1 DSCI 3023 Linear Programming Developed by Dantzig in the late 1940’s A mathematical method of allocating scarce resources to achieve a single objective The objective may be profit, cost, return on investment, sales, market share, space, time LP is used for production planning, capital budgeting, manpower scheduling, gasoline blending By companies such as AMD, New York Life, Chevron, Ford

2 DSCI 3023 Linear Programming Decision Variables –symbols used to represent an item that can take on any value (e.g., x 1 =labor hours, x 2 =# of workers) Parameters –known constant values that are defined for each problem (e.g., price of a unit, production capacity) Decision Variables and Parameters will be defined for each unique problem

3 DSCI 3023 Linear Programming A method for solving linear mathematical models Linear Functions f(x) = 5x + 1 g(x 1, x 2 ) = x 1 + x 2 Non-linear functions f(x) = 5x g(x 1, x 2 ) = x 1 x 2 + x 2

4 DSCI 3023 Formulating an LP Define the decision variables –identifying the key variables whose values we wish to determine Determine the objective function –determine what we are trying to do –Maximize profit, Minimize total cost Formulate the constraints –determine the limitations of the decision variables

5 DSCI Parts of a Linear Program Objective Function Constraints Non-negativity assumptions

6 DSCI Parts of a Linear Program Objective Function –linear relationships of decision variables describing the problems objective –always consists of maximizing or minimizing some value –e.g., maximize Z = profit, minimize Z = cost

7 DSCI Parts of a Linear Program Constraints –linear relationships of decision variables representing restrictions or rules –e.g., limited resources like labor or capital Non-negativity assumptions –restricts decision variables to values greater than or equal to zero

8 DSCI 3023 Objective Function Always a Max or Min statement –Maximize Profit –Minimize Cost A linear function of decision variables –Maximize Profit = Z = 3x 1 + 5x 2 –Minimize Cost = Z = 6x x 2

9 DSCI 3023 Constraints Constraints are restrictions on the problem –total labor hours must be less than 50 –x 1 must use less than 20 gallons of additive Defined as linear relationships –total labor hours must be less than 50 x 1 + x 2 < 50 –x 1 must use less than 20 gallons of additive x 1 < 20

10 DSCI 3023 Non-negativity Assumptions Negative decision variables are inconceivable in most LP problems Minus 10 units of production or a negative consumption make no sense The assumptions are written as follows –x 1, x 2 > 0

11 DSCI 3023 LP Standard Form Each LP must contain the three parts A standard form for a LP is as follows: MaximizeZ = c 1 x 1 + c 2 x c n x n subject toa 11 x 1 + a 12 x a 1n x n < b 1. a m1 x 1 + a m2 x a mn x n < b m x 1, x 2, x n > 0

12 DSCI 3023 Methods of Solving LP Problems Two basic solution approaches of linear programming exist The graphical Method –simple, but limited to two decision variables The simplex method –more complex, but solves multiple decision variable problems

13 DSCI 3023 Graphical Method 1. Construct an x-y coordinate plane/graph 2. Plot all constraints on the plane/graph 3. Identify the feasible region dictated by the constraints 4. Identify the optimum solution by plotting a series of objective functions over the feasible region 5. Determine the exact solution values of the decision variables and the objective function at the optimum solution