17.2 Quantitative Techniques for Decision Making

Quantitative Techniques for Decision Making

Management Services includes quantitative techniques that support planning and decision making (syllabus topic 1.4): decisions under risk and uncertainty, linear programming, and network scheduling. Board questions keep the arithmetic short, so the skill being tested is setting up the table, the constraints, or the network correctly.

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1. Decisions Under Risk: Expected Value

When probabilities can be assigned to the possible states of nature, compute each action's expected value (EV) = sum of (payoff x probability) and choose the action with the highest EV (or the lowest expected cost).

Example (stocking decision). Batangas Bakeshop buys a seasonal cake for PHP 60 and sells it for PHP 100; unsold cakes are sold to a caterer for PHP 20. Demand will be 100, 200, or 300 cakes with probabilities 0.3, 0.5, and 0.2. Profit = 100 x units sold + 20 x unsold units - 60 x units stocked.

The best action under risk is to stock 200 cakes (EV PHP 5,600).

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2. Expected Value of Perfect Information (EVPI)

EVPI is the most a decision maker should pay for a perfect forecast.

$\text{EVPI} = \text{EV with perfect information} - \text{EV of the best action without it}$

With perfect information, the bakeshop would always stock exactly the demand: 0.3 x 4,000 + 0.5 x 8,000 + 0.2 x 12,000 = PHP 7,600. EVPI = 7,600 - 5,600 = PHP 2,000. A market survey costing more than PHP 2,000 is not worth buying even if it were perfectly accurate, and an imperfect survey is worth less.

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3. Decisions Under Uncertainty (No Probabilities)

Regret for each cell = best payoff in that state - payoff of the action. For stock 100 the regrets are 0, 4,000, and 8,000; for stock 200 they are 4,000, 0, and 4,000; for stock 300 they are 8,000, 4,000, and 0.

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4. Decision Trees

A decision tree lays out sequential choices (squares) and chance events (circles). Work backward (rollback): compute the EV at each chance node, then choose the best branch at each decision node.

Example: Building a large plant costs PHP 500,000 and yields present-value inflows of PHP 1,200,000 if demand is high (0.6) or PHP 400,000 if low (0.4): EV = 720,000 + 160,000 - 500,000 = PHP 380,000. A small plant costs PHP 200,000 and yields PHP 600,000 (high) or PHP 450,000 (low): EV = 360,000 + 180,000 - 200,000 = PHP 340,000. The large plant is preferred on expected value, although it has a wider range of outcomes, and that risk can be measured with the coefficient of variation covered under risk and return.

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5. Linear Programming

Linear programming (LP) allocates scarce resources among products to maximize contribution margin (or minimize cost), subject to linear constraints.

Example. Product X earns a contribution margin of PHP 30 per unit and Product Y earns PHP 20. X needs 2 machine hours and 1 labor hour; Y needs 1 machine hour and 2 labor hours. Available: 100 machine hours and 80 labor hours.

• Objective: Maximize Z = 30X + 20Y
• Constraints: 2X + Y <= 100 (machine); X + 2Y <= 80 (labor); X, Y >= 0

The optimal solution lies at a corner point of the feasible region:

Produce 40 units of X and 20 units of Y for a total contribution margin of PHP 1,600. A shadow price (dual value) is the increase in the objective from one more unit of a binding resource; a non-binding constraint has slack and a shadow price of zero. With a single scarce resource, LP reduces to ranking products by contribution margin per unit of the scarce resource, as covered in relevant costing.

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6. Network Scheduling: PERT and CPM

• CPM (critical path method) uses single time estimates. The critical path is the longest path through the network; it sets the minimum project duration, and its activities have zero slack.
• PERT uses three estimates: optimistic (a), most likely (m), and pessimistic (b).

$t_e = \frac{a + 4m + b}{6} \qquad \sigma^2 = \left(\frac{b - a}{6}\right)^2$

Example: a = 4, m = 6, b = 14 gives t_e = (4 + 24 + 14) / 6 = 7 days.

Crashing shortens the project by adding resources to critical activities with the lowest crash cost per day, and it stops when crashing costs more than the benefit of finishing early or when a new critical path emerges.

Other tools in this topic include sensitivity analysis, simulation (Monte Carlo), queuing theory for waiting-line trade-offs, and the learning curve and regression methods covered under cost behavior.