# The Simplex Methoding Method: The Simplex Solution Method

*Register to read the introduction…*It is inserted into the equation simply to give a positive solution at the origin; we are artificially creating a solution: 2x1 + 4x2 - s1 + A1 = 16 2(0) + 4(0) - 0 + A1 = 16 A1 = 16 The artificial variable is somewhat analogous to a booster rocket—its purpose is to get us off the ground; but once we get started, it has no real use and thus is discarded. The artificial solution helps get the simplex process started, but we do not want it to end up in the optimal solution, because it has no real meaning. When a surplus variable is subtracted and an artificial variable is added, the phosphate constraint becomes 4x1 + 3x2 - s2 + A2 = 24 The effect of surplus and artificial variables on the objective function must now be considered. Like a slack variable, a surplus variable has no effect on the objective function in terms of increasing or decreasing cost. For example, a surplus of 24 pounds of nitrogen does not contribute to the cost of the objective function, because the cost is determined solely by the number of bags of fertilizer purchased (i.e., the values of x1 and x2). Thus, a coefficient of 0 is assigned to each surplus variable in the objective function. By assigning a “cost” of $0 to each surplus variable, we are not prohibiting it from being in the final optimal solution. It would be quite realistic to have a final solution that showed some surplus nitrogen or phosphate. Likewise, assigning a cost of $0 to an artificial variable in the objective function would not prohibit it from being in the final optimal solution. However, if the artificial variable appeared in the solution, it would render the final solution meaningless. Therefore, we must ensure that an artificial

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b. c. d. e.

Formulate the dual of this model. Define the dual variables and indicate their value. Determine the optimal ranges for c1 and c2. Determine the feasible ranges for q1 (pounds of brass) and q2 (labor hours). What is the maximum price the company would be willing to pay for additional labor hours, and how many hours could be purchased at that price?

43. The Southwest Foods Company produces two brands of chili—Razorback and Longhorn—from several ingredients, including chili beans and ground beef. The number of 100-gallon batches of Razorback chili (x1) and Longhorn chili (x2) that can be produced daily is constrained by the availability of chili beans and ground beef, as shown in the following linear programming model: maximize Z = 200x1 + 300x2 (profit, $) subject to 10x1 + 50x2 … 500 (chili beans, lb.) 34x1 + 20x2 … 800 (ground beef, lb.) x1, x2 Ú 0 The final optimal simplex tableau for this model is as follows:

cj 300 200

Basic Variables x2 x1 zj cj - zj

200 Quantity 6 20 5,800 x1 0 1 200 0

300 x2 1 0 300 0

0 s1 17/750 -1>75 310/75