1. Supply Chain Management By Dr. Asif Mahmood drasif@uet.edu.pk Chapter 9: Aggregate Planning

2. Role of Aggregate Planning in a Supply Chain Capacity has a cost, lead times are greater than zero Aggregate planning: –process by which a company determines levels of capacity, production, subcontracting, inventory, stockouts, and pricing over a specified time horizon –goal is to maximize profit –decisions made at a product family (not SKU) level –time frame of 3 to 18 months –how can a firm best use the facilities it has?

3. Linear programming (LP) techniques consist of a sequence of steps that will lead to an optimal solution to problems, in cases where an optimum exists –Used to obtain optimal solutions to problems that involve restrictions or limitations, such as: Materials, Budgets, Labor, Machine time –Graphical linear programming Vs Simplex method Aggregate Planning through Linear Programming

4. Objective Function: mathematical statement of profit or cost for a given solution Decision variables: amounts of either inputs or outputs Feasible solution space: the set of all feasible combinations of decision variables as defined by the constraints Constraints: limitations that restrict the available alternatives Parameters: numerical values Linear Programming Model (Components)

5. Linearity: the impact of decision variables is linear in constraints and objective function Divisibility: noninteger values of decision variables are acceptable Certainty: values of parameters are known and constant Nonnegativity: negative values of decision variables are unacceptable Linear Programming Model (Assumptions)

6. Model Formulation—Example Maximize Z=5X 1 +8X 2 +4X 3 (profit) (Objective function) Subject to Labor 2X 1 + 4X 2 +8X 3 ≤ 250 hours Material 7X 1 + 6X 2 +5X 3 ≤ 100 pounds Product 1 X 1 ≥ 10 units X 1, X 2, X 2 ≥ 0 X 1 =Quantity of product 1 to produce X 2 =Quantity of product 2 to produce X 3 =Quantity of product 3 to produce Decision variables (Constraints) (Non-negativity Constraints)

7. 1.Set up objective function and constraints in mathematical format 2.Plot the constraints 3.Identify the feasible solution space 4.Plot the objective function 5.Determine the optimum solution Graphical Linear Programming Graphical method for finding optimal solutions to two-variable problems

8. assembly timeinspection time storage space A firm that assembles computers and computer equipment is about to start production of two new types of microcomputers. Each type will require assembly time, inspection time and storage space. The amounts of each of these resources that can be devoted to the production of the microcomputers is limited. The manager of the firm would like to determine the quantity of each microcomputers to produce in order to maximize the profit generated by sales of these microcomputers. Graphical Linear Programming Example

9. Type 1Type 2 Profit per unit$60$50 Assembly time per unit4 hours10 hours Inspection time per unit2 hours1 hours Storage space per unit3 cubic feet Additional Information ResourceAmount Available Assembly time100 hours Inspection time22 hours Storage space39 cubic feet

10. Objective - profit Maximize Z=60X 1 + 50X 2 Subject to Assembly 4X 1 + 10X 2 ≤ 100 hours Inspection 2X 1 + 1X 2 ≤ 22 hours Storage3X 1 + 3X 2 ≤ 39 cubic feet X 1, X 2 ≥ 0 Example—Mathematical Model

11. Example—Plotting Constraints

13. Assembly Storage Inspection Feasible solution space Example—Plotting Constraints

14. Z=300 Z=900 Z=600 Example—Plotting Objective Function Line As we increase the value for the objective function: The isoprofit line moves further away from the origin The isoprofit lines are parallel

15. The optimal solution is at the intersection of the inspection boundary and the storage boundary Solve two equations in two unknowns 2X 1 + 1X 2 = 22 3X 1 + 3X 2 = 39 X 1 = 9 X 2 = 4 Z = $740 Solution

16. Redundant constraint: a constraint that does not form a unique boundary of the feasible solution space Binding constraint: a constraint that forms the optimal corner point of the feasible solution space Constraints

17. Solutions and Corner Points Feasible solution space is usually a polygon Solution will be at one of the corner points Enumeration approach: Substituting the coordinates of each corner point into the objective function to determine which corner point is optimal.

18. Surplus: when the optimal values of decision variables are substituted into a greater than or equal to constraint and the resulting value exceeds the right side value Slack: when the optimal values of decision variables are substituted into a less than or equal to constraint and the resulting value is less than the right side value Slack and Surplus

19. Exercise Solve the following problem using graphical linear programming. MinimizeZ=8x 1 +12x 2 Subject to 5x 1 +2x 2 ≥ 20 4x 1 +3x 2 ≥ 24 x 2 ≥ 2 x 1, x 2 ≥ 0

20. Simplex: a linear-programming algorithm that can solve problems having more than two decision variables Simplex Method

21. MS Excel Worksheet for Microcomputer Problem

22. MS Excel Worksheet Solution

23. Loading the Solver Add-in Step 1

24. Step 2

25. Step 3 Step 4