1 Linear Programming: Assumptions and Implications of the LP Model updated 18 January 2006 SMU EMIS 8374 Network Flows

2 Assumptions of the LP Model Proportionality –The contribution of any decision variable to the objective function is proportional to its value. –For example, in the diet problem, the contribution to the cost of the diet from one pound of apples is $0.75, $1.50 from two pounds of apples, $3.00 for four pounds, and $ for four hundred pounds. –In many cases, a volume discount is available whereby the price per pound goes down as more apples are purchased. Such discounts are often nonlinear which means that a linear programming model is either inappropriate or is really just an approximation of the real world.

3 Assumptions of the LP Model Additivity –The contribution to the objective function for any variable is independent of the other decision variables. For example in the NSC production problem, the production of P 2 tons of steel in Month 2 will always contribute $4000 P 2 regardless of how much steel is produced in Month 1.

4 Assumptions of the LP Model Proportionality and Additivity are also implied by the linear constraints. –In the diet problem, you can obtain 40 milligrams of protein for each gallon of milk you drink. It is unlikely, however, that you would actually obtain 4,000 milligrams of protein by drinking 100 gallons of milk. –Also, it may be the case due to a chemical reaction, you might obtain less than 70 milligrams of Vitamin A by combining a pound of cheese with a pound of apples.

5 Assumptions of the LP Model Divisibility –The LP model assumes that the decision variables can take on fractional variables. Thus, it allows for a solution to the GT Railroad problem that sends 0.7 locomotives from Centerville to Fine Place. In many situations, the LP is being used on a large enough scale that one can round the optimal decision variables up or down to the nearest integer and get an answer that is feasible and reasonably close to the optimal integer solution. –Divisibility also implies that the decision variables can take on the full range of real values. For example in the diet problem the LP may tell you to buy apples. For large problems, rounding or truncating of the optimal LP decision variables will not greatly affect the solution.

6 Assumptions of the LP Model Certainty –The LP model assumes that all the constant terms, objective function and constraint coefficients as well as the right hand sides, are know with absolute certainty and will not change. –If the values of these quantities are not known with certainty, for example if the demand data given in the NSC are forecasts that might not be 100% accurate, then this assumption is violated.