Resource Allocation and Product-Mix Models

Every hour on the wrong SKU is a hour stolen from the right one

Product-mix and resource allocation problems are the workhorse applications of linear programming in operations and finance. The pattern repeats: several activities compete for scarce resources (machine hours, labor, cash, warehouse slots, analyst time). Each activity earns a contribution or saves a cost per unit. Leadership asks how much of each activity to run this period.

When a plant runs without a mix model, sales incentives often push high-revenue SKUs while finance wonders why margin misses plan. The conflict is structural: revenue and contribution diverge when variable costs differ. Product-mix LP aligns incentives with contribution per scarce hour, not with commission schedules based on revenue alone. Implementation therefore requires both a correct model and a conversation with sales about why a capped SKU is optimal.

Lessons 1–3 gave vocabulary, geometry, and Solver mechanics. This lesson builds reusable templates for product mix, blending, multi-period inventory, and capital rationing, with emphasis on managerial interpretation rather than algebra alone. Unit 6 sensitivity (Lesson 2) and communication (Lesson 5) apply when you present mix recommendations to sales teams who fear SKU rationing.

Managers fail when they copy a mix model from a textbook but leave minimum batch constraints out, when they treat transfer prices inconsistently across plants, or when they optimize weekly mix while changeover time eats days. Good modelers iterate: start simple, validate optimum against intuition, add complexity only when slack proves it irrelevant.

Consider a contract manufacturer serving retail chains. Each chain demands minimum shelf presence for store-brand items even when national-brand SKUs earn higher margin per hour. A mix model without minimums tells production to ignore store-brand lines entirely. Finance celebrates higher contribution until account managers report breach penalties and lost renewals. The fix is not abandoning optimization. The fix is encoding contractual reality as constraints and letting the solver show the true cost of those mins in reduced contribution and binding shadow prices (Lesson 5).

Product-mix models also interact with pricing. When a demand cap binds because price is low and volume is high, the lever may be price increase rather than capacity expansion. Optimization clarifies whether the bottleneck is physical or commercial. That distinction saves capital: you do not add a line for demand you should not pursue at current margin.

Product-mix template

Standard form:

Maximize Σ (contribution_i × quantity_i)

Subject to:

• Σ (hours_{i,j} × quantity_i) ≤ capacity_j for each resource j
• quantity_i ≤ demand_i (market caps)
• quantity_i ≥ minimum_i (contracts, service levels)
• quantity_i ≥ 0 (or integer)

FreshPack (Lessons 1–3) is the canonical three-SKU example. Generalizations:

Multiple plants: index resources by plant; shipping variables link sites.

Byproduct yields: blending constraints (next section).

Effective capacity: subtract scheduled maintenance from gross hours upfront.

Always label contribution clearly: price − variable cost, excluding fixed overhead unless incremental fixed this period.

Sales may push revenue-max mix; finance pushes contribution-max. The objective encodes policy.

Build the template in a spreadsheet with explicit rows for each SKU: contribution, variable cost components, hours on Line 1, hours on Line 2, cold storage pallets, demand cap, contract minimum. Totals row computes LHS for each constraint. This layout mirrors Lesson 3 Solver setup and makes audit easy when commodity costs update midweek.

When multiple resources bind (hours and warehouse both tight), shadow prices on each (Lesson 5) rank which expansion investment pays first. A mix model is therefore a capital planning input, not only a weekly schedule.

Changeovers reduce effective capacity. If switching from SKU A to SKU B consumes four hours of cleaning, either subtract four hours from RHS for each switch (requires binary setup variables in advanced models) or net changeovers into average hours per unit when mix is stable. Ignoring changeovers makes the optimum an fantasy the floor cannot run.

Blending and proportion constraints

Blending models appear in refining, nutrition, finance (portfolio weights), and marketing mix percentages.

Example: feed mix must be at least 20% protein across tonnage. Variables are tons of ingredients; constraint linear in totals:

Protein_content = 0.30A + 0.10B

Total = A + B

Protein_content ≥ 0.20 * Total

Rearrange to linear form: 0.30A + 0.10B ≥ 0.20(A+B) → 0.10A - 0.10B ≥ 0.

Portfolio: weights sum to 1; each asset weight ≥ 0; expected return weighted sum ≥ target.

Blending introduces proportion constraints; still LP if linear after algebra.

Check units: protein percent on dry matter vs as-fed changes coefficients.

Nutrition labeling provides a consumer-facing blending case. A meal kit must deliver at least 25g protein per serving while minimizing ingredient cost. Ingredients are chicken, lentils, and cheese with different protein grams and cost per gram. Variables are grams of each ingredient; serving size fixed. Protein constraint linear; cost objective linear. Marketing may add maximum sodium proportion, another linear proportion row after rearrangement.

Finance blending appears in index fund replication: choose weights on liquid ETFs to track a benchmark while minimizing tracking error subject to weight bounds. Variables are fractions summing to 1. Regulatory constraints (no single name above X%) are linear. The product-mix logic is identical even without a factory.

Test blending models with extreme points: all ingredient A vs all B. If optimum sits in interior, both constraints and objective trade off; if at corner, one nutrient or cost dominates.

Multi-period and inventory linkage

Single-period mix ignores inventory carry and setup. Multi-period adds variables x_{i,t} and balance:

EndingInv_{i,t} = EndingInv_{i,t-1} + x_{i,t} - Sales_{i,t}

Storage caps constrain ending inventory. Holding cost enters objective (minimize variable cost including carry).

Setup binaries (MIP): produce SKU or not each period; fixed cost if positive run. MBA intro often uses fixed setup as lump subtracted outside LP or approximated.

Capital rationing across projects: binary select projects; budget constraint Σ cost ≤ B; max NPV sum. Continuous relaxation estimates upper bound; integer required for yes/no projects.

Link to Unit 6: EMV of projects can feed objective coefficients when uncertainty resolved.

Inventory balance prevents myopic mix that starves next week. Suppose SKU A is high contribution but inventory is low and replenishment takes two weeks. Add constraint ending inventory ≥ safety stock. That may force production of lower contribution SKU B today to protect service level. The shadow price on inventory balance shows value of holding stock, connecting operations to customer fill rate.

Capital rationing with binary variables is knapsack logic: limited budget, discrete projects, choose subset. Continuous LP relaxation (allow 0.3 of a project) overstates feasibility; integer Solver or enumeration for small N suffices. Report both NPV sum and strategic exclusions (Lesson 4 MCDM from Unit 6) when projects tie on financial score.

Transfer pricing and multi-plant allocation

When plants trade intermediate goods, transfer prices set internal cost coefficients. If Plant North sells substrate to Plant South at $12/unit but marginal cost is $9, South's mix model overstates substrate cost and under-produces substrate-heavy SKUs. Align transfer prices with marginal cost for short-run allocation unless corporate policy intentionally shifts margin.

Multi-plant models add variables for shipment flows. Balance constraints: production at North = domestic sales + shipments South. Capacity at each site. Shipping cost adds linear terms to objective (minimize total system cost or maximize contribution minus logistics). A manager seeing only single-plant optimum may overload one site while sister plant has slack.

Consolidated planning beats local heroes optimizing parochially. Still, publish plant-level shadow prices so local managers understand why headquarters capped their favorite SKU.

Organizational rollout and change management

Optimization changes who gets capacity. Expect sales pushback when demand cap binds a hero SKU.

Rollout steps:

1. Validate model on last week's actuals (shadow prices Lesson 5 should explain observed bottlenecks).
2. Publish assumption ledger (Unit 6 Lesson 5): contributions, hours, mins.
3. Run scenario demand ±10% (Unit 6 Lesson 2) and re-optimize.
4. Set review cadence weekly; contributions update with commodity prices.

Pair quantitative mix with pricing and marketing levers when caps bind market-side, not factory-side.

Ethical note: rationing customer orders requires customer communication; model is internal allocation, not customer promise without sales sign-off.

Demand uncertainty and rolling horizons

Deterministic mix models assume point estimates for demand caps and contributions. Unit 6 Lesson 2 sensitivity wraps the optimum: re-run Solver at demand −15% and margin −10% jointly. A robust rollout runs three optima (Base, Downside, Upside) and compares mix stability. If optimal SKU mix flips completely between scenarios, the operational plan is fragile; stage inventory or flexible capacity instead of betting one mix.

Rolling horizon practice: solve weekly with updated demand caps; compare shadow price trends. Rising shadow price on hours week over week signals impending bottleneck before stockouts.

Cross-functional sign-off

Before publishing mix to the floor, obtain sales acknowledgment on demand caps, finance on contributions, legal on contract minimums. Sign-off table in workbook prevents "we never agreed to zero Dessert" disputes.

Nutrition blending worked sketch (proportion constraint)

Campus dining minimizes meal cost subject to protein ≥ 25g and calories ≤ 600 per serving. Variables: grams chicken, rice, broccoli. Coefficients from USDA tables. Protein constraint linear in grams; calorie constraint linear. Proportion constraints (≤30% calories from fat) rearrange to linear inequalities. Optimum often uses corner mix of two ingredients; third zero with positive reduced cost.

Retail planogram as product mix

Retail shelf space is hours analog: facings per SKU limited, margin per facing per week is contribution coefficient, minimum facings per contract mirror contract minimums. Same LP template applies though units are facings not factory units.

### Finance portfolio as allocation

Maximize expected return subject to Σ weights = 1, each weight ≤ cap, sector ≤ 40%. Variables are fractions; objective linear in expected returns; constraints linear. Blending and product mix are one math story with different labels.

Multi-plant numerical sketch

North plant can make A and B; South plant B and C. Variables: production at each site. Balance: South B exports to North demand. Transport cost $0.10 per unit per mile in objective. Capacity constraints per plant. Optimum may split B across plants even when North has slack if freight saves contribution net.

Check feasibility: total demand for B equals sum of North and South B production.

Seasonal demand caps

Quarterly caps differ: Q4 Dessert cap 500, Q1 cap 200. Index variables D_t for t=1..4 with inventory balance if allowed. Seasonal models explain why single-week FreshPack is snapshot; annual planning chains snapshots with inventory.

Service level as constraint not objective

Call centers sometimes maximize handled calls subject to average wait ≤ 60 seconds. Wait is nonlinear in staffing; approximations linearize or use simulation. MBA LP uses simplified minimum staff constraint per interval derived from historical table.

Rationing communication template

When Solver zeros a SKU customers expect, sales email template: "SKU C allocation 0 this week due to line hours binding on A and B contract minimums; projected C return week 12 per model v1.3." Analytics supplies numbers; sales owns customer message.

Apex Foods re-solve narrative (expanded)

Start from Lesson worked optimum A=500, B=100. Commodity shock cuts B contribution from 6 to 5.5. Re-solve: B hours drop toward contract floor; C may enter if A+D minimum forces low-hour filler. Document mix pivot in weekly ops note. Shadow price on hours may rise from 0 to positive when B contribution falls and hours become binding.

Blending quality control

After solve, lab verifies protein or sulfur content on composite sample. If blend nonlinearities exist (mixing losses), adjust coefficients 2% and re-solve. Optimization plus QC loop is standard in process industries.

Energy and carbon caps

Carbon constraint Σ carbon_i x_i ≤ Cap linear in output. Dual price on carbon is implicit carbon price for internal chargeback. ESG reporting links to same constraint row finance already uses.

Weekly operating cadence (product mix)

Deviations (machine down) trigger intra-week re-solve.

Contract manufacturer case (narrative)

CM produces for Brand A (minimum 10k units) and Brand B (margin priority). Shared line 220 hours. Model sets mins and caps; shadow price on hours decides whether to accept Brand C spot order. Spot order accepted only if incremental contribution per hour exceeds shadow price.

Apex runs four SKUs on 200 hours with contributions and hours below.

Contract: A+D ≥ 400 total units (retailer shelf commitment).

Part A: Model

Max 4A+6B+5C+3D

s.t. 0.25A+0.50B+0.40C+0.20D ≤ 200, caps, A+D ≥ 400, non-neg.

Part B: Solver-style optimum (representative)

Wait: A+D=500≥400 but D=0. Check hours: 0.25(500)+0.50(100)=125+50=175, slack 25.

Contribution: 4(500)+6(100)=2600

Try improving: add D low hours... objective may shift; full simplex yields (verify):

Better corner: A=500, B=60, C=0, D=0 → hours 125+30=155, Z=2300 lower.

Optimal likely A=500, B=100 as above Z=2600 ✓

Part C: Checks

Hours 175 ≤ 200 ✓; A cap bind; contract 500≥400 ✓

Part D: Managerial read

B limited by hours after A maxed; C and D idle though D helps contract with low hours: if D had positive role, minimum A+D forces D>0; here A alone satisfies contract. Sales on C should note zero production due to hour opportunity cost vs B.

Shadow price preview (Lesson 5): if hours constraint binds when A is below cap, shadow price on hours ranks whether overtime or Saturday shift is worth it. Here hours slack 25 at optimum, shadow $0 on hours: do not buy overtime to push B further until demand or contract changes.

Three projects:

Budget $9M. Binary variables y_i ∈ {0,1}.

Max 12y1+9y2+7y3 s.t. 5y1+4y2+3y3 ≤ 9.

Enumeration: {P1}=12 cost5; {P2,P3}=16 NPV cost7 infeasible; {P1,P3}=19 cost8 best; {P2,P3}=16 cost7.

Optimal P1+P3, NPV $19M, cost $8M ✓ slack $1M.

Managerial read: P2 dropped despite strong NPV per dollar because bundle with P1 consumes budget.

Part D: Integer Solver note

If P2 required pairwise exclusivity with P1 (same team), add constraint y1+y2 ≤ 1. Optimum may shift to P2+P3 or P1 alone. Integer constraints encode organizational limits, not only financial.

Check: 5+3=8 ≤ 9 budget ✓; NPV 12+7=19 ✓

Two SKUs: Max 7x+5y s.t. x+y≤100, 3x+2y≤240, x≤80, y≤70, x,y≥0.

1. Solve (corner enumeration).
2. Which resources bind?

Solution

Corners include (0,0)=0; (80,0)=560; (0,70)=350; (80,20) hours 240+40=280>240; intersect 3x+2y=240 and x+y=100 → subtract: 2x=140 → x=70,y=30 Z=640; (80,0) Z=560.

Check (70,30): 100 bind, 240 bind, Z=490+150=640 max ✓

Binding: total units and labor (3x+2y).

Why might Apex add C to the mix even with lower contribution per hour than B?

Solution

Contract, customer bundle, inventory buffer, or minimum run constraints can force C positive though B dominates on pure hour efficiency. Optimization with mins differs from pure efficiency ranking.

Embed mix model outputs in MRP (material requirements planning, production scheduling systems) where possible. Manual email of mix invites floor overrides without visibility.

Sales planning aligns forecasts to demand caps in model; forecast bias shows up as cap slack or bind patterns.

Commodity updates trigger contribution refresh and re-solve same day for protein-heavy SKUs.

Knapsack capital rationing links to Unit 6 MCDM when projects tie on NPV.

Blending models share template with mix; only proportion rows differ.

Carbon caps internalize ESG into same LP finance already solves.

Extended Apex sensitivity narrative

When B contribution falls $0.50, re-solved mix may increase D hours if A+D minimum binds. Sales must accept temporary D allocation despite brand deprioritization. Shadow price on minimum constraint becomes positive: relaxing minimum one unit has dollar value in objective loss avoided.

Knapsack with mutually exclusive projects: add y1+y2 ≤ 1 if same team executes. MIP solve required.

Multi-period sketch four weeks: variables S_t; inventory I_t = I_{t-1} + S_t - Sales_t; storage cap on I_t. Link weeks; explode size but same template.

Deep dive: sales and operations planning (S&OP) link

Monthly S&OP sets demand caps; weekly LP sets mix inside caps. Cap slack patterns feed next month forecast revision. Zero slack on hero SKU triggers pricing or capacity conversation.

Deep dive: contract penalties in constraints

Retailer penalty $50k if A+D < 400 becomes constraint A+D ≥ 400 with high shadow price when binding, quantifying penalty exposure at margin.

Product mix governance committee

Weekly 20-minute meeting: analytics presents mix, slack, shadow; sales confirms caps; ops confirms hour availability; finance confirms contributions. Conflicts resolved before release to floor.

Blending QC feedback loop

Lab sample fails protein spec → adjust coefficients 1% → re-solve → re-blend. Optimization is live control, not annual project.

Product-mix models fail socially when sales hears "Solver cut your SKU" without context. Pair every mix change with reason: binding cap, shadow price, contract minimum, or contribution update. Context prevents sabotage of the model on the floor.

Supplemental narrative: Apex Foods contract floor

Apex optimum zeros C until commodity shock lowers B contribution; then minimum A+D forces D positive despite low C preference. Sales lead sees D appear in mix memo with sentence: "Contract A+D minimum binding; shadow price $6 per unit on minimum explains cost of shelf commitment." Lead accepts because constraint economics are visible, not because analytics won an argument.

Supplemental narrative: knapsack capital week

Harbor Capital budget $9M, projects P1 P2 P3. Integer model selects P1+P3 for NPV $19M. Dropped P2 still strong per dollar; budget binds. CFO asks shadow price on budget: relaxing $1M might include P2 partially. Re-solve with $10M budget for sensitivity only; do not treat as approval. Capital rationing LP links finance committee conversation to dual price language.

Supplemental blending check

Feed mix 30% protein: ingredients A 30% protein, B 10%. Optimum may use only A if cost lower; proportion constraint forces B share. Verify linearized constraint 0.30A+0.10B ≥ 0.20(A+B) in sheet before Solve. Lab confirms composite protein 20.2% ✓.

Closing standards

Refresh contributions and caps weekly. Sign-off sales, finance, legal before floor release. Explain zero SKUs with binding and reduced cost language. Stress Downside jointly on demand and margin.

Extended review: product-mix template in production

Standard weekly workflow: sales updates caps in blue cells; finance updates contributions; analyst solves; governance committee reviews binding constraints; ops publishes schedule; floor executes; Friday compares actual hours to model hours used; deviations feed next week's RHS tweaks.

Blending, knapsack, and multi-plant models reuse template rows: activities, contributions, resource coefficients, caps, mins, balance rows.

Carbon and ESG constraints enter as linear rows; shadow prices become internal carbon fees in memos.

Sales communication template mandatory when any SKU goes to zero.

S&OP monthly resets caps; weekly LP executes inside S&OP plan.

Product-mix success is measured by fewer emergency changeovers and fewer contract misses, not by whether the spreadsheet exists.

Integration narrative: sales and ops quarterly review

Quarterly review agenda: (1) caps versus actual sell-through, bias discussion; (2) contract minimum compliance rate; (3) SKUs zeroed in model more than 50% weeks, reduced cost trends; (4) stress Downside scenario mix shift; (5) capital proposals referencing shadow prices. Sales learns caps are forecasts not wishes; ops learns mins are contracts not suggestions; finance learns contributions drive mix not revenue pride.

Blending, knapsack, and multi-plant extensions all inherit the same weekly cadence and governance committee.

Additional examples: mix decisions in prose

Promo week: temporary demand cap lift on Dessert requires re-solve; reduced cost may turn negative; sales must supply cap evidence. Supplier outage: zero B contribution for one week; re-solve may force C into mix; communicate substitution to customers. New contract minimum: retailer adds 50 units to A+D minimum; feasible interval check before signing; shadow on new minimum becomes negotiation lever.

Product-mix LP is operations strategy expressed as algebra: caps encode market belief, mins encode promises, contributions encode finance facts, shadow prices encode scarcity prices.

Lesson 4 mastery check: explain why Apex zeros SKU C in optimum, how contract minimum could force D positive, and how knapsack budget binds project choice.

Templates survive commodity shocks and promo weeks only when governance committee enforces weekly refresh instead of monthly heroics.

Unit 7 bridge paragraph

Product-mix models fail when treated as one-time projects. They succeed when embedded in S&OP, linked to sales caps, and explained to the floor with binding constraint language. Optimization changes who gets scarce hours; change management is part of the math.

Publish mix changes with reason codes: CAP_BIND, MIN_BIND, CONTRIB_SHOCK, SHADOW_OT. Reason codes speed sales and ops alignment.

Knapsack, blending, and multi-plant models differ in rows, not in weekly governance rhythm.

If the floor cannot explain why a SKU is zero, the model is not ready for release regardless of Solver status Optimal.

Weekly mix memo should list top three binding constraints even when hours are not scarce.

Lesson 4 completes when sales, finance, and ops sign the same cap and contribution table before the mix hits the floor.

Signed inputs beat heroic re-solves after the floor already ran the wrong SKU for six hours.

Treat product-mix LP as production infrastructure with uptime expectations, not as a quarterly consulting deliverable.

Governance committee attendance is a leading indicator of mix model survival.

Caps and mins are promises; the LP enforces them with shadow prices as the bill. Weekly signed inputs are cheaper than weekly fire drills on the production line.

When a knapsack model excludes a strong project because of budget pairing effects, document the next-best portfolio for the following budget cycle. Capital rationing models are path-dependent; yesterday's exclusion is tomorrow's first candidate when slack appears.

Key takeaways

• Product-mix LP maximizes contribution subject to capacities, demand caps, and mins.
• Blending and portfolio models add proportion constraints still linear after rearrangement.
• Multi-period and binary extensions handle inventory and project selection.
• Validate optima against last period actuals and scenario demand before rollout.
• Pair mix results with sales and pricing when market caps bind.

1. Build a four-SKU mix for your firm with one minimum contract constraint.
2. Which SKU hits demand cap first when capacity tightens 10%?
3. Continue to Lesson 5: Interpreting Solver Results and Stress-Testing Recommendations.