Establishing a mathematical model (Ch. ): 1. Define the problem, gathering data;. Formulate the model; 3. Derive solutions;. Test the solution; 5. Apply. Linear Programming: Introduction (Ch. 3). An example (p. p.7): The WYNDOR GLASS CO. has some production capacity available for two new products. The problem is to determine how many of each new products to produce using the available capacity so that the additional revenue can be maximized. 1. The data: The data gathered are summarized in the table:. The mathematical model: (a) The variables: Production Time per Batch, Hours Product Production Time Plant 1 Available per Week, Hours 1 1 0 0 1 3 3 1 Profit perbatch $3,000 $5,000 z = the profit, in thousands of dollars; (the objective variable) x i = the amount of product i to be produced. (the decision varibles) We want to We also must have Max z =3x 1 +5x x 1 x 1 3x 1 +x 1. x 1 0,x 0. All functions involved are linear. 1
Thereseemstobeinfinitely many (x 1,x ) satisfying the constraints. The graph: 10-1 0 1 3 5 7 10-1 0 1 3 5 7 Some observations: 3x 1 +5x =10and 3x 1 +5x =15added. 1. The maximum must appear at the boundary of the region;. The maximum must appear at least at one corner.. Therefore, even though there might be infinite many of candidates for the maximum, we only need to consider a finite many of them (i.e. the corner points). Note: At this point, these observations are only backed by this graph, unless we can prove them mathematically, we can not consider them true in general. The corners and the corresponding values of z are: (0, 0) 0 (, 0) 1 (0, ) 30 (, ) 3 (, 3) 7 Using the two observations, we find that the optimal solution is at (, ) with z =3. This method for solving simple linear programming systems with two variables is called the graphical method.
General form and some terminologies. For a standard linear programming model, we have An objective function: such as z =3x 1 +5x. A number of decision variables: x 1,x,... Functional constraints: x 1, x 1, and 3x 1 +x 1. Nonnegative constraints: x i 0 for all i. The standard form of a general linear programming model is Max z = c 1 x 1 + c x + + c n x n a 11 x 1 + a 1 x + + a 1n x n b 1 a 1 x 1 + a x + + a n x n b a m1 x 1 + a m x + + a mn x n b m x i 0 for all i =1,..., n. where c 1,..., c n and a 11,a 1,..., a mn are constants. feasible solution: a solution that satisfies all the constraints. feasible region: the set of all feasible solutions. optimal solution: a feasible solution that has the largest value (for a Max problem, for a Min problem it is the smallest value) of the objective function. For the linear programming model to work, a number of conditions have to be satisfied. Read Section 3.3 for more details. Proportionality assumption: The contribution of each decision variable x i to the value of z is proportional to their value. Additivity assumption: Every function is the sum of the individual contributions of the respective activities. Divisibility assumption: Every decision variable x i areallowedtohaveanyrealvalue. Certainty assumption: All the coefficients are known constants.. 3
Another example and other forms of LP LP models may have different forms than the standard form. The Max canbereplacedbymin and can be replaced by or =. Wewillseelaterthatsometimestheymaycausesomeproblems but the nature of the LP model remains the same. Example. An auto company believes that its most likely customers are high-income women and men. To reach these groups, the company developed a TV commercial to be aired on two types of programs: comedy shows and football games. It is estimated that each comedy is seen by 7 million women and million men in the targeted group and each football game is been watched by million women and 1 million men in the targeted group. An ad in a comedy show costs $50,000 and an ad in a football game costs $100,000. The company wants its ad to reach at least million women and million men in its targeted group with the minimum cost. The model: Let x 1 bethenumberoftimestheadtobeshownincomedyshowsandx be the number of times the ad to be shown in football games, z the total cost. Min z =50x 1 +100x 7x 1 +x x 1 +1x x 1,x 0 (To be precise, the divisibility assumption is not satisfied here. However if the campion is to be repeated in a longer period of time, then this is not a problem.) The difference here: Min,. 1 1 10 0 10 1 1 50x 1 +100x = 500 and50x 1 +100x = 1000 added. From the graph we see another difference: The feasible region is unbounded. Still, this problem has an optimal solution at (3., 1.). What happens if this is a Max problem? This can happen even for problems in the standard form: Max z =3x 1 +x 3x 1 +x x 1 +9x 3 x 1,x 0
5 3 1-1 0 1 3-1 In this case, we say that the model has no optimal solution because of the unbounded feasible region. In practice, that means some constraints have not been formulated into the model. Theoppositesituationisthatthefeasibleregionisempty: Max z = x 1 +x x 1 +x 5x 1 +3x 15 x 1,x 0 5 3 1-1 0 1 3-1 When the feasible region is bounded, then there is an optimal solution no matter it is a Max problem or a Min problem. Sometimes, it may have more than one optimal solution: Max z =x 1 +0.5x x 1 +5x 30 x 1 + x 1 x 1,x 0 5
10-1 0 1 3 5