Concept explainers
a)
To determine: The impact of adding an extra constraint to the feasible area of a solved linear programming problem.
Introduction:
Linear programming:
It is a linear optimization technique followed to develop the best outcome for the problem in hand. The outcome might be of maximum profit or less cost which is represented by a linear relationship. The outcome will take into consideration the constraints present in achieving the solution.
Constraints:
The constraints are the limitation for a situation within which the process must operate. The constraints are the limits within which the available resources can be utilized that will help to maximize or minimize an amount.
Feasible region:
A feasible region is a solution space which contains all the possible points of an optimization problem. The region will be formed after satisfying the constraints in the problem which includes inequalities, integer constraints and inequalities. It is the area that is bounded by the constraints of the problem.
b)
To determine: The impact of adding an extra constraint to the optimal value of the objective function of a solved linear programming problem.
Introduction:
Linear programming:
It is a linear optimization technique followed to develop the best outcome for the problem in hand. The outcome might be of maximum profit or less cost which is represented by a linear relationship. The outcome will take into consideration the constraints present in achieving the solution.
Constraints:
The constraints are the limitation for a situation within which the process must operate. The constraints are the limits within which the available resources can be utilized that will help to maximize or minimize a quantity.
Feasible region:
A feasible region is a solution space which contains all the possible points of an optimization problem. The region will be formed after satisfying the constraints in the problem which includes inequalities, integer constraints and inequalities. It is the area that is bounded by the constraints of the problem.
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OPERATIONS MGMT.(EBOOK)W/MYOMLAB>IC<
- Consider the following linear programming model with 4 regular constraints:Maximize 3X + 5Y (a) Draw your graph in the space below:subject to: 4X + 4Y ≤ 48 (constraint #1) 4X + 3Y ≤ 50 (constraint #2) 2X + 1Y ≤ 20 (constraint #3) X ≥ 2 (constraint #4) X, Y ≥ 0 (non-negativity constraints)(a) Which of the constraints is redundant? Constraint #______.Justify by drawing a graph similar to Figure 7.14 on p.263.(b) Is point (9,3) a feasible solution? _____. Explain your answer (by analyzing each of the constraints).Constraint #1: _______________________________________________________________Constraint #2: _______________________________________________________________Constraint #3: _______________________________________________________________Constraint #4: ______________________________________________________________arrow_forwardConsider the following LP problem: Min 6X+ 18Y; Subject to : 3 X + 9Y <= 47, and X + Y <= 141. Which one of the following is true?: a. Slack for each constraint is zero. b. Optimal Obj. function value is 94 c. X=70.5, Y=70.5 is the only optimal solution. d. Optimal Obj. function value is 0arrow_forwardConsider the following LP problem with two constraints: 18X + 8Y >= 144and 9X + 4Y= 36. The objective function is Min 14X + 30Y . What combination of X and Y will yield the optimum solution for this problem? a. infeasible problem b. unbounded problem c. 0 , 9 d. 4 , 0 e. 2 , 4.5arrow_forward
- Set up the simplex matrix used to solve the linear programming problem. Assume all variables are nonnegative.Maximize f = 5x + 9y subject to 8x + 5y ≤ 200 x + 6y ≤ 250. x y s1 s2 f first constraint second constraint objective functionarrow_forwardI will need the excel "solver" solution to be able to solve. #4) Solve the following LP Maximize $5x + $6y Subject to 2x + 3y ≤ 10 (labor, in hours) 6x + 6y ≤ 36 (materials, in pounds) 7x + 5y ≤ 40 (storage, in square feet) x, y ≥ 0 a) Write the original optimal solution and objective function value. b) What is the optimal solution and objective function value if you acquire 2 additional pounds of material? c) What is the optimal solution and objective function value if you acquire 1.5 additional hours of labor? d) What is the optimal solution and objective function value if you give up 1 hour of labor and get 1.5 pounds of material? e) What is the optimal solution and objective function value if you introduce a new product that has a profit contribution of $2? Each unit of this product will use 1 hour of labor, 1 pound of material, and 2 square feet of storage space.arrow_forwardGiven this linear programming model, solve the model and then answer the questions that follow.Maximize Z = 12x1 + 18x2 + 15x3 where x1 = the quantity of product 1 to make, etc.Subject toMachine 5x1 + 4x2 + 3x3 ≤ 160 minutes Labor 4x1 + 10x2 + 4x3 ≤ 288 hoursMaterials 2x1 + 2x2 + 4x3 ≤ 200 poundsProduct 2 x2 ≤ 16 units x1, x2, x3 ≥ 0 a. Are any constraints binding? If so, which one(s)?arrow_forward
- Set up the objective function and the constraints, but do not solve.Chemical Products makes two insect repellents, Regular and Super. The chemical used for Regular is 15% DEET, and the chemical used for Super is 25% DEET. Each carton of repellent contains 24 ounces of the chemical. In order to justify starting production, the company must produce at least 14,000 cartons of insect repellent, and it must produce at least twice as many cartons of Regular as of Super. Labor costs are $9 per carton for Regular and $4 per carton for Super. How many cartons of each repellent should be produced to minimize labor costs if 70,680 ounces of DEET are available? (Let x represent the number of cartons of Regular, y the number of cartons of Super, and z the labor costs in dollars.) z = , subject to total production ratio of carton type amount of DEET x ≥ 0, y ≥ 0arrow_forwardSolve the following Linear programming problem using the simplex method:Maximize Z = 10X1 + 15X2 + 20X3subject to:2X1 + 4X2 + 6X3 ≤ 243X1 + 9X2 + 6X3 ≤ 30X1, X2 and X3 ≥ 0(b) Suppose X1, X2, X3 in (a) refer to number of red, blue, and green balloons respectivelywhich are produced by a company per day. And Z is the total profit obtained afterselling these balloons. Interpret your answer obtained in (a) above(c) Write the dual of the following linear programming problem:Minimize Z = 2X1 − 3X2 + 4X3subject to:3X1 + 4X2 + 5X3 ≥ 96X1 + X2 + 3X3 ≥ 47X1 − 2X2 − X3 ≤ 105x1 − 2X2 + X3 ≥ 34X1 + 6X2 − 2X3 ≥ 3X1, X2 and X3 ≥ 0arrow_forwardUsing Excel Solve the following LP Maximize $4x + $5y Subject to 2x + 3y ≤ 20 (labor, in hours) 6x + 6y ≤ 36 (materials, in pounds) 4x + 4y ≤ 40 (storage, in square feet) x, y ≥ 0 a) Write the original optimal solution and objective function value. b) What is the optimal solution and objective function value if you acquire 2 additional pounds of material? c) What is the optimal solution and objective function value if you acquire 1.5 additional hours of labor?arrow_forward
- Construct one example for each of the following types of two-variable linear programs. •Feasible with a unique optimal solution. That is, one set of (x, y) that gives one optimal objective value. Provide an algebraic formulation, graph, decision variable quantities, and optimal objective value (if any).arrow_forwardsuppose a linear programming (maximation) problem has been solved and that the optimal value of the objective function is $300. Suppose an additional constraint is added to this problem. Explain how this might affect the optimal value of the objective function.arrow_forwardYou are given the tableau shown in Table 74 for amaximization problem. Give conditions on the unknowns a1, a2, a3, b, and c that make the following statements true: a The current solution is optimal. b The current solution is optimal, and there are alternativeoptimal solutions. c The LP is unbounded (in this part, assume that b0).arrow_forward
- Practical Management ScienceOperations ManagementISBN:9781337406659Author:WINSTON, Wayne L.Publisher:Cengage,