Part 3 S2 LP Reduced Cost

Reduced Cost
📌 Reduced cost is the sacrifice or loss in profit incurred if a product, currently produced at zero units in the optimal solution, is forced into production (one unit).
📉 For a product with a reduced cost of $2 (like Product B in Scenario 1), producing one unit reduces the total profit by $2.
🎯 Reduced cost is only applicable when the optimal solution for a variable is zero. If the objective is cost minimization, the reduced cost represents the increase in cost per unit.

Shadow Price (Dual Value)
🔗 The shadow price represents the impact on the objective function (profit increase or cost decrease) resulting from increasing the capacity of a binding constraint by one unit.
⬆️ In Scenario 1, Machine 1 capacity (24 hours) had a shadow price of $3; increasing capacity by one hour increases profit by $3.
🚫 If a constraint is non-binding (like Machine 2 in Scenario 1), its shadow price is zero.
📊 In Scenario 2, both machines were binding; Machine 1 had a shadow price of $2, and Machine 2 had a shadow price of $1, indicating Machine 1 capacity is twice as valuable for profit increase.

Post-Optimality Analysis: Objective Function
📈 This analysis assesses how changes in objective function coefficients (profit margins) affect the optimal solution (the mix of X1 and X2).
🎯 For Product A (initial profit $6), the optimal solution remains the same if the profit margin changes within the allowable range of a decrease of 1.33 or an increase up to $14 (6 + 8).
🔄 If a change in profit margin exceeds the allowable increase/decrease, the optimal solution will shift, requiring a re-run of the solver.

Post-Optimality Analysis: Constraints
⏳ Allowable increase/decrease for constraints defines the range where the shadow price remains valid.
🔗 For Machine 1 (initial capacity 24 hours), the shadow price of $2/hour is valid if capacity is increased up to 48 hours or decreased down to 16 hours.
➕ If increasing capacity by 12 hours (within the allowable range), the profit increase is $12 \times$2 = $24$. If the increase exceeds the range, the current shadow price is invalid.

Opportunity Cost and Decision Making
💰 When evaluating a new product (Product C, $10 profit) that consumes scarce resources, the cost is the opportunity cost derived from the shadow prices of the consumed constraints.
💲 If Product C requires 4 hours of Machine 1 (shadow price $2) and 3 hours of Machine 2 (shadow price $1), the cost is $(4 \times$2) + (3 \times $1) = $11$. Since the cost ($11) exceeds the benefit ($10), the decision should be No.

Capital Expenditure Justification (ROI)
💵 To justify a capital investment (e.g., $10,000 equipment adding 30 hours/week), calculate the projected profit increase using the shadow price, provided the added capacity is within the allowable increase.
✅ If the shadow price is $2.083/hour and 30 additional hours/week are added (within the allowable increase), the yearly profit increase (assuming 50 weeks) is $3,124.50. This yields an ROI of $3,124.50 / $10,000 = \mathbf{31.25%}$, which exceeds the required 20% hurdle rate, justifying the purchase.

Key Points & Insights
➡️ Reduced cost indicates the profit lost if a non-producing item is forced into production due to strategic reasons (demand, market entry, risk minimization).
➡️ Shadow price quantifies the benefit of addressing a binding constraint; it pinpoints the bottleneck resource (e.g., which machine or budget item) to focus improvement efforts on.
➡️ Post-optimality analysis allows for justification of resource allocation by comparing the benefit (profit increase) against the opportunity cost (loss of profit due to resource consumption).
➡️ Always check the allowable increase/decrease before applying a shadow price or objective function sensitivity range to ensure the calculated impact remains valid.

📸 Video summarized with SummaryTube.com on Mar 03, 2026, 12:38 UTC