Mathematical modeling

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2.5.3 Problems Having No Feasible SolutionIt is possible that there are no feasible solution for a given problem formulation. This mayoccur when constraints conflict with one another. Or it may occur due to errors inentering problem formulation to a solver. Thus means that there is no specific solutionthat satisfies all constraints together. Assume the following set of constraints:5x 1 + 5x 2 ≤ 50x 1 ≥ 8x 2 ≥ 6Figure 2.19 shows three constraints plotted in decision space in such a way that there isno feasible region formed; the problem is said to be infeasible.Figure 2.19: Graphical solution showing no-feasible region2.5.4 Problems that are UnboundedThe infeasible solution results from problems that are over constrained. In the currentcase, we may encounter a situation where the problem is under constrained. Consider thefollowing set of constraints:5x 1 + 5x 2 ≥ 50x 1 ≤ 8x 1 ≥ 6Systems Analysis 31 Dr. Emad Elbeltagi
In Fig. 2.20, the feasible region is shaded and bounded by three constraints. The objectivefunction line and its direction of improvement are also shown. It is clearly shown that forany feasible solution in this decision space, there is always another one that gives a bettervalue of the objective function such that the objective function for this problem could bemoved upwards and to the right without limit. Such a problem is said to be unbounded.5x 1 + 5x 2 ≥ 50x 1 ≥ 6Unbounded feasible regionx 1 ≤ 8Objective function lineFigure 2.20: Unbounded feasible region2.6 Limitations of Linear Programming ModelingHidden in our models of these problems are a number of assumptions. The usefulness ofa model is directly related to how close reality matches up with these assumptions. Thefirst assumption is due to the linear form of our functions. Since the objective is linear,the contribution to the objective of any decision variable is proportional to the value ofthe decision variable. Producing twice as much of a product produces twice as muchprofit; buying twice as many pages of ads costs twice as much. This is theProportionality Assumption.Furthermore, the contribution of a variable to the objective is independent of the valuesof the other variables. One notebook computer is worth LE750, independent of howmany desktop computers we produce. This is the Additivity Assumption. Similarly, sinceeach constraint is linear, the contribution of each variable to the left hand side of eachconstraint is proportional to the value of the variable and independent of the values of anyother variable.Systems Analysis 32 Dr. Emad Elbeltagi
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Page 12 and 13: Changeable cells(variables)Target c
Page 14 and 15: 2.4 Second Example: Market Allocati
Page 16 and 17: Figure 2.13: Marketing example show
Page 18 and 19: Model formulationFirst we have to d
Page 22 and 23: These assumptions are quite restric
Page 24 and 25: 2.7.2 Workforce PlanningProblem Def
Page 26 and 27: Now, let's look at the average risk
Page 28 and 29: Profit produced from each 240-m 2 h
Page 30 and 31: Monthly delivery to wholesaler 2Mon
Page 32 and 33: solve the following problem using t
Page 34 and 35: espectively. If 100 units of raw ma
Page 36 and 37: In a given period there are 600 hou
Page 38 and 39: Minimize z = - 4x 1 - 6x 2Subject t

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