| Course | BU 351 Introduction to Business Analytics |
|---|---|
| Unit | Unit 6 |
| Paper type | Prescriptive or optimization exercise |
| Length | About 990 words, 6 pages |
| Format | APA 7 student paper |
| School | Herzing University |
| Program | BS in Business Analytics |
| Updated | September 2026 |
Free sample paper for BU 351 Unit 6
Five Thousand Dollars, Three Retention Actions, Thirty-Six Members Kept: An Optimization Exercise for a Climbing Gym Chain, and the Constraint Holding It Back
Student Name
Business, Herzing University
BU 351: Introduction to Business Analytics
Instructor Name
April 1, 2026
Five Thousand Dollars, Three Retention Actions, Thirty-Six Members Kept: An Optimization Exercise for a Climbing Gym Chain, and the Constraint Holding It Back
The Decision
Basecamp Climbing, the invented climbing chain followed in this course, has set aside $5,000 a month for keeping members who might otherwise leave. The earlier units found that members who skip the introductory class, members with autopay failures and members who have stopped visiting are more likely to cancel. This exercise answers the prescriptive question framed in Unit 1: how should the monthly budget be divided among three retention actions to keep the most members? The problem is written twice, first in plain sentences and then in notation, so that anyone who doubts a piece of the model can find it in words before meeting the algebra.
The Problem in Plain Sentences
The chain can do three things. It can give new members a free introductory class, which costs about $25 in instructor time and waived fees. It can have staff call members whose autopay has failed, which takes about 10 minutes and costs about $6 in staff time. And it can send a free guest pass to members who have not visited in a month, which costs about $12 and takes about 5 minutes of staff time to arrange. The goal is to maximize the number of members kept each month.
The limits are the $5,000 budget; 60 hours, or 3,600 minutes, of front-desk staff time available for retention calls and passes each month; the number of eligible members, which each month is about 110 new joiners, about 180 members with a payment failure and about 520 members with no recent visit; and 95 seats a month in introductory classes, set by instructor schedules.
Each action's effect is an estimate. The introductory class is credited with keeping about 0.10 of a member per class given, based on the 10-point difference in early cancellation found in Unit 4, which may overstate the effect if members who choose the class differ from those who do not. The call is credited with 0.12 of a member per call, based on the staff's estimate that roughly one failed payment in eight is an expired card that a call fixes. The guest pass is credited with 0.04, a cautious assumption from the owner. These three figures are the weakest part of the model and are labeled as assumptions.
The Problem in Notation
Let x1 be the number of free introductory classes given, x2 the number of payment-failure calls made and x3 the number of guest passes sent in a month. The objective is to maximize 0.10 x1 + 0.12 x2 + 0.04 x3, the expected number of members kept. The constraints are: budget, 25 x1 + 6 x2 + 12 x3 is at most 5,000; staff time, 10 x2 + 5 x3 is at most 3,600; eligible new joiners, x1 is at most 110; class seats, x1 is at most 95; eligible failure members, x2 is at most 180; eligible low-visit members, x3 is at most 520; and x1, x2 and x3 are zero or more. This is a linear program, since the objective and every constraint are straight-line relationships (Hillier & Lieberman, 2015).
The Solution
The problem was solved by checking every corner of the feasible region, which for three variables and a handful of constraints is quick in Python and gives the same answer as a spreadsheet solver. The optimal plan is to give 95 introductory classes, make 180 payment-failure calls and send about 129 guest passes each month, keeping an expected 36.3 members. The plan spends the full $5,000 and uses 2,444 of the 3,600 available staff minutes.
The logic behind the answer is visible in the cost of keeping one member through each action: about $50 through a call, about $250 through a class and about $300 through a guest pass. The solution uses the cheapest action as far as the eligible members allow, then the next cheapest as far as class seats allow, and spends what remains on guest passes.
| Action | Amount | Cost | Staff minutes | Expected members kept |
|---|---|---|---|---|
| Free introductory classes | 95 | $2,375 | 0 | 9.5 |
| Payment-failure calls | 180 | $1,080 | 1,800 | 21.6 |
| Guest passes | 129 | $1,545 | 644 | 5.2 |
| Total | $5,000 | 2,444 | 36.3 |
Which Constraints Hold the Answer Back
Three constraints are binding, meaning the solution presses against them. The budget is binding: each additional $1,000 would keep about 3.3 more members, spent on guest passes. The limit on eligible failure members is binding: if the chain could identify one more member with a failing payment, it would keep about 0.10 more member, because calls are the cheapest action. The class seat limit is binding: each added seat would keep about 0.017 more member, because a class slightly outperforms the guest pass it would replace. Staff time is not binding; about 1,156 minutes a month go unused, so hiring more front-desk time would add nothing under this plan.
These results change the conversation with the owner. The first priority is not more money but more members identified for calls: a daily report of failed payments, rather than a monthly one, would widen the most efficient action. The second is to check whether the class's effect is real before adding seats, since the class's advantage over a guest pass is small and rests on the weakest estimate in the model.
If the Problem Had No Solution
Had the owner required, for example, that every new joiner receive a class while seats stayed at 95, the constraints would contradict each other and no plan could satisfy them. An infeasible model is itself a finding: it tells the owner that the targets and the resources do not fit, and one of them must move. Davenport and Harris (2007) describe optimization as most useful when managers treat its results as a way to test assumptions rather than as instructions, and the same spirit applies here: the model's value lies as much in showing what matters as in the plan it recommends. Basecamp Climbing, its costs and its estimates are composites created for this course (Provost & Fawcett, 2013).
References
Davenport, T. H., & Harris, J. G. (2007). Competing on analytics: The new science of winning. Harvard Business School Press.
Hillier, F. S., & Lieberman, G. J. (2015). Introduction to operations research (10th ed.). McGraw-Hill Education.
Provost, F., & Fawcett, T. (2013). Data science for business: What you need to know about data mining and data-analytic thinking. O'Reilly Media.
BU 351 Unit 6 instructions, in plain terms
The Unit 6 assignment asks students to formulate and solve a prescriptive or optimization problem for their case. Most versions want the objective, decision variables and constraints stated clearly, the problem solved with a spreadsheet solver or other tool, the solution reported and the binding constraints discussed. Many rubrics ask for the problem written in words as well as notation, and for sensitivity or shadow price interpretation. Instructors value honesty about where the model's numbers come from. If the problem is infeasible, explaining why earns credit. Expect a few pages with the formulation and a results table, citing optimization sources in APA 7.
How this BU 351 Unit 6 example is built
The sample opens with the decision and the evidence from earlier units behind it. The plain-language section describes the three actions, their costs and staff time, the limits and the estimated effect of each action, labeling the effects as assumptions. The notation section defines three variables, the objective and seven constraints. The solution section reports the optimal amounts, the spending and staff use, and explains the logic through the cost of keeping one member by each action. A section on binding constraints reports what relaxing each would be worth and which constraint does not matter. A final section discusses what an infeasible version would mean. The table of results lets a reader check every figure against the constraints.
BU 351 Unit 6 rubric: what full points look like
The rubric for this unit typically weighs the formulation, the solution, the interpretation of constraints and the discussion of assumptions. Formulation earns full points when the objective, variables and constraints are stated in words and in notation, each constraint with its source. The solution is judged by correct values and a clear table. Interpretation carries heavy weight and rewards identification of binding constraints and the value of relaxing them, turned into practical advice, as the sample does. Discussion of assumptions rewards honesty about uncertain coefficients. Instructors also credit explaining infeasibility. Clean presentation and APA citations complete the grade. Recognizing an assumption as the weakest link, and saying what would change if it were wrong, also earns credit.
Common BU 351 Unit 6 mistakes, and how to avoid them
A frequent Unit 6 mistake is presenting the solver's answer without explaining which constraints limit it. Another is writing constraints only in notation, so readers cannot tell what each represents. Students often treat estimated effects as facts, even when they come from rough judgments. Some formulations include decision variables no one could actually change. Others report a solution without checking it against the constraints. Finally, think about which limit to relax first; the sample shows that more money helps a little, more identified failure members helps a lot and more staff time helps not at all, which is the most useful output for a manager.
Get a BU 351 Unit 6 example written to your instructions
We start from the Unit 6 instructions, the BU 351 rubric in your classroom, the case parameters and whichever solver your section expects. We write a custom example that states the model in words before notation, separates choices from givens, sources every constraint, reports the allocation first and prices the binding limit. First custom sample free, returned in 24 to 48 hours.
More BU 351 papers and business analytics samples
- BU 351 Unit 1: Analytics types and framing
- BU 351 Unit 2: Data quality assessment
- BU 351 Unit 3: Descriptive analysis
- BU 351 Unit 4: Exploratory analysis
- BU 351 Unit 5: Predictive exercise
- BU 351 Unit 7: Reporting to a business audience
- BU 351 Unit 8: End-to-end analytics case
- BU 407 Unit 6: Transportation model record
- BU 451 Unit 6: Analysis script with output
- BU 453 Unit 6: Joins and aggregation queries
- BU 454 Unit 6: Accessibility and color
BU 351 Unit 6 questions, answered
Do I need a solver, or will a spreadsheet do?
A spreadsheet with its optimization add-in handles the size of problem these assignments set, and many sections assume exactly that. Whatever you use, print the formulation and the parameters into the document rather than only the result, since the marker is grading the model. Where a small problem can be solved by hand, doing so and showing the working reads well.
What if the problem has no feasible solution?
Then say so and identify the constraints that conflict, since infeasibility is itself a result and not a failure. Report the pair that cannot both hold, show what would have to give, and solve the relaxed version with the relaxation clearly labeled. Infeasibility handled this way usually earns more than a tidy answer produced by quietly dropping a limit.
How does this differ from a decision analysis course?
A decision course weighs a small number of named options against uncertain outcomes and picks one. This unit builds an allocation across many possible combinations under hard limits, and the uncertainty is usually held aside. If your draft compares three alternatives on expected value, it has answered a different question, and the formulation this criterion asks for is missing.
Where can I find a free BU 351 Unit 6 sample paper?
The finished BU 351 Unit 6 exercise, formulation through binding constraints, is free to read above. For an optimization problem built on your own case, pass along the prompt and the decision you face, and a free custom exercise will be worked for you, usually within two days.
Do I need a solver for BU 351 Unit 6?
A spreadsheet solver is enough for most problems of this size. The sample checked every corner of a three-variable problem in Python, which gives the same answer as Excel Solver. What matters is stating the model clearly and interpreting the constraints.