| Course | BU 407 Quantitative Decision-Making |
|---|---|
| Unit | Unit 5 |
| Paper type | Sensitivity table analysis |
| Length | About 980 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 5
$48.84 for Each Extra Thousand Pounds of Milk, Up to a Point: A Sensitivity Analysis That Answers a Dairy Plant's Purchasing Question With a Figure
Student Name
Business, Herzing University
BU 407: Quantitative Decision-Making
Instructor Name
March 18, 2026
$48.84 for Each Extra Thousand Pounds of Milk, Up to a Point: A Sensitivity Analysis That Answers a Dairy Plant's Purchasing Question With a Figure
The Question
A neighboring cooperative has offered to sell Northwoods' Eau Claire plant additional raw milk at $30 per thousand pounds above Northwoods' standard price, in any quantity up to 400 thousand pounds a week. The plant manager wants to know whether to accept, and if so, how much to buy. This analysis answers from the sensitivity report for the product mix model in Unit 4, which maximized weekly contribution from fluid milk, yogurt and cheese subject to limits on raw milk, pasteurizer hours, packaging hours, cheese vat hours, yogurt demand and a fluid milk contract.
The Constraint Table
The table reproduces the constraint section of the sensitivity report. For each constraint it shows the final value used by the optimal plan, the shadow price, the right-hand side and the allowable increase and decrease. The shadow price is the change in the objective, weekly contribution in dollars, for each one-unit increase in the right-hand side, and it holds only within the allowable range (Hillier & Lieberman, 2015).
| Constraint | Final value | Shadow price | Right-hand side | Allowable increase | Allowable decrease |
|---|---|---|---|---|---|
| Raw milk (thousand lb) | 2,400 | $48.84 | 2,400 | 260.0 | 428.0 |
| Pasteurizer hours | 261.8 | $0 | 300 | Unlimited | 38.2 |
| Packaging line hours | 383.7 | $0 | 420 | Unlimited | 36.3 |
| Cheese vat hours | 120 | $137.21 | 120 | 128.4 | 78.0 |
| Yogurt demand (thousand cases) | 90 | $856.98 | 90 | 31.2 | 90.0 |
| Fluid milk contract minimum (thousand gallons) | 169.8 | $0 | 120 | 49.8 | Unlimited |
Reading the Binding Constraints
Raw milk is binding, with a shadow price of $48.84. Each additional thousand pounds of raw milk, at Northwoods' standard cost, would raise weekly contribution by $48.84, because the plan would send it to fluid milk, which earns $420 for every 8.6 thousand pounds it consumes, and $420 over 8.6 gives $48.84. The price holds for an increase of up to 260 thousand pounds. Beyond that, the packaging line, which currently has 36.3 hours of slack, becomes full, because fluid milk uses 1.2 packaging hours per thousand gallons; 36.3 hours divided by 1.2 is 30.2 thousand gallons, which requires 260 thousand pounds of raw milk.
Cheese vat hours are binding, with a shadow price of $137.21 per hour, valid for an increase of up to 128.4 hours. The figure is less than cheese's full contribution of $300 per vat hour because each extra thousand pounds of cheese uses 10 thousand pounds of raw milk that would otherwise have become fluid milk; the shadow price is the net gain after that trade.
Yogurt demand is binding, with a shadow price of $856.98 per thousand cases, valid for an increase of up to 31.2 thousand cases. That is yogurt's contribution of $1,150 less the fluid milk given up for its raw milk, 6.0 thousand pounds times $48.84. The sales team would add $857 of weekly contribution for every additional thousand cases it could sell, within that range.
The pasteurizer, packaging line and fluid milk contract are not binding, and their shadow prices are zero. Adding pasteurizer or packaging hours, or relaxing the contract minimum, would earn nothing under current conditions.
The allowable decreases matter too. Raw milk's shadow price holds for a decrease of up to 428 thousand pounds, which means a drop in member deliveries of that size would cost $48.84 in contribution for every thousand pounds lost, taken entirely out of fluid milk, until fluid milk falls to the 120 thousand gallon contract minimum. Beyond that point the plant would have to cut yogurt or cheese to honor the contract, and each thousand pounds lost would cost more.
Answering the Purchasing Question
The offered milk costs $30 per thousand pounds more than the standard price already built into the model's contributions. Within the allowable increase, each thousand pounds is worth $48.84, so the net gain is $48.84 minus $30, or $18.84 per thousand pounds. The plant should therefore buy up to 260 thousand pounds a week. At that quantity, the net gain is about 260 times $18.84, or $4,898 a week. Buying more than 260 thousand pounds would add nothing, because the packaging line would then be full and extra milk could not be turned into more product; the model, re-solved with 300 thousand pounds added, shows exactly the same contribution as with 260.
The answer is a figure, not an opinion: buy 260 thousand pounds a week and gain about $4,900 a week, or roughly $255,000 a year.
The same reasoning sets a ceiling on what the plant should ever pay. Any premium below $48.84 per thousand pounds makes the purchase worthwhile up to 260 thousand pounds; a premium above $48.84 makes it a loss at any quantity. If the neighboring cooperative raised its premium to $50, the right answer would change from buy 260 to buy nothing.
Shadow Price Versus Reduced Cost
The constraint table answers questions about resources; the variable table, with its reduced costs, answers questions about products. A reduced cost tells how much a product's contribution would have to improve before it would enter the optimal plan, or, for a product already in the plan, how much its contribution could change before the plan would shift. In this model all three products are produced, so their reduced costs are zero; the purchasing question depends only on the constraint table. Winston (2004) stresses that confusing the two reports is a common error, since a shadow price values an extra unit of a resource while a reduced cost values a change in a product's contribution.
Limits of the Answer
Three cautions apply. First, the shadow price holds only while the rest of the model is unchanged; if yogurt demand or cheese vat capacity changed at the same time, the value of extra milk would change too. Second, the model assumes contributions are constant per unit, so it does not capture discounts the grocery chain might demand for extra fluid milk volume. Third, the answer beyond the allowable increase was checked by re-solving rather than assumed; a re-solve is the only safe way to evaluate changes outside the range. Northwoods Dairy Cooperative, the offer and all coefficients are composites created for this course (Render et al., 2018).
References
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.
Winston, W. L. (2004). Operations research: Applications and algorithms (4th ed.). Thomson Brooks/Cole.
What the BU 407 Unit 5 instructions ask for
The Unit 5 assignment asks students to interpret a sensitivity report from a solved linear program and use it to answer a business question. Most versions want the constraint table reproduced, the meaning of each column explained, shadow prices on binding constraints interpreted with their allowable ranges and a specific question, such as whether to buy more of a resource, answered with numbers. Many rubrics also ask about reduced costs and what happens outside the allowable range. Instructors reward answers that check results by re-solving when a change falls outside the range. Most submissions run a few pages around the reproduced report, cite the course text in APA 7 and give the answer as a quantity and a dollar figure. A bare yes or no rarely earns full credit.
How the BU 407 Unit 5 example is put together
The sample states the purchasing question and the model it relies on. The constraint table is reproduced with a short explanation of what the shadow price and allowable range mean. The next section works through each binding constraint, explaining its shadow price with arithmetic and the reason its range ends where it does, and adds a note on allowable decreases. The purchasing answer subtracts the premium from the shadow price, multiplies by the allowable increase and confirms the range limit with a re-solve, then sets a ceiling premium. A section explains the difference between shadow prices and reduced costs, and a final section lists three limits of the answer.
Reading the BU 407 Unit 5 rubric
Herzing's rubric for sensitivity analysis usually includes accurate reading of the report, interpretation of shadow prices and ranges, the answer to the business question and verification. Accurate reading earns full points when each column is explained correctly. Interpretation is judged by shadow prices tied to the model's logic and ranges explained, not just copied. The answer scores highest when it gives a quantity and a dollar figure derived from the report, as the sample's does. Verification rewards re-solving for changes near or beyond the range, since the report alone cannot say what happens there. Instructors also credit a clear distinction between shadow prices and reduced costs. Organized tables and citations complete the grade.
BU 407 Unit 5 help: mistakes that cost points
The most frequent Unit 5 mistake is applying a shadow price to a change larger than its allowable range. Another is forgetting that shadow prices already reflect the costs built into the objective, which leads to double-counting a purchase price. Students often confuse shadow prices with reduced costs. Some papers read the report correctly but never answer the question asked, which leaves the manager exactly where they started. Others assume several changes can be evaluated by adding shadow prices without re-solving. Finally, give the answer in the manager's terms: how much to buy, what it is worth and where the value stops.
Get a BU 407 Unit 5 example written to your instructions
Send the Unit 5 instructions from your BU 407 classroom with the rubric and the solved model or report the assignment builds on. We write a custom example that marks the binding rows, pairs every shadow price with its range, keeps the two ranging questions apart and prices one real decision. First custom sample free, back in 24 to 48 hours.
More BU 407 papers and business analytics samples
- BU 407 Unit 1: Payoff table write-up
- BU 407 Unit 2: Expected value comparison
- BU 407 Unit 3: Forecast selection record
- BU 407 Unit 4: Optimization model summary
- BU 407 Unit 6: Transportation model record
- BU 407 Unit 7: Simulation output brief
- BU 407 Unit 8: Quantitative recommendation file
- BU 351 Unit 5: Predictive exercise
- BU 451 Unit 5: Aggregation and joins
- BU 453 Unit 5: Single-table queries
- BU 454 Unit 5: Distribution and relationship charts
BU 407 Unit 5 questions, answered
What is the difference between a shadow price and a reduced cost?
A shadow price belongs to a constraint and says what one more unit of that resource is worth to the objective. A reduced cost belongs to a variable sitting at zero and says how much its coefficient must improve before the model would use it. Mixing them produces advice about buying a resource when the report was talking about a product.
What happens outside the allowable increase?
The shadow price stops applying, because a different set of constraints becomes binding and the model changes shape. Past that point the honest answer is that the report cannot say, and the model has to be resolved with the new right hand side. Saying exactly that, and naming the quantity where it happens, is usually worth more than an extrapolation.
Does the analysis need the model rebuilt to check a result?
Not usually, and that is the argument for sensitivity output existing at all. Where a change sits inside the stated ranges, the report answers the question directly. Where it sits outside, or where the output looks degenerate with a zero allowable movement, resolving the model is the safe route and a line explaining why you resolved it reads well.
Where can I find a free BU 407 Unit 5 sample paper?
The complete BU 407 Unit 5 analysis, with the sensitivity table and purchasing answer, is laid out in full above, free. For an analysis of your own solver report, send the prompt with the report, and a free custom sensitivity analysis will follow, normally within two days.
What happens outside the allowable increase in BU 407 Unit 5?
The shadow price no longer holds, and the model must be re-solved to find the new value. The sample re-solves with 300 thousand pounds added and shows the contribution is the same as with 260, because the packaging line becomes the binding limit.