From a Description to a Mathematical Model
A design or resource problem often begins in words: a workshop has limited labor and materials, a mixture must meet a target concentration, or a plan must stay under a budget. To analyze such a problem mathematically, first express its choices and requirements precisely. The goal here is to build that model, not to solve it for the best design.
A basic optimization model has three parts: decision variables describing choices, an objective measuring what is to be maximized or minimized, and constraints specifying which choices are allowed. A model is useful only if its equations and inequalities represent the stated situation accurately.
Read the Wording for Mathematical Clues
Words such as “exactly,” “at most,” and “at least” indicate different mathematical relationships. An exact total is an equality. A capacity that cannot be exceeded is an upper-bound inequality. A minimum requirement is a lower-bound inequality.
| Wording | Typical mathematical form |
|---|---|
| Exactly \(T\) units | \(q=T\) |
| At most \(C\) units; no more than \(C\) | \(q\le C\) |
| At least \(M\) units; no fewer than \(M\) | \(q\ge M\) |
| Between \(L\) and \(U\), inclusive | \(L\le q\le U\) |
| A quantity cannot be negative | \(q\ge 0\) |
These are translation patterns, not substitutes for reading carefully. For example, “use all available material” may call for an equality, whereas “material use cannot exceed the stock” calls for an inequality. The two descriptions impose different requirements.
Also distinguish a requirement from a goal. “Keep spending below the budget” is a constraint. “Minimize spending” is an objective. A budget can restrict allowable plans without requiring that every dollar be spent.
Worked Example: Planning Production With Limited Resources
Worked Example: Planning Production With Limited Resources
A workshop makes two products. Each unit of product A earns \(18\) dollars and uses \(2\) labor-hours and \(3\) kilograms of material. Each unit of product B earns \(25\) dollars and uses \(4\) labor-hours and \(2\) kilograms of material. The workshop has at most \(160\) labor-hours and \(150\) kilograms of material. It must make at least \(10\) units of B, and it can make no more than \(50\) units of A or \(35\) units of B. Both products are made in whole units. Formulate the model and check the plan of \(30\) units of A and \(20\) units of B.
Let \(x\) be the number of units of A and \(y\) the number of units of B. Since products are made in whole units, \(x\) and \(y\) must be nonnegative integers.
The earnings are \(18x\) dollars from A and \(25y\) dollars from B. Thus the total earnings, in dollars, are
and the objective is to maximize \(P\).
Labor use is \(2x+4y\) hours. Since no more than \(160\) hours are available, the labor constraint is \(2x+4y\le160\). Material use is \(3x+2y\) kilograms, so the material constraint is \(3x+2y\le150\). The stated production limits and minimum requirement give \(x\le50\), \(y\le35\), and \(y\ge10\).
The full model is therefore:
Check \(x=30,\ y=20\). Labor use is \(2(30)+4(20)=60+80=140\) hours, and \(140\le160\). Material use is \(3(30)+2(20)=90+40=130\) kilograms, and \(130\le150\). The individual bounds hold because \(0\le30\le50\) and \(10\le20\le35\); both values are integers. Earnings are \(18(30)+25(20)=540+500=1040\) dollars. Thus this plan satisfies every stated constraint and has earnings of \(1040\) dollars. This check does not establish that the plan is best; it verifies only that it is allowed and evaluates its objective.
Why a Faithful Translation Matters
A mathematical model should admit exactly the choices described as allowable in the original problem. The following simple justification explains the standard for a correct translation.
To see why both directions matter, consider two products with \(x\) and \(y\) units, where each unit uses \(a\) and \(b\) units of a resource, respectively, and the capacity is \(C\). The total resource use is \(ax+by\). The verbal requirement “use no more than \(C\)” means that total use does not exceed capacity; by the meaning of “does not exceed,” this is exactly the condition \(ax+by\le C\).
If a plan meets the verbal requirement, its resource use is at most \(C\), so it satisfies the inequality. Conversely, if the plan satisfies the inequality, its resource use is at most \(C\), so it meets the verbal requirement. Thus the inequality includes every allowed plan and excludes every plan that exceeds capacity. Replacing it with \(ax+by=C\) would fail the reverse implication: a plan using less than the capacity is allowed by the words but would not satisfy the equality.
This “if and only if” check is a useful modeling test. A missing constraint may allow a plan that violates the description. An unnecessary or overly strict constraint may rule out a plan that the description permits.
Worked Example: Blending to Meet an Exact Target
Worked Example: Blending to Meet an Exact Target
A tank must contain exactly \(100\) liters of a solution with \(30\) liters of active ingredient. One input is a \(10\%\) solution and costs \(2\) dollars per liter; the other is a \(50\%\) solution and costs \(5\) dollars per liter. Formulate a model that minimizes input cost, and check a proposed blend using \(50\) liters of each input.
Let \(x\) be the liters of \(10\%\) solution and \(y\) the liters of \(50\%\) solution. The inputs cannot have negative volumes, so \(x\ge0\) and \(y\ge0\). The final volume must be exactly \(100\) liters:
A \(10\%\) solution contributes \(0.10x\) liters of active ingredient, because \(10\%=0.10\). The \(50\%\) input contributes \(0.50y\) liters. The exact active-ingredient target therefore gives the separate equality
The cost is \(2x+5y\) dollars, so the objective is to minimize that quantity subject to both equalities and nonnegativity.
For the proposed blend, \(x=50\) liters and \(y=50\) liters. Its total volume is \(50+50=100\) liters. Its active-ingredient volume is \(0.10(50)+0.50(50)=5+25=30\) liters. Its cost is \(2(50)+5(50)=100+250=350\) dollars. Thus it meets both exact targets, has nonnegative input volumes, and costs \(350\) dollars. The calculation checks the proposed blend; it does not show that its cost is minimal.
Notice that the two equalities encode different facts. Meeting the total-volume requirement does not, by itself, guarantee the correct concentration. The ingredient equation is needed as well. Percentages must be converted to decimal fractions when used to calculate ingredient volume.
Worked Example: Meeting Minimums Under a Budget
Worked Example: Meeting Minimums Under a Budget
An organization buys two kinds of outreach packages. Package A reaches an estimated \(400\) people and costs \(300\) dollars; package B reaches \(900\) people and costs \(500\) dollars. The plan must reach at least \(12{,}000\) people and cost no more than \(9{,}000\) dollars. Staff time is limited to \(80\) hours; each A package uses \(2\) hours and each B package uses \(5\) hours. Package counts must be whole numbers. Formulate a model to maximize estimated reach, then check \(x=10,\ y=10\).
Let \(x\) and \(y\) be the numbers of packages A and B. The objective, estimated reach in people, is \(R(x,y)=400x+900y\), which is to be maximized. The minimum reach requirement is \(400x+900y\ge12{,}000\). The budget condition is \(300x+500y\le9{,}000\), with costs measured in dollars. The staff-time condition is \(2x+5y\le80\), measured in hours. Since package counts are whole and cannot be negative, \(x,y\) are nonnegative integers.
For the proposed plan, reach is \(400(10)+900(10)=4{,}000+9{,}000=13{,}000\) people, which is at least \(12{,}000\). Cost is \(300(10)+500(10)=3{,}000+5{,}000=8{,}000\) dollars, which is at most \(9{,}000\). Staff time is \(2(10)+5(10)=20+50=70\) hours, which is at most \(80\). The counts are nonnegative integers. Therefore the plan satisfies all constraints and gives an estimated reach of \(13{,}000\) people. As before, feasibility and objective value do not prove optimality.
Units as a Modeling Check
Units help reveal errors in a model. In an objective such as \(18x+25y\), if \(x\) and \(y\) count products and the coefficients are dollars per product, each term is in dollars. Adding them gives total dollars. In a resource constraint such as \(2x+4y\le160\), the coefficients are hours per product, so the left side is in hours, matching the capacity on the right.
By contrast, an equation that adds a dollar amount to a number of hours has incompatible units and cannot represent a total of either quantity. Check the units of the terms being added and make sure the two sides of an equality or inequality have the same units. For percentages, use a consistent representation: a \(50\%\) ingredient fraction is \(0.50\), not \(50\).
A Common Mistake: Turning a Capacity Into an Exact Requirement
Suppose a workshop has at most \(160\) labor-hours available. A common incorrect translation is \(2x+4y=160\). It says that all \(160\) hours must be used, which is stronger than the stated capacity. For example, the production plan \(x=30,\ y=20\) uses \(2(30)+4(20)=140\) hours. Since \(140\le160\), the plan respects the capacity, even though it does not use every available hour. The incorrect equality rejects it.
The correction is \(2x+4y\le160\). Use equality only when the description requires an exact total, such as making exactly \(100\) liters of mixture. Similarly, “at least \(12{,}000\) people” requires \(\ge\), not \(\le\). Translating the direction of each comparison correctly is essential.
A Practical Translation Sequence
When turning a description into a model, proceed in a deliberate order:
Introduce a variable for each quantity that can be chosen, and state what it measures.
Write the quantity to be maximized or minimized, including its units.
Use equalities for exact amounts, upper bounds for capacities, and lower bounds for minimum requirements.
Include nonnegativity and, when choices are indivisible, integer conditions.
Test a candidate choice against the original wording and verify that each equation or inequality compares compatible quantities.
The resulting model is a precise account of the design problem. Writing it carefully separates two tasks: first determine which choices meet the requirements, then use the objective to compare allowable choices. Keeping those tasks distinct prevents an objective from being mistaken for a constraint, or a resource limit from being mistaken for a requirement to use every resource.
Check Your Understanding
For each question, focus on what the wording requires before choosing a mathematical symbol.
- A process must produce exactly \(250\) liters. What kind of relation should represent the total-volume condition?
- A machine can run for no more than \(90\) hours. Why would setting machine time equal to \(90\) impose an extra requirement?
- A plan must deliver at least \(500\) units. Which direction should the inequality have, and what quantity belongs on its left side?
- If \(x\) counts indivisible items, what additional restriction should accompany \(x\ge0\)?
- In a resource constraint, one term is measured in hours and another in dollars. What should be checked before adding them?
