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Problem 387

Solve the following linear programming problem: Maximize \(\quad 6 \mathrm{~L}_{1}+11 \mathrm{~L}_{2}\) subject to: \(\quad 2 \mathrm{~L}_{1}+\mathrm{L}_{2} \leq 104\) \(\mathrm{L}_{1}+2 \mathrm{~L}_{2} \leq 76\) and \(L_{1} \geq 0, L_{2} \geq 0\)

Short Answer

Expert verified
The solution of the given linear programming problem is \(L_1 = 20\) and \(L_2 = 28\), yielding a maximum objective function value of 504.
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Step 1: Graph the Constraints

To graph the constraints, first transform each inequality into an equation: \(2L_1+L_2 = 104\) and \(L_1 + 2L_2 = 76\). Now, plot these two lines on the Cartesian coordinate system together with the conditions \(L_1\geq 0\) and \(L_2\geq 0\). You should get a feasible region (polygon) enclosed by these lines and the axes.

Step 2: Find the Vertices of the Feasible Region

Next, find the vertices of the feasible region by finding the points where the constraint lines intersect each other and the axes. 1. Intersection with the x-axis: Solve \(2L_1+L_2 = 104\) with \(L_2 = 0\). 2. Intersection with the y-axis: Solve \(L_1+2L_2 = 76\) with \(L_1 = 0\). 3. Intersection between constraint lines: Solve the system of equations \(2L_1+L_2 = 104\) and \(L_1+2L_2 = 76\). Using these methods, we obtain the following vertices of the feasible region: - Vertex A: \((0,0)\) - Vertex B: \((0,38)\) - Vertex C: \((52,0)\) - Vertex D: \((20,28)\)

Step 3: Evaluate the Objective Function at Each Vertex

Now that we have found the vertices of the feasible region, we need to evaluate the objective function at each vertex: 1. Vertex A: (0,0) - Objective Function Value: \(6(0)+11(0)=0\) 2. Vertex B: (0,38) - Objective Function Value: \(6(0)+11(38)=418\) 3. Vertex C: (52,0) - Objective Function Value: \(6(52)+11(0)=312\) 4. Vertex D: (20,28) - Objective Function Value: \(6(20)+11(28)=504\)

Step 4: Determine the Maximum Value

Now that we have evaluated the objective function at each vertex, we can compare the values to determine which vertex yields the maximum value. The results are: - Vertex A: 0 - Vertex B: 418 - Vertex C: 312 - Vertex D: 504 The maximum value, 504, is achieved at the vertex D with coordinates (20, 28). Therefore, the solution of the given linear programming problem is \(L_1 = 20\) and \(L_2 = 28\), yielding a maximum objective function value of 504.

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Most popular questions from this chapter

Chapter 17

The Brown Company has two warehouses and three retail outlets. Warehouse number one (which will be denoted by \(\mathrm{W}_{1}\) ) has a capacity of 12 units; warehouse number two \(\left(\mathrm{W}_{2}\right)\) holds 8 units. These warehouses must ship the product to the three outlets, denoted by \(\mathrm{O}_{1}, \mathrm{O}_{2}\), and \(\mathrm{O}_{3} \cdot \mathrm{O}_{1}\) requires 8 units. \(\mathrm{O}_{2}\) requires 7 units, and \(\mathrm{O}_{3}\) requires 5 units. Thus, there is a total storage capacity of 20 units, and also a demand for 20 units. The question is, which warehouse should ship how many units to which outlet? (The objective being, of course, to accomplish this at the least possible cost.) Costs of shipping from either warehouse to any of the outlets are known and are summarized in the following table, which also sets forth the warehouse capacities and the needs of the retail outlets: $$ \begin{array}{|c|c|c|c|c|} \hline & \mathrm{O}_{1} & \mathrm{O}_{2} & \mathrm{O}_{3} & \text { Capacity } \\\ \hline \mathrm{W}_{1} \ldots \ldots \ldots & \$ 3.00 & \$ 5.00 & \$ 3.00 & 12 \\\ \hline \mathrm{W}_{2} \ldots \ldots . . & 2.00 & 7.00 & 1.00 & 8 \\ \hline \text { Needs (units) } & 8 & 7 & 5 & \\ \hline \end{array} $$

Chapter 17

Give an example of a two-person non-zero-sum game.

Chapter 17

Players \(\mathrm{A}\) and B simultaneously call out either of the numbers 1 and 2 . If their sum is even, \(B\) pays \(A\) that number of dollars, if odd, A pays B. What kind of strategy should both players adopt?

Chapter 17

According to the Fundamental Theorem of linear programming, if either a linear program or its dual has no feasible point, then the other one has no solution. Illustrate this assertion with an example.

Chapter 17

Find nonnegative numbers $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4}\( which maximize \)3 \mathrm{x}_{1}+\mathrm{x}_{2}+9 \mathrm{x}_{3}-9 \mathrm{x}_{4}$ and satisfy the conditions $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}-5 \mathrm{x}_{4}=4$ \(\mathrm{x}_{1}-\mathrm{x}_{2}+3 \mathrm{x}_{3}+\mathrm{x}_{4}=0\)

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