1. STDM - Linear Programming 1 By Isuru Manawadu B.Sc in Accounting Sp. (USJP), ACA, AFM 31130

2. Learning Outcomes After studying this session you will be able to: Why linear programming Assumptions of linear programming Graphical method of linear programming Cost Minimization Problem Using Linear Programming Shadow price calculation

3. Why Linear Programming?  Programming formulations gives solutions for the problems.  The output generated by linear programs provides useful “what-if” information.  Improves quality of decision.  Utilized to analyze numerous economic, social, military and industrial problem.

4. Linear-Programming Applications  Constrained Optimization problems occur frequently in economics: maximizing output from a given budget; or minimizing cost of a set of required outputs.  A number of business problems have inequality constraints.

5. Profit Maximization Problem Using Linear Programming  Constraints of production capacity, time, money, raw materials, budget, space, and other restrictions on choices. These constraints can be viewed as inequality constraints  A "linear" programming problem assumes a linear objective function, and a series of linear inequality constraints

6. Linear Programming Assumptions: 1.Constant prices for outputs (as in a perfectly competitive market). 2.There are no interactions between the décision variables. 3.The parameters are know with certainly. 4.Constant returns to scale for production processes.

7. Linear Programming Assumptions: 6. Typically, each decision variable also has a non-negativity constraint. For example, the time spent using a machine cannot be negative. The décision variable are continuons..

8. Graphical Analysis – The Feasible Region The non-negativity constraints

9. Solution Methods  Linear programming problems can be solved using graphical techniques, SIMPLEX algorithms using matrices, or using software, such as ForeProfit software.  In the graphical technique, each inequality constraint is graphed as an equality constraint. The Feasible Solution Space is the area which satisfies all of the inequality constraints.

10. Solution Methods Cont….  The Optimal Feasible Solution occurs along the boundary of the Feasible Solution Space, at the extreme points or corner points.  The corner point that maximize the objective function is the Optimal Feasible Solution.

11. Solution Methods Cont….  There may be several optimal solutions. Examination of the slope of the objective function and the slopes of the constraints is useful in determining which is the optimal corner point.  One or more of the constraints may be slack, which means it is not binding.

12. GRAPHICAL X1X1 X2X2 A B C CONSTRAINT # 1 CONSTRAINT # 2 Corner Points A, B, and C Feasible Region OABC O

13. Extreme points and optimal solutions If a linear programming problem has an optimal solution, an extreme point is optimal. 13

14. Multiple optimal solutions 14 For multiple optimal solutions to exist, the objective function must be parallel to one of the constraints Any weighted average of optimal solutions is also an optimal solution.

15. Shadow Prices Assuming there are no other changes to the input parameters, the change to the objective function value per unit increase to a right hand side of a constraint is called the “Shadow Price” l One more of the constraints may be slack, which means it is not binding. l Each constraint has an implicit price, the shadow price of the constraint. If a constraint is slack, its shadow price is zero.

16. 16 1000 500 X2X2 X1X1 2X 1 + 1x 2 <=1000 When more plastic becomes available (the plastic constraint is relaxed), the right hand side of the plastic constraint increases. Production time constraint Maximum profit = 4360 2X 1 + 1x 2 <=1001 Maximum profit = 4363.4 Shadow price = 4363.40 – 4360.00 = 3.40 Shadow Price – graphical demonstration The Plastic constraint

17. Complexity and the Method of Solution  The solutions to primal and dual problems may be solved graphically, so long as this involves two dimensions.  With many products, the solution involves the SIMPLEX algorithm, or software available in FOREPROFIT

18. Cost Minimization Problem Using Linear Programming  Multi-plant firms want to produce with the lowest cost across their disparate facilities. Sometimes, the relative efficiencies of the different plants can be exploited to reduce costs.  A firm may have two mines that produces different qualities of ore. The firm has output requirements in each ore quality.

19. Cost Minimization Problem Using Linear Programming  Scheduling of hours per week in each mine has the objective of minimizing cost, but achieving the required outputs

20. Thank you