Chapter 3: Section 3-3 Solutions of Linear Programming Problems

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1 Chapter 3: Section 3-3 Solutions of Linear Programming Problems D. S. Malik Creighton University, Omaha, NE D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 1 / 21

2 Geometric Approach to Solve Linear Programming (LP) Problems The objective of a linear programming problem is to maximize or minimize an objective function, subject to certain constraints. We have considered several examples illustrating how to formulate the constraints and the objective function. In the constraints, the inequalities use either the symbol or the symbol, i.e., none of the inequalities are strict. This is important because the rst step in solving a linear programming problem using the geometric approach is to draw the graphs of the inequalities and determine the region (solution set) that satis es the linear inequalities. Next determine the corner points of the solution set. If an inequality is strict, then a corner point given by that inequality may not be in the solution set. Thus, in all of the linear programming problems that we consider, the linear inequalities will not be strict. D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 2 / 21

3 Feasible Region Feasible Region (Solution Set) Bounded Unbounded D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 3 / 21

4 Bounded Feasible Region Theorem ( Bounded Region) Let F be the feasible region of a linear programming problem and F 6=. Let f be the objective function of the linear programming problem. Suppose that F is bounded. Then (i) f attains its maximum value at a corner point of F, (ii) f attains its minimum value at a corner point of F.. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 4 / 21

5 Procedure to nd a solution of a linear programming problem when the feasible region is bounded. 1 Graph the linear inequalities and determine the feasible region (solution set). 2 Find the corner points of the feasible region. 3 Evaluate the objective function at each of the corner points. 4 If the problem is to maximize the objective function, then choose the largest value of the objective function. The coordinates of the corner point that gives the largest value is a solution of the linear programming problem. (If there is more than one such corner point, then choose one of those points.) 5 If the problem is to minimize the objective function, then choose the smallest value of the objective function. The coordinates of the corner point that gives the smallest value is a solution of the linear programming problem. (If there is more than one such corner point, then choose one of those points.) D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 5 / 21

6 Example Consider the following constraints of a linear programming problem: x 3, x + y 0, x y 0, x + 2y 5. The region in the x-y plane that satis es these inequalities is: x - y = 0 y 4 x = 3 B(3, 4) 3 C(3,3) A(-(5/3),(5/3)) 2 1 O(0,0) x + y = 0 -x + 2y = 5 x Corner points of the feasible region are: O(0, 0), A 5 3, 5 3, B(3, 4), and C (3, 3).. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 6 / 21

7 Example The corner points of the feasible region are: O(0, 0), A 5 3, 5 3, B(3, 4), and C (3, 3). Let us minimize 6x 3y subject to the given constraints. Because the feasible region is bounded, the minimum occurs at a corner point. Let us evaluate 6x 3y at the corner points. Now Corner Point Value of the Objective function: 6x 3y O(0, 0) = 0 A 5 3, ( 3 ) = 10 5 = 15 B(3, 4) = = 6 C (3, 3) = 18 9 = 9 The minimum value of 6x 3y is 15 and it occurs at A 5 3, 5 3. D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 7 / 21

8 Example The corner points of the feasible region are: O(0, 0), A 5 3, 5 3, B(3, 4), and C (3, 3). Let us maximize 5x + 2y subject to the given constraints. Because the feasible region is bounded, the maximum occurs at a corner point. Let us evaluate 5x + 2y at the corner points. Now Corner Point Value of the Objective function: 5x + 2y O(0, 0) = 0 A 5 3, ( 3 ) = = 15 3 B(3, 4) = = 23 C (3, 3) = = 21 The maximum value of 5x + 2y is 23 and it occurs at B(3, 4).. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 8 / 21

9 Exercise: Consider the following exercise described earlier. A pet store specializes in cats and bunnies. Each cat costs $9 and each bunny costs $6. The pro t on each cat is $12 and on each bunny is $9. The store cannot house more than 30 animals and cannot spend more than $216 to buy the pets. Under a special agreement the pet store must house at least 2 cats. How many pets of each type should be housed to maximize the pro t? Let x be the number of cats and y be the numbers of bunnies. The linear programming problem describing this problem is: The linear programming problem is: maximize: 12x + 9y subject to: x 2, y 0 x + y 30 3x + 2y 72 D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 9 / 21

10 The graph of the inequalities x 2, y 0, x + y 30, 3x + 2y 72 is: 40 x + y = y x = 2 B(2, 28) C(12, 18) A(2, 0) D(24, 0) x 3x + 2y = 72 The corner points are A(2, 0), B(2, 28), C (12, 18), D(24, 0). D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 10 / 21

11 The corner points are A(2, 0), B(2, 28), C (12, 18), D(24, 0). Example Let us evaluate 12x + 9y at the corner points. Now Corner Point Value of the Objective function: 12x + 9y A(2, 0) = 24 B(2, 28) = 276 C (12, 18) = 306 D(24, 0) = 288 The maximum value of 12x + 9y is 306 and it occurs at C (12, 18). To maximize the pro t, the pet store should house 12 cats and 18 bunnies.. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 11 / 21

12 Unbounded Feasible Set Theorem ( Unbounded Feasible Region) Let F be the feasible region of a linear programming problem and F 6=. Let f be the objective function of the linear programming problem. Suppose that F is unbounded. (i) If f attains its maximum value, then the maximum value occurs at a corner point of F, (ii) If f attains its minimum value, then the minimum value occurs at a corner point of F... S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 12 / 21

13 Unbounded Feasible Set Remark If the feasible region is bounded, then the preceding theorem does not guarantee that an objective function will have a maximum or minimum. It only guarantees that, if the maximum or minimum occurs, then it would occur at a corner point. So how do we know that a maximum or minimum will occur? To give a general answer to this problem, we will state a theorem at the end of this section. However, there are some special cases. Let us consider some special cases that you will encounter frequently when solving a real-world problem.. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 13 / 21

14 Suppose that the feasible region is restricted to the rst quadrant and it extends inde nitely in the rst quadrant. Consider the object function 2x + 3y. Because the objective function extends inde nitely in the rst quadrant, the value of 2x + 3y increases without bounds when the values of x and y increase in the feasible region. That is, 2x + 3y has no xed largest value. So 2x + 3y has no maximum in the feasible region. In fact, if the objective function is ax + by, where a 0, b 0, and both a and b are not zero, then the objective function ax + by has no maximum. Suppose that the feasible region is restricted to the rst quadrant and it extends inde nitely in the rst quadrant. However, the objective function, say 3x + 5y, is to be minimized. In this case, the minimum will occur at a corner point, because the value of 3x + 5y cannot be made arbitrarily small. Thus, in general, in this case, if the objective function is ax + by, where a 0, b 0, and both a and b are not zero, then the objective function ax + by has the minimum, and the minimum will occur at a corner point. D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 14 / 21

15 Suppose that the feasible region is restricted to the third quadrant and it extends inde nitely in the third quadrant. Suppose that the objective function is ax + by, where a 0, b 0, and both a and b are not zero. Then ax + by has the maximum, and the maximum will occur at a corner point. Furthermore, ax + by has no minimum. D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 15 / 21

16 Example A hiker wants to take a snack mix of peanuts and raisins. The hiker wants 1000 calories and 120 grams of fat from the mix. Each gram of peanuts contains 8 calories and 0.4 grams of fat, and costs 7 cents. Each gram of raisins contains 2 calories and 0.6 grams of fat, and costs 4 cents. How many grams of each food should the hiker take so that the cost of the snack is the minimum? Suppose that the hiker takes x grams of peanuts and y grams of raisins. Then Variable Calories Fat Cost Each gram of peanuts x Each gram of raisins y Total (required) D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 16 / 21

17 The linear programming problem is: minimize: 7x + 4y subject to x 0 y 0 8x + 2y x + 0.6y 120. D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 17 / 21

18 x 0, y 0, 8x + 2y 1000, 0.4x + 0.6y 120. We graph these inequalities and obtain the following graph: y A(0, 500) x + 3y = B(90, 140) x + y = 500 C(300, 0) x The corner points are: A(0, 500), B(90, 140), C (300, 0) D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 18 / 21

19 The corner points are: A(0, 500), B(90, 140), C (300, 0) Let us evaluate the objective function at the corner points. Thus, Corner point Value of the objective function: 7x + 4y A(0, 500) = 2000 B(90, 140) = = 1190 C (300, 0) = 2100 From this table, the minimum value is 1190 and this value occurs at B(90, 140). Thus, the minimum occurs when x = 90 and y = 140. Thus, the hiker must take 90 grams of peanuts and 140 grams of raisins to minimize the cost.. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 19 / 21

20 Theorem (Solutions of Linear Programming Problems with Unbounded Feasible Sets). Let L be a linear programming problem and S be the feasible region of L. Let f be the objective function of L. Suppose that S is unbounded. (i) Suppose that f is to be maximized. Let T be the set of all corner points of S. Let A be a corner point in T that gives the maximum value of f. Let l 1 and l 2 be the boundary lines of S such that l 1 and l 2 intersect at A. On each line l 1 and l 2, choose a point in the feasible region that is not a corner point and evaluate f at that point. If the value of f at that point is less than or equal to the value of f at A, then the linear programming problem has a solution, and it is the largest value that is attained at a corner point. Otherwise the problem has no solution.. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 20 / 21

21 Theorem (ii) Suppose that f is to be minimized. Let T be the set of all corner points of S. Let A be a corner point in T that gives the minimized value of f. Let l 1 and l 2 be the boundary lines of S such that l 1 and l 2 intersect at A. On each line l 1 and l 2, choose a point in the feasible region that is not a corner point and evaluate f at that point. If the value of f at that point is greater than or equal to the value of f at A, then the linear programming problem has a solution, and it is the smallest value that is attained at a corner point. Otherwise the problem has no solution.. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming Problems 21 / 21