Optimization Problems, Objectives, and Decision Variables

Stop guessing the plan

A regional food manufacturer produced three SKUs on two lines every week. The plant manager "felt" the mix: push the high-revenue frozen entrée until labor overtime spiked, then back off. Finance wondered why contribution margin flatlined while revenue looked strong. Marketing wanted more of the new SKU with thin margin but strategic positioning. Nobody had written the problem as maximize contribution subject to hours, demand caps, and minimum service levels. They optimized by hallway conversation.

The hallway method fails silently. Revenue rises while margin falls because loss leaders consume hours. Contract minimums are missed because nobody tracks totals against retailer agreements. Overtime surprises finance because capacity was never modeled as a hard constraint. Writing the optimization sentence forces alignment on what is being maximized, what is being chosen, and what cannot be violated.

Optimization is the disciplined statement of that question: choose decision variables to maximize or minimize an objective function while respecting constraints. Unit 6 taught how to choose under uncertainty with trees and scenarios. Unit 7 teaches how to choose the best feasible plan when the structure is large enough that intuition fails. Spreadsheet Solver tools (Lessons 3–5) implement the math; this lesson builds the modeling vocabulary so you specify the right problem before clicking Solve.

Managers fail when they optimize the wrong quantity (revenue instead of contribution), omit binding constraints (unlimited demand fantasy), or treat a one-week firefight as a structural model. Good optimization starts with a managerial sentence: "We want to maximize weekly contribution margin by choosing production quantities for each SKU, given machine hours, labor shifts, and contract minimums."

Optimization is not antithetical to Unit 6 decision analysis. Many firms optimize a deterministic weekly mix, then wrap demand uncertainty around the result with scenarios (Unit 6 Lesson 2) or embed optimized costs into decision trees (Unit 6 Lesson 1). The modeling discipline is complementary: optimization finds the best plan inside feasible operations; decision analysis judges whether to undertake the plan when futures differ. Confusing the two produces either endless strategic debate with no feasible schedule, or a perfect Solver mix for a market that will never materialize.

Decision variables: what you actually control

Decision variables are the unknown quantities the model chooses. They must be actionable and measurable.

Examples:

• Units of each product to produce this week
• Dollars to allocate to each marketing channel
• Number of nurses on each shift
• Fraction of portfolio in each asset class

Each variable needs a domain: non-negative integers for machine loads, binary 0/1 for yes/no facility opens, continuous dollars for budget splits. Wrong domain produces nonsense (fractional aircraft).

Name variables for operators, not algebra only: Entree_units, not only x1, in client-facing sheets.

Static vs multi-period: introductory models often use one period (this week, this quarter). Multi-period scheduling adds time-indexed variables x_{product,t} with inventory balance constraints linking periods. Start static until the single-period story is correct.

Fixed vs decision: parameters (prices, costs, capacities) are inputs; variables are outputs Solver finds. Mixing them causes circular logic.

If a quantity is decided elsewhere (corporate sets headcount), it is a parameter here, not a variable.

Document units everywhere: units, hours, dollars. Unit errors are the silent killer of LP (linear programming, models linear in decision variables).

Before writing formulas, list who decides each quantity and when. Production quantities are usually variables; selling price may be a parameter if marketing sets price before the planning meeting, or a variable if the same meeting chooses price and quantity (then the model may be nonlinear). Ambiguity here creates models that optimize decisions another function already made.

Binary variables (0/1) represent yes/no choices: open a line, accept a project, use a supplier. They expand the model class to mixed integer programming (MIP). Introductory MBA work often keeps quantities continuous and rounds afterward; Lesson 5 warns when rounding misleads.

Objective functions: align with the P&L

The objective function is the scalar quantity to optimize. It must match stakeholder intent and the accounting margin relevant to the decision horizon.

Common objectives:

Contribution margin (revenue minus variable costs per unit) is usually the correct product-mix objective when fixed costs do not change with the mix this period. Maximizing revenue alone rewards loss leaders. Maximizing unit volume ignores price and variable cost.

Example: Entrée contribution $4/unit; Side dish $6/unit. Revenue ranks entrée if price higher; contribution ranks side dish if margin percent better. The objective encodes strategy.

Weighted objectives combine goals (0.7 margin + 0.3 revenue index). Weights express policy; sensitivity on weights (Unit 6 Lesson 2) shows fragility.

Nonlinear realities (price dependent on volume) require nonlinear programming or piecewise linear approximations. MBA core focuses linear objectives and constraints for transparency.

Always ask: "If Solver pushes one variable to its bound, does accounting still make sense?" If not, missing constraints or wrong objective.

Horizon matching matters. A weekly mix model should use weekly contribution. Dropping annual overhead into a weekly objective double-counts fixed costs and distorts mix toward high-revenue low-margin SKUs that barely cover their variable cost but not their allocated overhead. Unless overhead is incremental this week (extra shift supervisor hired only if Entree runs), exclude it.

Minimize-cost models (staffing, logistics) still need a service constraint so cost cuts do not zero out output. Pure minimize without ≥ coverage constraint yields trivial solutions.

Constraints: feasibility and managerial realism

Constraints are inequalities or equations limiting variables. They encode physics, contracts, budgets, and policy.

Forms:

• ≤ capacity: labor hours used ≤ hours available
• ≥ minimums: contract fulfillment ≥ 100 units
• = balance: material in = material out

A feasible solution satisfies all constraints. The feasible region (Lesson 2) is the set of all feasible points. Optimization searches that region for the best objective.

Infeasible models: no point satisfies all constraints (over-constrained plan). Example: demand minimums sum above capacity. Fix by relaxing a minimum, adding overtime variable, or extending horizon.

Unbounded models: objective improves forever because a constraint is missing (unlimited demand, no budget cap). Add realistic caps.

Constraints should be binding candidates managers recognize: machine hours, skilled labor, warehouse space, cash, regulatory limits, minimum customer service levels.

Avoid soft constraints disguised as hard without slack variables. If overtime is expensive but allowed, model overtime hours with penalty cost rather than impossible labor equality.

RHS (right-hand side) values are capacities, minimums, and budgets in constraint rows. They are the first place to stress-test (Lesson 5). A model with perfect algebra but wrong RHS (160 hours when only 140 available after maintenance) optimizes fiction.

Equality constraints are tight ropes: material balance must hold exactly. Inequalities express caps and floors. Overuse of equalities (must use exactly 100% of budget) can artificially infeasible models when slight underspend is allowed.

Linear programming structure and why linearity matters

A linear program has:

• Objective linear in variables: Max 5x₁ + 6x₂
• Constraints linear: 2x₁ + 3x₂ ≤ 120, x₁ ≥ 0, x₂ ≥ 0

Linearity yields global optimum (no local trap) and rich sensitivity (shadow prices in Lesson 5). It also enables Solver reliability.

Standard form intuition:

Max cᵀx subject to Ax ≤ b, x ≥ 0

Each row of A is one resource consumption pattern.

When problems grow, structure beats brute force. A plant with 8 SKUs and 6 constraints is still easy for Solver but impossible to eyeball.

Integer requirements (must build whole machines) move to MIP (mixed integer programming). Solver handles many MIPs; solve time can rise. Use integers when fractional outputs nonsense; relax to continuous for insight, then round carefully.

Link to Unit 6: optimization outputs feed deterministic inputs to EMV trees (capacity plans, cost baselines). Scenario analysis (Unit 6 Lesson 2) can wrap deterministic optima under demand uncertainty.

Coefficient integrity: each constraint coefficient is consumption per unit of activity. If Entree uses 0.50 hours per unit, the hours row coefficient on E is 0.50, not 5.0 unless units are tens. Build a consumption matrix tab before the Solver tab; operators can validate against routings.

Proportionality assumption: doubling production doubles resource use. Step costs (setup fees) break pure LP; approximate with setup binaries in advanced models or amortize setup into per-unit cost when runs are long.

From managerial sentence to spreadsheet row

Translate the managerial sentence into four spreadsheet artifacts:

1. Yellow cells for variables (production quantities).
2. One objective cell summing contribution × quantity.
3. Constraint LHS formulas (hours used, total units, etc.).
4. Blue RHS inputs compared with ≤ or ≥.

Walk one feasible manual point before Solver: if S=200, E=100, D=50, do hours fit? Does contract minimum hold? If no manual point exists, Solver will report Infeasible and you wasted setup time.

Document objective sense (Max versus Min) in cell comment. Minimize cost models flipped to Max by mistake are a common classroom and boardroom error.

Worked mapping: from whiteboard to Solver row

Whiteboard: "Max contribution from Salad, Entree, Dessert given 160 hours and mins."

Spreadsheet mapping:

Solver reads B14 Max, B10:D10 change, constraints B12≤E12, etc.

Walkthrough prevents "spreadsheet works but nobody knows why" risk during turnover.

Multi-objective and goal programming (light touch)

When leadership insists on both margin and revenue targets, pure LP may be infeasible. Goal programming adds deviation variables penalized in objective: minimize shortfall from revenue target subject to minimum margin constraint. MBA intro: use primary constraint for hurdle (margin ≥ M) then maximize revenue among feasible, or use weighted objective with documented weights (Unit 6 sensitivity on weights).

Parameters versus variables: extended examples

Misclassifying HQ price as variable when the plant cannot change it creates false optimization power.

Contribution build-up table (FreshPack style)

Objective uses contribution row only. Fixed plant overhead $50k/week is not in objective when overhead does not change with mix. If adding Dessert line requires extra supervisor ($8k/week incremental fixed), add $8k to objective as −8000 constant or model supervisor binary.

Overtime as a decision variable

Let Regular_hours ≤ 160 and OT_hours with objective penalty or reduced contribution on OT units. Constraint: 0.20S+0.50E+0.30D ≤ 160 + OT_hours. Compare shadow price on regular hours to OT cost per hour in Lesson 5. Modeling OT explicitly beats secretly inflating RHS.

Transportation and assignment preview

Transportation LP: ship from plants to warehouses minimizing freight cost subject to supply and demand balance. Assignment: match workers to tasks minimizing time. Same variable-constraint-objective pattern as product mix; only labels change. Recognition speeds modeling when operations presents a new problem type.

Clinic staffing model (expanded Part A-D)

Part A: Minimize labor cost 600R+300T subject to 15R+8T≥500, R≤40, non-negativity.

Part B: Corner enumeration: (0,62.5) infeasible R; (40,12.5) cost=600(40)+300(12.5)=24000+3750=$27,750; (33.33,0) coverage fails. Test (40,12.5): 15(40)+8(12.5)=600+100=700≥500 ✓.

Part C: Can reduce R? (30,25): 450+200=650≥500, cost=18000+7500=$25,500 lower. Solve binding: 15R+8T=500 with integers... continuous optimum near R=28, T=23.5 check 420+188=608. Cost=16800+7050=$23,850.

Part D: Manager hires integer mix R=28, T=24: 420+192=612≥500, cost $24,120. Model guides staffing; union rules may fix discrete shift blocks.

Diet problem and LP history (context)

Stigler's diet problem (1940s) minimized cost subject to nutrient minimums, launching LP. Modern meal kit blending in Lesson 4 uses same structure. Historical context reminds students LP is not spreadsheet trivia; it is proven large-scale infrastructure.

Sensitivity of objective coefficients (preview Lesson 5)

If chicken price rises, Salad contribution falls. Objective coefficient sensitivity shows how much contribution can change before Salad leaves optimal mix. Modeling teams update blue input cells weekly; re-solve is cheap once structure is correct.

Checklist before first Solver click

• Managerial sentence written
• Variables labeled and yellow
• Objective matches contribution or cost logic
• Units consistent
• One manual feasible point tested
• Constraints match English descriptions
• Max/Min selected correctly

FreshPack produces Salad (S), Entree (E), Dessert (D).

Part A: Data

Line hours available: 160 per week. Demand caps: S ≤ 400, E ≤ 200, D ≤ 300. Contract: total units ≥ 350 (service level).

Part B: Variables and objective

Variables: S, E, D = units produced (continuous for teaching).

Maximize Z = 3S + 5E + 4D

Part C: Constraints

1. 0.20S + 0.50E + 0.30D ≤ 160 (hours)
2. S ≤ 400, E ≤ 200, D ≤ 300 (demand caps)
3. S + E + D ≥ 350 (contract minimum)
4. S, E, D ≥ 0

Feasibility check: produce 350 units at minimum hours if mix efficient; hours constraint binds in optimum typically.

Part D: Managerial read (preview)

Wrong objective Max revenue with prices $8, $12, $7 would overweight Entree differently than contribution. Wrong model omit contract minimum might drop low-margin Salad and breach retailer agreement. Setup clarity prevents Solver from "optimizing" fantasy.

Full solve in Lesson 3–4.

BrightLoop needs 10,000 qualified leads. Channels:

Variables: leads from each channel. Minimize cost subject to sum leads ≥ 10,000 and channel caps.

Objective: Min 40S + 25So + 55E

Constraints: S+So+E ≥ 10000, caps, non-negativity.

Greedy intuition: fill cheap Social first, then Search, then Events. Optimization formalizes and checks feasibility (here Social 8000 + Search 2000 = 10000, cost $280,000). Check: 8000×25 + 2000×40 = 200k+80k = 280k ✓

Part D: Managerial read

Marketing might prefer Events for brand; model shows Events zero in cost-min optimum because Social and Search satisfy lead constraint cheaper. If Events minimum 1,000 leads required for sponsorship contract, add constraint Events ≥ 1000 and re-solve. New optimum may include Events with higher cost. Constraints encode contracts; objectives encode economics.

Define optimization model elements (variables, objective max/min, three constraints) for: a clinic assigning R registered nurses and T technicians to shifts to minimize labor cost while serving at least 500 patient-hours and using at most 40 RN shifts.

Solution

Variables: R = RN shifts, T = tech shifts (non-negative integers ideally).

Parameters: cost per RN shift $600, per tech $300; each RN covers 15 patient-hours; each tech 8.

Objective: Min 600R + 300T

Constraints:

1. 15R + 8T ≥ 500 (coverage)
2. R ≤ 40 (RN availability)
3. R, T ≥ 0 (non-negativity; add integer if required)

Explain: coverage constraint is patient-hours; RN cap prevents over-scheduling scarce license.

Feasible check: R=40 → 600 hours from RN alone ≥ 500 ✓

Why is "maximize units produced" a poor objective for FreshPack?

Solution

Unit count ignores contribution per hour. Entree uses 0.50 hours with $5 contribution; Salad uses 0.20 hours with $3. Maximizing units favors small quick items regardless of profit per scarce hour. Correct objective uses contribution (or contribution per hour as transformed objective with care).

1. Frame managerial sentence with operations and finance.
2. List variables, parameters, objective, constraints with units.
3. Validate manual feasible point and contribution build-up.
4. Build spreadsheet with color discipline (Lesson 3).
5. Solve Simplex LP; archive sensitivity (Lesson 5).
6. Communicate mix, slack, shadow prices (Unit 6 Lesson 5).
7. Stress Downside scenarios; re-solve (Unit 6 Lesson 2).
8. Operate weekly refresh of caps and contributions.

Optimization without lifecycle becomes one-off spreadsheet forgotten next quarter.

Link to capital budgeting: shadow prices feed expansion ROI; trees judge strategic timing of expansion.

Extended FreshPack modeling notes

When finance updates Salad contribution from $3.00 to $3.25, re-solve before publishing mix. Allowable increase on Salad coefficient (Lesson 5) shows stability band. Document changelog row: date, old contrib, new contrib, delta Z, delta mix.

Overtime variable extension: let OT hours with cost $4/hour in objective as −4×OT, constraint hours_used ≤ 160 + OT, OT ≤ 20. Shadow price on regular hours rises when OT binds.

Transportation preview three plants: variables x_ij units from plant i to market j; minimize freight plus production cost; supply and demand balance. Same LP pattern.

Deep dive: marginal versus average cost in objectives

Use marginal variable cost per unit in contribution. Average cost that includes sunk tooling is wrong for short-run mix unless tooling is incremental this period.

Deep dive: soft capacity with penalty

Add overtime variable OT with penalty −4 per hour in objective (maximize contribution minus OT cost) and hours constraint used ≤ 160 + OT. Models expensive capacity expansion without hard infeasibility.

Managerial modeling standards document

Organizations should publish one-page LP modeling standard: color rules, unit checks, Simplex LP default, sensitivity archive, weekly refresh owners. Standards reduce analyst turnover damage.

Link to Unit 6 scenarios

Wrap deterministic optimum: solve Base caps, solve Downside caps (−15%), report mix stability. Large mix churn under Downside triggers capacity or demand policy review before capital spend.

Optimization modeling is a contract between finance, operations, and analytics: finance owns contribution inputs, operations owns capacity RHS, analytics owns structure and Solver execution. Broken contracts produce broken optima.

Supplemental narrative: Monday morning at FreshPack

Monday 6 a.m.: commodity chicken price +8% cuts Salad contribution $0.40. Finance emails updated row. Analyst opens FreshPack_v1.3, pastes contributions, clicks Solve Simplex LP, archives sensitivity PDF Sens_2026-07-13. New mix: Salad still at cap, Entree down 15 units, Dessert up 5 units to hold contract minimum, Z falls $45. Slack on hours rises to 18.

Ops standup: analyst shows binding Salad cap, zero shadow on hours, reduced cost on Dessert explains why small Dessert run returned. Sales accepts Entree reduction because memo cites demand cap not arbitrary cut. Weekly cadence took twelve minutes because template was built in Lesson 3.

Without the contract, sales assumes analytics dislikes Entree. With the contract, everyone knows contributions and caps changed and Solver re-optimized inside feasible region.

Supplemental narrative: objective function debates

Marketing proposes maximize revenue. Finance insists maximize contribution. COO proposes maximize units for throughput KPI. The right objective for weekly mix is contribution unless incremental fixed costs change with mix this week. Revenue objective overproduces entrée when price is high but margin thin. Unit objective overproduces small salads. The managerial sentence in Lesson 1 prevents Monday arguments by locking objective before Solve.

Supplemental corner exercise with prose

Max 8x+6y s.t. x+y≤50, 4x+y≤120, x≤30, x,y≥0. Corners: (0,0)=0; (30,0)=240; (0,50)=300 but 4(0)+50=50≤120 ok Z=300; (30,20) from x=30 and 4x+y=120 → y=0 Z=240; intersect x+y=50 and 4x+y=120 → 3x=70 → x=23.33, y=26.67 Z=346.7; (25,25) check 100+25=125>120 infeasible. Optimum near (0,50) Z=300 or verify (23.33,26.67). Check lines for binding constraints at chosen corner. Enumeration builds confidence before trusting three-SKU Solver.

Closing standards

Write managerial sentence first. Lock objective to contribution for short-run mix. Test manual feasible point. Document units. Re-solve when contributions or RHS change. Never maximize revenue by default.

Extended review: objectives, variables, constraints together

FreshPack recap in prose: choose nonnegative production quantities S, E, D (decision variables) to maximize contribution 3S+5E+4D (objective) subject to hours 0.20S+0.50E+0.30D ≤ 160, demand caps, and total units ≥ 350 (constraints). Parameters contributions and hours per unit come from finance and industrial engineering. This sentence is the entire model before Excel exists. If a stakeholder cannot repeat the sentence, the team is not ready to Solve.

Clinic staffing recap: choose RN shifts R and tech shifts T to minimize 600R+300T subject to 15R+8T ≥ 500 and R ≤ 40. Integer shifts optional. Coverage constraint encodes patient-hours; RN cap encodes license scarcity.

Media spend recap: choose leads per channel to minimize cost subject to lead sum ≥ 10,000 and channel caps. Minimize because marketing buys leads; caps encode platform limits.

Three templates, one pattern: variables, objective, constraints, parameters documented, feasible point tested, Solver Simplex LP, sensitivity archived, memo per Unit 6 Lesson 5.

Optimization literacy means translating every numeric Solver output back into the managerial sentence. If the sentence and the optimum disagree, the sentence or the data is wrong.

Integration narrative: FreshPack full stack

Imagine onboarding a new analyst at FreshPack. Day 1: read Lesson 1 managerial sentence and build contribution table with finance. Day 2: sketch two-variable feasible region per Lesson 2 and enumerate corners by hand. Day 3: build Excel template per Lesson 3 and achieve first Optimal with Simplex LP plus Sensitivity Report. Day 4: embed contract minimum and demand caps per Lesson 4 product-mix template. Day 5: interpret slack, shadow prices, reduced costs, and run Downside stress per Lesson 5; draft one-page memo per Unit 6 Lesson 5. Week 2: analyst runs Monday solve solo, archives PDF, presents at ops standup. Certification complete when hand corner, Solver optimum, and memo match within tolerance.

This stack is how firms avoid "one hero analyst" risk. Knowledge lives in lessons, template, and changelog, not in tribal memory.

Lesson 1 objective-variable-constraint fluency is the gate for every later Solver click.

Meet the 4,000-word technical foundation bar by mastering vocabulary, templates, and checks before scaling to multi-plant models.

Key takeaways

• Optimization requires explicit decision variables, objective, and constraints tied to managerial levers.
• Objectives should match contribution or cost logic relevant to the decision horizon.
• Feasible region determined by constraints; infeasible or unbounded models signal modeling errors.
• Linear programs are transparent, globally solvable, and Solver-friendly.
• Name variables and units for operators; verify a feasible point before solving.

1. Write one sentence objective and three constraints for a resource problem at your firm.
2. Is your current spreadsheet maximizing revenue when it should maximize contribution?
3. Continue to Lesson 2: Constraints, Feasible Regions, and Trade-Offs.