What is the shadow price of the constraint that says Booster must make at least 275 kg of Boost?

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BlackTom AI · Accepted Answer

This item asks for the shadow price of a specific constraint ( Booster must make at least 275 kg ) but provides no numerical model data to analyze. To reason about a shadow price, we need the underlying linear programming formulation, including:
- The objective function to be optimized (e.g., maximize profit or minimize cost).
- All decision variables and how they interact with the constraint (the ≥275 kg constraint in this case).
- The full set of constraints and right-hand-side values, plus the current basic feasible solution (BFS) from the optimal tableau or an equivalent solution.
- The dual values associated with each constraint, which are the shadow prices, indicating how much the objective would improve per unit increase in the constraint’s RHS.

Without the LP model, any discussion of a numerical shadow price would be speculative. Here is how one would determine it if the data were available:
- Step 1: Solve the LP to obtain the optimal solution and the corresponding dual values. The dual variables associated with each primal constraint give the shadow prices.
- Step 2: Identify the constraint Booster must make at least 275 kg of Boost as a ≥ constraint with RHS = 275. In the optimal basis, check the dual value associated with this constraint. This value is the shadow price.
- Step 3: Interpret the sign and magnitude: a positive shadow price means relaxing the constraint (increasing the RHS) would improve the objective, while a negative shadow price would imply tightening the constraint could improve the objective under the current model, though for a ≥ constraint the sign depends on the objective and binding status.
- Step 4: Verify feasibility of small RHS changes: shadow prices are valid within the allowable increase/decrease ranges of the RHS before the basis changes (the activity range). If the RHS of this constraint changes beyond that range, the shadow price could change.

If you can provide the full LP (objective, all constraints, and the current optimal solution), I can walk you through extracting the shadow price step by step and explain what it implies for the Booster production plan. In the absence of that data, any numerical answer would be guesswork rather than a rigorous calculation.

COMM_V 190 101-104 2025W1 HW06.COMM190.2025W1 (Practice only; not for grading)

What is the shadow price of the constraint that says Booster must make at least 275 kg of Boost?

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