Max 4A + 3B s.t. 1A <= 120 1B <= 90 2A + 3B <= 360 A,B >= 0 What is the optimal solution value for A?
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| Max 4A + 3B |
| s.t. |
| 1A <= 120 |
| 1B <= 90 |
| 2A + 3B <= 360 |
| A,B >= 0 |
What is the optimal solution value for A?
Linear programming is one of the mathematical techniques that is widely used by companies. This technique is basically used to get the maximum or minimum of the problems. There are various tools and procedures to use this technique and each of them is used on the basis of different factors.
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- We have 60 meters of fence and want to fence a triangular shaped area. Please formulate an NLP (do not try to solve) that will enable us to maximize the fenced area (Hint: The area of a triangle with sides of length a, b, and c is ( s (s – a) (s – b) (s – c))1/2, where s is half the parameter of the triangle).A firm decides to invest x units of capital in project A and y units in project B. The expected return for 1 unit of investment is $400 in project A and $800 in project B. However, in order to meet the expectations of the firm’s ethical and environmental policy, the values of x and y must satisfy the constraintx2 + y2 − 4x − 6y = 67How many units of each type should the firm buy in order to maximize total return?If there is a maximum of 4,000 hours of labor available per month and 300 ping-pong balls (x1) or 125 wiffle balls (x2) can be produced per hour of labor, which of the following constraints reflects this situation? a. 300x1 + 125x2 = 4,000 b. 425(x1 + x2) < 4,000 c. 300x1 + 125x2 > 4,000 d. 300x1 + 125x2 < 4,000
- Find the values of x1 and x2 where the following two constraints intersect. (Round your answers to 3 decimal places.) (1) 10x1 + 5x2 ≥ 50 (2) 1x1 + 2x2 ≥ 12True or False 1. Given three corner points A, B, and C of a linear programming problem, if A is adjacent to B and B is adjacent to C, then A can be determined from C by interchanging exactly two basic and two nonbasic variables. 2. For a set of primary and dual solutions to be feasible, the objective function value of the primary (Z) can never exceed that of the dual (W) no matter which problem is to maximize and which is to minimize. 3. If the current iteration is degenerate, the next iteration will also be degenerate. 4. The optimal (dual) primary solution can be obtained from the dual (primary) optimal tableau.For the products A, B, C, and D, which of the following could be a linear programming objective function? Select one: a. Z = 1A + 2BC + 3D b. Z = 1A + 2AB + 3ABC + 4ABCD c. Z = 1A + 2B + 3C + 4D d. Z = 1A + 2B/C + 3D
- A. What is the optimal solution and what is the optimal value of the objective function? B. Which constraints are binding? C. What are the dual values? Interpret each. D. If you could change the right-hand side of one constraint by one unit, which one would you choose? Why?Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular model and a catcher's model. Letting R = number of regular gloves C = number of catcher's mitts leads to the following formulation: Max 6R + 9C s.t. R + 3 2 C ≤ 1,000 Cutting and sewing 1 2 R + 1 3 C ≤ 300 Finishing 1 8 R + 1 4 C ≤ 100 Packaging and shipping R, C ≥ 0 The computer solution is shown below. Optimal Objective Value = 4350.00000 Variable Value Reduced Cost R 500.00000 0.00000 C 150.00000 0.00000 Constraint Slack/Surplus Dual Value 1 275.00000 0.00000 2 0.00000 4.50000 3 0.00000 30.00000 Variable ObjectiveCoefficient AllowableIncrease AllowableDecrease R 6.00000 7.50000 1.50000 C 9.00000 3.00000 5.00000 Constraint RHSValue AllowableIncrease AllowableDecrease 1 1000.00000 Infinite 275.00000 2 300.00000 100.00000 166.66667 3…On the other hand, the point (x1 ϭ 15, x2 ϭ 70) is not in the feasible region, because eventhough x1 ϭ 15 and x2 ϭ 70 satisfy (2), (4), (5), and (6), they fail to satisfy (3): 15 ϩ 70is not less than or equal to 80. Any point that is not in an LP’s feasible region is said tobe an infeasible point. As another example of an infeasible point, consider (x1 ϭ 40,×2 ϭ Ϫ20). Although this point satisfies all the constraints and the sign restriction (5), itis infeasible because it fails to satisfy the sign restriction (6), x2 Ն 0. The feasible regionfor the Giapetto problem is the set of possible production plans that Giapetto must consider in searching for the optimal production plan.DEFINITION sFor a maximization problem, an optimal solution to an LP is a point in thefeasible region with the largest objective function value. Similarly, for aminimization problem, an optimal solution is a point in the feasible…
- In a capital budgeting problem, a decision maker having a limited amount of budget is considering to give financial support to research project A, B, C, D and E. Suppose is a binary variable, which equals 1 if project is chosen and 0 otherwise. Which of the following logical constraints correctly models the requirement that “If Project C is not chosen, Then Projects A and B must be chosen”? A. Xa +Xb <= Xc B. 1- Xc <= Xa +Xb C. 1- Xc <= Xa , 1- Xc <= Xb D. Xa +Xb <= 1 + XcFind solution using BigM (penalty) method.Maximize Z = x1 + 2x2 + 3x3 - x4subject to the constraintsx1 + 2x2 + 3x3 = 152x1 + x2 + 5x3 = 20x1 + 2x2 + x3 + x4 = 10and x1, x2, x3, x4 ≥ 0