Public Television Campaign Optimization: How Many ✓ Solved

Public Television Campaign Optimization: Determine how many umbrella

Determine how many umbrella spots and magazine spots to air during a 7-day fundraising campaign to maximize net income, given the following facts: The campaign runs 7 days, 12 hours per day, with one sales pitch each hour; each pitch offers only one gift (umbrella for $45 donation or magazine subscription for $50 donation). Gifts must be purchased in advance at $25 each. Typical audience per hour is 110,000 homes (different each hour). Response rates: 1% per umbrella spot, 1.5% per magazine spot. 25% of umbrella recipients will later donate an additional $25; 10% of magazine recipients will later donate an additional $25. Management requires at least 30 spots for each gift. Determine the number of umbrella and magazine spots to run to maximize net income. Provide: (1) the optimal allocation and its net income with binding constraints identified (as would be produced by Solver); (2) three additional distinct recommendations, each derived from the binding constraints, including the projected net income or change in net income for each recommendation.

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Executive summary

This analysis finds the optimal airtime allocation and three alternative recommendations to increase net income for the public television fundraising campaign. Using the campaign assumptions and straightforward linear revenue-cost arithmetic (consistent with Solver linear programming approach), the optimal allocation under current constraints is 30 umbrella spots and 54 magazine spots (total 84 available spots). This allocation yields an estimated net income of $3,316,500. The solver’s binding constraints are the total-spot capacity (84 spots) and the management minimum of 30 umbrella spots. Three additional recommendations—(A) relax the minimum umbrella-spot requirement to 10, (B) negotiate a reduced gift cost from $25 to $20, and (C) extend campaign airtime by one day dedicated to magazine pitches—are quantified below with projected net income deltas.

Modeling approach and per-spot economics

Define U = number of umbrella spots, M = number of magazine spots. Total available spots = 7 days × 12 hours/day = 84, so U + M ≤ 84, with U ≥ 30 and M ≥ 30 per management policy. Each hourly spot reaches 110,000 homes; response rates are 1.0% for umbrella and 1.5% for magazine. Gifts cost $25 each and must be purchased in advance. Donations: umbrella = $45; magazine = $50. In addition, a fraction of gift recipients donate an extra $25 later: 25% for umbrella recipients and 10% for magazine recipients.

Compute net contribution per purchase (donation minus gift cost, plus expected spontaneous donation):

  • Umbrella: net per purchase = (45 − 25) + 0.25×25 = 20 + 6.25 = $26.25
  • Magazine: net per purchase = (50 − 25) + 0.10×25 = 25 + 2.50 = $27.50

Per spot purchases and per-spot net:

  • Umbrella purchases per spot = 110,000 × 0.01 = 1,100 → net per umbrella spot = 1,100 × $26.25 = $28,875
  • Magazine purchases per spot = 110,000 × 0.015 = 1,650 → net per magazine spot = 1,650 × $27.50 = $45,375

Because magazine spots produce higher net per spot, the objective (maximize total net income) is to allocate as many spots as possible to magazine subject to constraints.

Solver-style optimal allocation and binding constraints

With U + M = 84, U ≥ 30, M ≥ 30, maximizing Net = 28,875·U + 45,375·M yields the optimal corner solution:

  • U* = 30 (management minimum binds)
  • M* = 54 (fills remaining spots; total equals 84 so total-spot capacity binds)
  • Total net income = 30×28,875 + 54×45,375 = $3,316,500

Binding constraints reported by Solver would be: (1) total spots used = 84 (capacity fully used); (2) umbrella minimum U = 30 is at its lower bound. The magazine minimum is not binding (M = 54 > 30).

Recommendation 1 — Baseline (Solver) recommendation

Run 30 umbrella spots and 54 magazine spots. Rationale: magazine spots generate the highest net income per spot; with the total-hour cap and the management minimum for umbrellas, the Solver optimum is U=30, M=54. Projected net income: $3,316,500.

Recommendation 2 — Reduce the minimum umbrella-spot requirement to increase magazine spots

Policy constraint analysis shows the umbrella minimum is binding and is limiting higher-value magazine spots. If management can be persuaded to relax the umbrella minimum to U_min = 10 (from 30), the optimal allocation becomes U = 10, M = 74 (filling 84 spots). Net income then becomes:

  • 10×28,875 + 74×45,375 = $3,646,500

Increase vs. baseline = $3,646,500 − $3,316,500 = $330,000. Recommendation: lower the internal fairness/commitment policy (if acceptable) to allow more magazine pitches, yielding a substantial increase in net funds.

Recommendation 3 — Negotiate lower gift acquisition cost (reduce per-unit gift cost from $25 to $20)

The per-unit gift cost is a controllable expense that directly affects per-purchase net contribution. If procurement negotiates gifts at $20 each for both items, per-purchase nets become:

  • Umbrella net per purchase = (45 − 20) + 0.25×25 = 25 + 6.25 = $31.25 → per-spot = 1,100×31.25 = $34,375
  • Magazine net per purchase = (50 − 20) + 0.10×25 = 30 + 2.5 = $32.50 → per-spot = 1,650×32.5 = $53,625

Keeping the original solver allocation (U=30, M=54) produces total net = 30×34,375 + 54×53,625 = $3,927,000 — an increase of $610,500 over baseline. Recommendation: pursue vendor negotiations, bulk discounts, or alternate suppliers to reduce unit cost; this improves margin significantly and is directly supported by the cost binding logic.

Recommendation 4 — Add one extra day of advertising focused on magazine spots

The total-spot-capacity constraint is binding. Increasing available airtime is an effective lever. Adding one day (12 additional hours) dedicated to magazine pitches increases magazine spots by 12 (if U remains at 30), so new M = 54 + 12 = 66 and total spots = 96. New net (with original $25 gift cost) is:

  • 30×28,875 + 66×45,375 = 866,250 + 2,993, = 3,859,? Compute: 66×45,375 = 45,375×60=2,722,500 +×6=272,250 → 2,994,750. Sum = 866,250 + 2,994,750 = $3,861,000

Increase vs. baseline = $544,500. Recommendation: extend campaign airtime when possible (additional day devoted to higher-yield magazine spots) to exploit higher per-spot returns; operational feasibility and marginal airtime cost should be measured against projected gain.

Discussion and implementation notes

All recommendations are constrained by real organizational considerations: audience fatigue, regulatory/contractual airtime costs, donor relationships and fairness, and procurement realities (Sargeant & Jay, 2014; Bekkers & Wiepking, 2011). The arithmetic above is linear and deterministic; in practice, response rates can vary (Kotler & Keller, 2016). It is recommended that management run a small pilot to validate higher expected response or negotiate supplier terms before full-scale policy changes. Use Solver (or similar LP tool) to re-run with any modified parameters; Microsoft Excel Solver and standard linear-programming texts provide the implementation framework (Microsoft, 2020; Hillier & Lieberman, 2015).

In summary: the highest immediate uplift comes from reallocating airtime to magazine spots, lowering gift acquisition costs, or expanding airtime. Each lever maps directly to a binding constraint identified in the solver-style solution: the total-spot capacity and the umbrella minimum. Addressing those constraints yields measurable revenue improvements.

References

  • Goldratt, E. M., & Cox, J. (2012). The Goal: A Process of Ongoing Improvement (30th anniversary ed.). North River Press.
  • Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research (10th ed.). McGraw-Hill Education.
  • Winston, W. L. (2004). Operations Research: Applications and Algorithms (4th ed.). Thomson Brooks/Cole.
  • Kotler, P., & Keller, K. L. (2016). Marketing Management (15th ed.). Pearson.
  • Sargeant, A., & Jay, E. (2014). Fundraising Management: Analysis, Planning and Practice. Routledge.
  • Bekkers, R., & Wiepking, P. (2011). A literature review of empirical studies of philanthropy: Eight mechanisms that drive charitable giving. Nonprofit and Voluntary Sector Quarterly, 40(5), 924–973.
  • Microsoft. (2020). Excel Solver documentation. Microsoft Support. https://support.microsoft.com
  • Association of Fundraising Professionals (AFP). (2018). Fundraising effectiveness benchmarking and trends reports. AFP.
  • PBS. (2020). PBS fundraising and pledge drives: best practices. Public Broadcasting Service.
  • Rothschild, M., & Russell, J. (2017). Nonprofit Finance and Fundraising: A practical guide. Oxford University Press.

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