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Bus 480 Management Science

Bus 480 Management Science. Office Hours : Tuesday 7:30-8:30 Or by request DA 210 – Adjunct Faculty Office. Approach. 1. Observation 2. Definition of the problem 3. Model Construction What are the independent variables? What are the dependent variables?

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Bus 480 Management Science

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  1. Bus 480Management Science Office Hours : Tuesday 7:30-8:30 Or by request DA 210 – Adjunct Faculty Office

  2. Approach • 1. Observation • 2. Definition of the problem • 3. Model Construction • What are the independent variables? • What are the dependent variables? • What type of problem is it? Minimization?Maximization? • What are the constraints? Limitations? • 4. Model Solution • 5. Implementation

  3. Break Even Analysis • Fixed Cost - Cf – • Independent of how much volume is produced • Constant over time • Rent, taxes, salaries, building maintenance, etc… • Variable Cost - vcv - volume * cost per volume • Determined on a per unit basis • Changes by how much product is produced • Raw materials, labor, material handling, etc….

  4. Break Even Analysis (continued) • Total Cost = Cf + vcv • Total profit = Total Revenue – Total Cost • Where total revenue = vp = volume * price per unit • Break even occurs when total profit=0 • 0 = Total Revenue – Total Cost • Total Revenue = Total cost

  5. Example #6 • What are the fixed costs? • What are the variable costs? • What is the revenue? • What is break even?

  6. Example #7 • What are the fixed costs? • What are the variable costs? • What is the revenue? • What is break even?

  7. Linear Programming (LP) • Method for building a mathematical model based on linear relationships • Based on an objective function and constraints

  8. Model Formulation • Decision Variables • Independent variables • Objective function • Linear combination of decision variables • Maximization or minimization • Constraints • Restrictions of the model e.g. – only so much warehouse space • Linear equations of the decision variables • Typically inequalities

  9. Solution Types • Feasible – • There is a numeric solution • May have multiple feasible solutions • What is the optimal solution? • Infeasible • There are no solutions • Unbounded • An infinite number of solutions

  10. Setting Up LP • Identify Decision Variables • Define what x1, x2, x3, etc….. • What are we making? • What are the raw materials needed? • Where are we transporting merchandise?

  11. Setting Up LP • Formulate the objective function • Minimizing or maximizing • Will be a linear combination of decision variables • Defined as z = f(x) • E.g. - maximize z = 2x1 + 5x2 • Typically the coefficients represent costs per product, or revenue, or distance

  12. Setting Up LP • Determine the Constraints • Each constraint should involve one process, the same process or resource • Painting, wood supply, • Will typically be inequalities but can be equality depending on the constraint • 3x1 + 7x2≤ 100 • Based on the decision variables

  13. Setting Up LP • Write out the entire problem max z=7x1 + 3x2 subject to 3x1 + 7x2≤ 100 x1 - 3x2≤ 75 x1 , x2≥ 0 (non-negative constraints)

  14. Setting Up LP • Solve problem to determine the optimal solution • Graphically • Plot all constraints • Solutions will be the intersection points of the constraints • Find intersection points • Plug in the intersection points to the objective function and choose the point that maximizes or minimizes the objective • Excel • Set up cells to be the decision variables • Set up cells to represent the coefficients of constraints and the objective function • Utilize the sumproduct function • Use solver

  15. Example • Kelson Sporting company makes two types of baseball gloves: a regular model and a catcher’s model. The firm has 900 hours of production time in it’s cuting and sewing department, 300 hours available in it’s finishing department and 100 hours available in it’s packaging and shipping department. The production time requirements and profit contribution per glove is in the following table

  16. Example (cont) • What are the decision variables? What are we making? • Let x1 = Total number of regular model to produce x2 = Total number of catcher’s model to produce • What is the objective? • Maximize profit • z = 5*x1 +8*x2 • 5 times the number of regular model plus 8 times the number of catcher’s model

  17. Example (cont) • What are the constraints? • Cutting/Sewing hours • 1*x1 + 3/2*x2 ≤ 900 • Finishing hours • 1/2*x1 + 1/3*x2 ≤ 300 • Packaging/Shipping hours • 1/8*x1 + 1/4*x2 ≤ 100

  18. Example (cont) • The total LP Maximize z = 5*x1 +8*x2 Subject to • 1*x1 + 3/2*x2 ≤ 900 • 1/2*x1 + 1/3*x2 ≤ 300 • 1/8*x1 + 1/4*x2 ≤ 100 • x1,x2 ≥ 0

  19. Excel Set Up

  20. Excel Set Up - Formulas

  21. Solver Window

  22. Solver Options

  23. Solution

  24. Sensitivity Analysis • Sensitivity on the Objective Function Coefficients • How much can you change before the current solution is no longer optimal • RHS of Constraints • Shadow price – increase in the objective function for every unit increase in resource • Keep the same “mix” of the solution

  25. Example - Sensitivity

  26. Add one hour to finishing

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