Sample 340 Homework 8
.pdf
School
University Of Arizona**We aren't endorsed by this school
Course
SIE 340
Subject
Economics
Date
Dec 16, 2024
Pages
4
Uploaded by BailiffBravery9533
What Are Ghost Prices?Ghost prices, commonly referred to asshadow prices, represent the marginal value of relaxinga constraint in an optimization problem. They quantify how much the objective function (e.g.,profit, cost) would improve or deteriorate with a one-unit change in the right-hand side (RHS) ofa binding constraint while keeping all other variables constant.2. Key Characteristics of Ghost Prices1.Applies Only to Binding Constraints:○A constraint has a ghost price only if it is actively restricting the solution (i.e.,binding).2.Marginal Impact:○Measures therateof change in the objective function value for a unit increase inthe RHS of the constraint.3.Positive or Negative Values:○Positive ghost prices indicate an increase in the objective value with a relaxationof the constraint.○Negative ghost prices indicate a decrease in the objective value.4.Zero for Non-Binding Constraints:○If a constraint is not binding (i.e., there is slack), its ghost price is zero, asrelaxing it further has no impact.3. Applications of Ghost Prices1.Resource Allocation:○Identifies the value of adding scarce resources, such as labor hours or materials.2.Cost-Benefit Analysis:○Helps determine whether acquiring additional resources is cost-effective.3.Decision Making:○Guides managerial decisions on resource prioritization and operational planning.4.Scenario Analysis:○Evaluates the impact of changes in constraints due to external factors like policyshifts or supply chain disruptions.4. Examples of Ghost Prices in PracticeExample 1: Manufacturing Problem
Problem: A company produces two products (A and B) using a limited resource (labor hours).Objective Function: Maximize profit = $5A + $4BConstraints:1.Labor hours available = 8 hours.2.Each product requires 2 hours of labor.Ghost Price Analysis:●Shadow Price of labor hours = $2/unit.●Interpretation: Adding 1 more hour of labor increases the total profit by $2, up to aspecific allowable range.Example 2: Transportation ProblemProblem: Minimize transportation cost while meeting delivery requirements.Constraint: Warehouse supply limit.Ghost Price Analysis:●Shadow Price = $10/unit for an additional unit of supply capacity.●Interpretation: Increasing the warehouse's supply capacity by one unit reduces thetransportation cost by $10.5. Key Insights from Ghost Prices1.Range of Validity:○Shadow prices are valid only within the allowable range of RHS values. Beyondthis range, the solution structure may change.2.Economic Interpretation:○Provides insight into the marginal worth of constrained resources.3.Cost Implications:○Helps in evaluating whether to invest in additional resources or relax constraints.4.Dual Relationship:○Ghost prices correspond to dual variables in linear programming, reflecting thevalue of constraints in the dual problem.6. Limitations of Ghost Prices
1.Local Perspective:○Reflects marginal changes and does not account for large or non-linear shifts.2.Non-Binding Constraints:○Provides no information about constraints that are not currently limiting thesolution.3.Dependence on Problem Structure:○Changes in the problem (e.g., adding variables or constraints) may alter ghostprices.4.Static Assumptions:○Assumes other parameters remain constant during the sensitivity analysis.7. How to Calculate Ghost Prices1.Solve the Primal Problem:○Use linear programming techniques (e.g., Simplex method).2.Identify Binding Constraints:○Determine which constraints are active in the optimal solution.3.Analyze the Dual Problem:○Shadow prices are the optimal values of the dual variables corresponding to theprimal constraints.4.Software Tools:○Excel Solver: Provides shadow prices directly in the sensitivity report.○Python Libraries: Use Pyomo, Gurobi, or PuLP for detailed analysis.8. Software Tools for Ghost Prices1.Excel Solver:○Generates shadow price information in the sensitivity report.2.Python Libraries:○Pyomo, Gurobi, and PuLP for computational solutions and sensitivity analysis.3.MATLAB Optimization Toolbox:○Offers tools to compute and interpret shadow prices.4.AMPL:○Provides shadow price details in dual problem solutions.9. Further Reading●Linear Programming and Network Flowsby Bazaraa et al.●Operations Research: Applications and Algorithmsby Winston.
●Online Tutorials and Courses on platforms like Coursera and edX.