C5 for more detailed results)
2) “What if we set the minimum amount for the Grand Cypress Premium models to be built at 25% (instead of 25% being the maximum limit for the number of Grand Cypress Premium models )?”
Another interesting result for the planned community at Lake Saddleback was the recommendation of building no Grand Cypress Premium models at all. We have attempted to better understand the reasons for this result by setting a constraint on the minimum number of models of this type to be produced.
Although this ‘what if’ analysis affects the recommended number of units for models other than the Grand Cypress, we shall focus only on the changes to the Grand Cypress model as these are the most significant, and these are illustrated in Tables 5 and 6 below.
(Please see pages D0-D5 for more detailed results)
Table 5
Table 6
As illustrated in the Table 6, although changing this constraint increases the variety of housing in the planned community, it is not a profitable model; total profits for LSDC decreased by $17,866.20 as a result of increasing the number of Grand Cypress Premium units built. Further supporting the idea that this model is unprofitable is the fact that the revised number of units for the Grand Cypress Premium model is 53, a figure which is 25% of the total number of Grand Cypress models built: the bare minimum.
3) “What if there was no minimum or maximum limit to the percentage of each type of product built in the complex?”
The large number of constraints affecting the planned community at Lake Saddleback has means that it is sometimes difficult to recognize exactly which models are most profitable to LSDC. By removing the maximum and minimum constraints to the percentage of each type of model in the community, we explored the results of focusing more on maximizing the organisation’s profits, rather than increasing variety.The table below summarises the results of this ‘What if’ analysis:
(Please see pages E0-E5 for more detailed results)
Table 7
Key
(L) = Model built on the lake
(NL) = Model not built on the lake
(P) = Premium model
(NP) = non-premium model
Table 8
As illustrated by Table 7, if the maximum and minimum constraints to the percentage of each type of model in the community were to be removed, the recommended number of units of the Grand Cypress Premium models would remain zero, and the entire Lakeview Patio Homes product would be eliminated. This indicates that these are LSDC’s least profitable models, as their elimination results in an increase in total profit of $3,619,2219.90 compared to the original total profit figure.
Final Recommendations
After considering both the recommendations from the original planned community problem and the results of the ‘What If’ analyses, we have formulated final recommendations to LSDC which we believe to be optimal because they allow LSDC to offer a variety of different housing models to its customers while still maximizing profit. The feasibility of these recommendations has been tested by solving it as a linear programming model on an Excel Spreadsheet (Please see pages F0-F6 for detailed results).
If LSDC were to follow our recommendations, it would generate a net profit of $141,484,938.71, which shows a $14,782,235.20 increase on the total profit generated from following the original planned community problem.
The recommendations are as follows:
- To build the sports/recreation complex
- To increase the maximum limit of the number of Grand Estate Models to be built on the lake to 100 units
- To build the following number of units for each model:
Table 9
We hope this report has been useful, and we would happy to undertake more analyses should LSDC require any further assistance on this matter
QUESTION 2
Management science is a discipline which is crucial to the overall running of organizations because it allows managers to see how they can best allocate their resources and plan their projects to meet their goals. In this essay we shall be discussing how various management science principles and techniques can help improve the running of City University for the benefit of its staff and students.
The Linear Programming Model is an approach used by managers to guide decision-making when the objective involves the maximization or minimization of a linear function, taking into consideration any linear constraints which will affect the outcome of the original objective.
This model can easily be applied to a number of issues facing City University, and provide a useful analysis of the problem via the construction of a linear programming formulation. For example, when planning how many computer terminals should be available at the university, a model can be formulated incorporating constraints such as the budget available to finance additional computers, or the floor space available to accommodate the extra terminals. Such formulations can be easily resolved using spreadsheets, making linear programming a very useful approach to guiding decisions made by the university.
Project scheduling is a management science technique used by organizations to ensure that projects are planned and conducted resourcefully. City University could use this technique in its planning for events such as the annual summer ball, or in establishing a new department.
Using project scheduling would allow City University to plan the building of the new law department that is currently taking place more effectively; as this technique would allow City to determine a schedule of the earliest and latest start and completion dates for each task in the project, and therefore also the earliest completion time for the entire project. Also, the University would be able to efficiently allocate resources, approximate the probability of the project being finished within a specific time period, and would also be able to monitor the progress of the project in terms of being on time and within budget.
Project scheduling allows organizations to analyse the impact of possible delays on the project’s completion time, giving them the opportunity to act accordingly. Overall, project scheduling could prove very beneficial to City University as it would mean that the University would be better organized in the development of new projects, minimizing the inconvenience felt by staff and students and ensuring that all projects are carried out as efficiently as possible.
Network Flow Models, such as assignment models, transportation models and capacitated transshipment models, deal with the delivery of a measurable quantity of items from a supply node to a demand node.
-
Assignment models allow organizations to allocate individuals to tasks in the most efficient manner possible, and do so by minimizing total costs or maximizing profit. City University could use an assignment model to assign the optimal number of catering workers at its refectories by allocating each staff member to a job that he/she would be best at. A disadvantage of assignment models would be that certain workers would no be able to perform certain jobs. However, this problem could be dealt with by ideas such as job combination.
-
Transportation models allow organizations to determine the total cost of directly shipping supplies from source nodes to destination nodes. City University could use such a model to calculate the minimum cost of transporting books requested from the Cyril Kleinwort Library to the Northampton Square Library. This idea could also be expanded upon by using a Capacitated Transshipment Model to calculate the cost of transporting books from the Cyril Kleinwort Library to the Northampton Square Library via an intermediary node, for example, the Nursing School Library at St Barts.
Connectivity Models such as Spanning Trees allow organizations to minimize the total distance between all of the ‘nodes’ in their network. City University could use a spanning tree to connect all course offices within a department such as the Business School and find the shortest and most efficient routes between them so that the transport of information, materials and booklets could be facilitated. The main advantage of this approach is that it would save the university time and money. However, the solution provided by spanning trees is not always optimal.
Decision Analysis allows an organization such as City University to select an option from a set of possible alternatives in situations where the future is uncertain. Decision trees are diagrams which illustrate the decision process, and represent how a decision is likely to affect the future via a series of arcs and nodes. City University could use a decision tree to decide things such as whether or not to have built Cass Business School, as illustrated below:
The decision tree’s main advantage is that if offers a straightforward ‘yes’ or ‘no’ answer to the question of whether it would be a good idea for an organization to carry out a project. Decision trees are also simple to construct and allow decisions to be easily identified. However, decision trees often require lengthy calculations in order to analyse the problem at hand.
Queuing theory is a management science principle that deals with measuring the performance of the queuing systems present in an organization, and allows organizations to use this information to make recommendations and improvements to their queuing systems. City University could use a queuing model to find the optimal number of library staff that should be available at the library reception which would allow for both a reduction in waiting time for customers, while still keeping costs down for the university. There are two approaches to queuing models: the analytical approach and the simulation approach. The main advantage of the analytical approach is that it is relatively simple to carry out calculations as output and input variables can easily be changed. However, unless simplifying, and often unrealistic, assumptions are made, the mathematics can become quite complex. The simulation approach’s main advantage is that it provides more flexibility and lets the organizations ‘see’ how the queue(s) in question would be affected, but it is not an approach which is well-suited to spreadsheets.
Dynamic programming is a process for problem solving that is applied to problems that can be separated into a number of separate stages, and allows organizations to make a number of decisions that will ultimately benefit the entire organization. In the case of City University, dynamic programming could be used for deciding the maximum total number of new jobs that should be created in one department from the allocation of a specific sum of money to that department by the university. The main advantage of this approach is that it eliminates combinations under consideration by an organisation that could never be optimal.
Time Series Forecasting is a management science technique used to predict future trends by looking at patterns in data generated from an organisation’s past. Time series foreacasting can help organizations such as City University prepare for the future in many different respects, from predicting this year’s graduating classes’ grades, to deciding how many of each type of book should be provided by the university’s libraries.
Using a specific example, by collecting data on the number of students in the past that applied for the Management and Systems course, and plotting this data against time, City University could predict the number of applicants that will apply for a place in the Management and Systems degree course in the next academic year. Using time series forecasting can greatly improve the running of City University as it would allow the university to be better prepared for the future, and therefore encourages better resource management and efficiency within the University.
Goal programming models allow organizations to find ways of meeting numerous goals simultaneously, provided that the set of goals follow a set of constraints. City University could use goal programming when considering its future advertising campaign for the newly-opened Cass Business School, for example by setting itself the goals:
Goal 1- Spend no more than £25,000 on advertising
Goal 2- Reach 30,000 potential students
Goal 3- Cover at least 10 underground stations
In this way, the university could see how it could most efficiently use the resources available to it to reach its target audience.
Quality management models allow organizations to monitor and maintain the quality of the goods and services which they provide. City University could use quality management models to do things such as creating control charts to test lecture quality, or to ensure that lecture theatres are neither overcrowded or below optimal capacity.
Quality management models include fishbone diagrams, which are diagrams that analyse processes, and identify areas within these processes which are in need of improvement. If City University was to use a fishbone diagram to analyse its teaching quality for example, it could investigate the following factors:
-
Methods/ Procedures: Various procedures can affect teaching quality (e.g. a lecturer’s inability to provide quality teaching due to large volumes of students asking too many questions)
-
Manpower issues: These relate to employee training (e.g. employing postgraduate students to teach when they haven’t been trained to do this)
-
Materials: Obstructive teaching material will cause lower lecture quality (e.g. confusing and illegible handouts provided by lecturers are more of an hindrance than an aid to students)
-
Machines: Malfunctioning computers and electrical equipment (e.g. microphones) can affect the quality of teaching that staff provide.
Simulation is a management science idea which evaluates a proposed system over a defined period of time via the use of a model in order to estimate features of the system, so that an organization can then choose the most appropriate policy from the alternatives being considered. City University could use a simulation model to determine factors such as the average waiting time in registration queues at the beginning of the academic year, and the time it would take for a new first year student to complete the registration process.
Simulation’s main advantages are that it allows the performance of a system to be tested and altered without causing disruption to the system as a whole, and that it allows organizations to conduct ‘what-if’ analyses to test out ideas without actually having to implement them in real life. However, simulation also has its disadvantages: the simulation models are often expensive and time-consuming to develop, they only provide approximations of a model’s real parameter values, and the ‘optimal policy’ determined by the simulation may not be so in real life.
As has been illustrated, there are numerous management science principles and techniques that can aid organizations in the service sector, such as City University, in improving the service that they provide. However, as identified by Targett, there also exist some limitations to applying management science to the service sector:
Firstly, measurement of output and performance is difficult as the data which such organizations use to measure quality of service is often qualitative. Also, the product offered by service-sector organisations is not tangible; which also causes measuring the quality of performance to be difficult. It must also be noted that production and consumption are usually simultaneous in organisations such as City University, which leads to their being no inventory of the service, and causing systematic observations to be difficult.
Fourthly, the ‘product’ offered by organizations such as City University is time-perishable, which also causes difficulties in systemic observation. Finally, site selection is governed by customer demand, which means that operations are often decentralized, a fact which goes against the management science belief that planning and control should be centralized. However, in the case of City University, although different buildings are located in numerous sites, centralization does exist in the form of the University’s numerous departments each being responsible for their own activities. Also, the number of departments itself is decreasing, exemplified by departments such as the Actuarial science department merging with the Business School.
References-
-
Lawrence J. A. & Paternack, B. A. (2002) Applied Management Science, 2ND Edition, USA: Wiley
-
Anderson D.R., Sweeney D.J. & Williams T.A. (2002) An Introduction to Management Science- Quantitative Approaches to Decision Making (10th Edition) USA: Thompson Learning- SouthWestern
-
Winston, W.L, & Albright S.C. (2001) Practical Management Science (2nd Edition) USA: Thompson Learning- Duxbory
