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Decision Support System ICS425

Decision Support System ICS425. Unit 4. Last time…. Mental Model => is an explanation in someone's thought process for how something works in the real world It is a kind of internal symbol or representation of external reality, hypothesized to play a major part in cognition. Mental Model.

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Decision Support System ICS425

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  1. Decision Support SystemICS425 Unit 4

  2. Last time… • Mental Model => is an explanation in someone's thought process for how something works in the real world • It is a kind of internal symbol or representation of external reality, hypothesized to play a major part in cognition

  3. Mental Model • Mental models are representations in the mind of real or imaginary situations • Suggested by Kenneth Craik (1914 - 1945)

  4. Introduction This unit will cover; • Mathematical model • Linear Programming • Simplex Method

  5. Mathematical Models • is an abstract model that uses mathematical language to describe the behavior of a system • Eykhoff (1974) defined a mathematical model as 'a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in usable form'.

  6. Mathematical Models • For example, we can develop a mathematical model to determine the pay of a salesperson who received a commission of 20 pounds on each sale. So, how can we develop a mathematical model for this situation?

  7. Mathematical Models Write down the data that describe the relationship between the salesperson’s commission and the number of sale.

  8. Mathematical Models If we let x represent the number of sales and y represent the pounds of income, then the mathematical function between sales and income is expresses: y= 20x

  9. Mathematical Models • This functional relationship can be viewed mentally as representing a processing operation much in the same manner as we would visualize a data processing operation. • The various values of x (0,1,2,3,4,…) can be thought of as input, with the corresponding values of y (0,20,40,60,…) being outputs. “ The inputs and outputs are commonly called variables”

  10. Mathematical Models Using conventional mathematical terminology, the input variable is referred to as the independent variable and the output variable as the dependent variable. The numerical value is referred to by several labels: constant, coefficient, and parameter

  11. Linear Programming

  12. Linear Programming • In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear. • The term “linear optimization and linear programming are equivalent”

  13. Linear Programming • Linear programming is an application to find the maximum or minimum value of a linear function f(x,y) subjected to a set of constraints which can be expressed as a system of linear inequalities i.e. LP is a solution method to problems where an objective has to be optimized subject to constraints

  14. Linear Programming • A feasible solution is a set of variables which satisfies all the constraints. • The set of all solutions is called either the set of feasible solutions or the feasible region • The optimal solution, the solution that minimize or maximize f(x,y) is at one of the vertices of the feasible region • Or any solution that optimize the objective function f(x,y) among the set of all solutions is called an optimal solution.

  15. Linear Programming A method of solving; • Analyze the problem and set up the inequalities for the constraints • Write down the objective function f(x,y) to be maximized or minimized • Draw the graphs pf the inequalities and find out the feasible region • Find the optimal solution either by substituting the coordinates of the vertices of the feasible solution into the function f(x,y) or by drawing lines touching the feasible region and parallel to the line f(x,y) = 0

  16. Linear Programming Note: All factors concerned must be numeric and there must be linear relationships

  17. Formulation of the LP Problem Example: An engineering company is considering its idle time on the machines for manufacturing one or more of three products 1, 2 and 3. The available time in a week on the machine is as follows:

  18. Formulation of the LP Problem The hours required for each unit of the product are as follow:

  19. Formulation of the LP Problem Product1 and product 2 can be sold up to any amount but product 3 can be sold at the most up 10 units per week only. The profile per unit will be Rs.10, Rs.3, Rs.4 respectively on product 1, 2. and 3. What will the product-mix be for the company for maximum profit? Formulate the problem.

  20. Formulation of the LP ProblemSolution!!

  21. Formulation of the LP ProblemSolution continue..

  22. Linear ProgrammingGraphical Solution • Graphical methods of solving LP problems can be used for problems with two decision variables. The purpose of the graphical method is not to provide a “practical” method for solving linear programs, since practical problems usually include a large number of variables

  23. Linear ProgrammingGraphical Solution • In stead, the method demonstrates the basic concepts for developing the general algebraic technique for linear programs with more than two variables • Problems with three or more variables must be solved by techniques such as the simplex method

  24. Linear ProgrammingGraphical Solution • The idea of method is to plot the (feasible) solution space, which is defined as the space enclosed by constraints. The optimum solution is the point (in the solution space) which maximizes/minimizes the value of the objective function

  25. LP: an exampleTwo Dimensional Problem • The daily minimum requirement of an animal is 60g of protein, 48 g of fat, and 120 g. carbohydrates. The contents of scraps A and B per kg are as follows:

  26. Continue… The cost of scraps A and B are 9 bht/kg and 13 bht/kg respectively. How to give a diet to the animal which will deliver adequate nutrition at the lower cost?

  27. Solution of an example

  28. ANY QUESTIONS??

  29. How will you do if you need to work with variables more than two? Simplex Method

  30. Simplex method • Used for standard linear programming problems; Step 1: Using slack variables, convert the LP problem to a system of linear equations Step 2: Set up the initial tableau Step 3: Select the pivot column Step 4: Select the pivot in the pivot column Step 5: Use the pivot to clear the pivot column in the normal manner. This gives the next tableau Step 6: Repeat Steps 3-5 until there are no more negative numbers in the bottom row (with the possible exception of the Answer column)

  31. Conclusion • Mathematical model • Linear Programming • Simplex Method

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