Linear Programming Developed by Dantzig in the late

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 DSCI 3023 1

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 DSCI 3023 2

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

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 DSCI 3023 4

3 Parts of a Linear Program • Objective Function • Constraints • Non-negativity assumptions DSCI 3023 5

3 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 DSCI 3023 6

3 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 DSCI 3023 7

Objective Function • Always a Max or Min statement – Maximize Profit – Minimize Cost • A linear function of decision variables – Maximize Profit = Z = 3 x 1 + 5 x 2 – Minimize Cost = Z = 6 x 1 - 15 x 2 DSCI 3023 8

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 DSCI 3023 9

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 DSCI 3023 10

LP Standard Form • Each LP must contain the three parts • A standard form for a LP is as follows: Maximize subject to DSCI 3023 Z = c 1 x 1 + c 2 x 2 +. . . + cnxn a 11 x 1 + a 12 x 2 +. . . + a 1 nxn < b 1. am 1 x 1 + am 2 x 2 +. . . + amnxn < bm x 1, x 2, xn > 0 11

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 DSCI 3023 12

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 DSCI 3023 13