| Course | BU 407 Quantitative Decision-Making |
|---|---|
| Unit | Unit 4 |
| Paper type | Optimization model summary |
| Length | About 960 words, 6 pages |
| Format | APA 7 student paper |
| School | Herzing University |
| Program | BS in Business Analytics |
| Updated | September 2026 |
Free sample paper for BU 407 Unit 4
Milk, Yogurt or Cheese From 2.4 Million Pounds a Week: A Linear Programming Model Summary for a Composite Dairy Plant, Read Back as a Production Plan
Student Name
Business, Herzing University
BU 407: Quantitative Decision-Making
Instructor Name
March 4, 2026
Milk, Yogurt or Cheese From 2.4 Million Pounds a Week: A Linear Programming Model Summary for a Composite Dairy Plant, Read Back as a Production Plan
The Problem
Each week, Northwoods' Eau Claire plant receives about 2.4 million pounds of raw milk from member farms and must decide how much of it to turn into fluid milk, yogurt and cheese. Each product earns a different contribution and uses the plant's equipment differently, and several limits apply at once: the milk supply itself, pasteurizer hours, packaging line hours, cheese vat hours, the yogurt the market will take and a contract that requires a minimum volume of fluid milk for a regional grocery chain. This summary builds a linear programming model of that weekly choice, reports what the solver returned and reads the answer as a production plan a plant supervisor could follow. All coefficients come from the plant's cost and engineering records and belong to the composite case.
Decision Variables
Three decision variables describe the weekly plan. The first, x1, is the quantity of fluid milk produced each week, in thousands of gallons. The second, x2, is the quantity of yogurt produced each week, in thousands of cases. The third, x3, is the quantity of cheese produced each week, in thousands of pounds. Each is a weekly quantity, not a rate or a share, and each must be zero or more.
Objective Function
Maximize weekly contribution, in dollars: 420 x1 + 1,150 x2 + 900 x3. Each coefficient is the contribution per unit after all variable costs, including raw milk at the cooperative's standard price, packaging and energy. A thousand gallons of fluid milk contributes $420, a thousand cases of yogurt $1,150 and a thousand pounds of cheese $900.
Constraints
Six constraints limit the plan, each written with the resource it limits.
1. Raw milk: 8.6 x1 + 6.0 x2 + 10.0 x3 is at most 2,400, in thousands of pounds of raw milk per week. A thousand gallons of fluid milk uses about 8,600 pounds of raw milk, a thousand cases of yogurt about 6,000 pounds and a thousand pounds of cheese about 10,000 pounds.
2. Pasteurizer: 1.0 x1 + 0.8 x2 + 0.5 x3 is at most 300 hours per week.
3. Packaging line: 1.2 x1 + 2.0 x2 is at most 420 hours per week; cheese is packaged on separate equipment.
4. Cheese vats: 3.0 x3 is at most 120 vat hours per week.
5. Yogurt demand: x2 is at most 90 thousand cases per week, the sales team's estimate of what customers will take.
6. Fluid milk contract: x1 is at least 120 thousand gallons per week.
Non-negativity: x1, x2 and x3 are each at least zero. Hillier and Lieberman (2015) note that a model's usefulness depends on whether its constraints capture the limits that actually bind in practice, so each was checked with the plant manager before solving.
What the Solver Returned
The model was solved with a linear programming routine written in Python; Excel Solver's simplex method, the algorithm Dantzig (1963) developed for exactly this kind of problem, returns the same answer. The optimal plan is to produce about 169.8 thousand gallons of fluid milk, 90 thousand cases of yogurt and 40 thousand pounds of cheese each week, for a weekly contribution of about $210,802.
| Constraint | Limit | Used by the plan | Slack | Binding? |
|---|---|---|---|---|
| Raw milk (thousand lb) | 2,400 | 2,400 | 0 | Yes |
| Pasteurizer hours | 300 | 261.8 | 38.2 | No |
| Packaging line hours | 420 | 383.7 | 36.3 | No |
| Cheese vat hours | 120 | 120 | 0 | Yes |
| Yogurt demand (thousand cases) | 90 | 90 | 0 | Yes |
| Fluid milk contract (thousand gallons, minimum) | 120 | 169.8 | 49.8 above minimum | No |
The Plan in a Supervisor's Terms
Read as a production plan, the answer says three things. First, make all the yogurt the market will take, 90 thousand cases a week, because yogurt earns the most contribution for each pound of raw milk it uses. Second, run the cheese vats at full capacity, 40 thousand pounds a week, the most the vats can produce in 120 hours. Third, send all the remaining milk to fluid milk, about 169.8 thousand gallons a week, which comfortably meets the grocery contract's minimum of 120 thousand gallons.
The plan also shows which limits matter. Raw milk, cheese vat time and yogurt demand are all used up; the plant could earn more if it had more of any of them. The pasteurizer and packaging line each have about 36 to 38 hours of spare time a week, so adding capacity there would earn nothing under current conditions. A supervisor who wanted to raise contribution should look at milk supply, cheese vats or the yogurt market, not at the pasteurizer or packaging line.
The plan also has a practical scheduling meaning. Running the cheese vats at their full 120 hours means the cheese room works every available hour, leaving no margin for a vat breakdown; the plant should keep maintenance windows outside production hours or accept that any lost vat hour costs contribution. Producing yogurt at the demand ceiling means the sales forecast is doing real work in the model, and if actual orders fall short of 90 thousand cases, the freed milk should go to fluid milk, which the model shows is the next-best use once cheese is at capacity.
If the Model Had Come Back Infeasible
Had the fluid milk contract required, for example, 300 thousand gallons a week, the raw milk supply could not meet it alongside any yogurt or cheese at all, and the solver would report that no feasible plan exists. An infeasible result is information: it means the commitments and resources contradict each other, and one of them must change. Render et al. (2018) recommend checking such results by relaxing constraints one at a time to find which combination causes the conflict. In this case the model is feasible with room to spare on the contract. The next unit examines the solver's sensitivity report to answer a purchasing question about buying extra raw milk. Northwoods Dairy Cooperative and its coefficients are composites created for this course.
References
Dantzig, G. B. (1963). Linear programming and extensions. Princeton University Press.
Hillier, F. S., & Lieberman, G. J. (2015). Introduction to operations research (10th ed.). McGraw-Hill Education.
Render, B., Stair, R. M., Jr., Hanna, M. E., & Hale, T. S. (2018). Quantitative analysis for management (13th ed.). Pearson.
What the BU 407 Unit 4 instructions ask for
For Unit 4, students formulate and solve a linear programming model for a business problem. Most versions want decision variables defined in words with units and time periods, an objective function stated in money, every constraint written with the resource it limits, non-negativity stated, the model solved in Excel Solver or similar software and the solution interpreted in operational terms. Many rubrics ask which constraints are binding and what the plan means for the people who would carry it out. Instructors expect the algebra to appear even when a solver does the work. Plan on a few pages holding the formulation and a solution table, with APA 7 citations to the course text and an operations research reference. Check the solution against every constraint before reporting it, since a solver will happily return an answer to a model that was typed incorrectly.
How the BU 407 Unit 4 example is put together
The sample describes the plant's weekly choice and the limits that apply. A variables section defines three quantities with units and time periods. The objective function states contributions per unit and what they include. Six constraints are listed with coefficients explained, followed by non-negativity and a note that constraints were checked with the plant manager. The solver section reports the optimal plan and a table of each constraint's use, slack and binding status. The interpretation section turns the solution into three instructions for the supervisor and draws scheduling implications, such as which lines run full and which have spare hours. A final section shows what an infeasible version would look like and how to diagnose it.
Where the points sit in the BU 407 Unit 4 rubric
The rubric for the model summary typically weighs variable definitions, the formulation, the solution and its interpretation. Variable definitions earn full points when each names a unit and a time period. The formulation is judged by a correct objective, constraints that match the problem and non-negativity. The solution scores highest when it is reported with each constraint's use and slack and checked for feasibility. Interpretation carries heavy weight and rewards translating the answer into actions and identifying which resources limit the plan, as the sample does. Instructors also credit a short discussion of infeasibility and how to diagnose it. Clean presentation and citations complete the grade.
BU 407 Unit 4 help before you submit
The most common Unit 4 mistake is defining variables with symbols only, so readers cannot tell whether x1 is gallons, dollars or a rate. Another is writing constraints in inconsistent units, such as mixing pounds and thousands of pounds. Students often report the solver's output without saying what it means for production. Some forget non-negativity or a minimum commitment such as a contract. Others do not identify binding constraints, which is where the useful insight lives. Finally, read the answer back in plain words; a supervisor needs to know what to make, not the value of x3. A plan that cannot be explained in a sentence or two probably has not been understood yet.
Get a BU 407 Unit 4 example written to your instructions
Send the Unit 4 instructions with your BU 407 rubric and the problem data the assignment provides. We write a custom example that defines every variable with its units, states constraints in words and symbols, describes the sheet layout and reads the solution back as a workable plan. First custom sample free, back in 24 to 48 hours.
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BU 407 Unit 4 questions, answered
Do I still show the algebra if the solver did the work?
Almost always, and the instructions will normally say so outright. The algebraic model is what proves you understood the problem, while the solver only proves the sheet was filled in. Write the objective and constraints out, then show the layout and the output beside them. Where a section asks for output alone, label every figure so nothing has to be inferred.
What if the model comes back infeasible?
Report it and diagnose it rather than deleting a constraint until something solves. Infeasibility means the limits you wrote cannot all hold at once, so name the pair that conflicts and say which one the case would actually relax. A summary that finds and explains a conflict often scores above one that quietly removed it.
How many constraints does the model need?
As many as the situation genuinely imposes, and no decorative ones. Follow any count the instructions set. Where the choice is yours, include the resources that could realistically bind and leave out limits so generous they never matter, since an inactive constraint adds length without adding anything a solver or a marker can use.
Where can I find a free BU 407 Unit 4 sample paper?
The finished BU 407 Unit 4 model summary, from variables to the production plan, is posted above for free. If your optimization problem differs, send the case and its figures, and a free custom model summary will be written for you, generally within 48 hours.
Do I still show the algebra in BU 407 Unit 4 if the solver did the work?
Yes. Instructors grade the formulation, and the solver's output is meaningless without it. The sample writes the objective and every constraint in words and symbols before reporting what the solver returned.