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<h2 class="hd hd-2 unit-title">Practice Problem 1</h2>
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PART A
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<p>Charles Watts Electronics manufactures the following four peripheral devices used in computers especially designed for jet fighter planes: cameras, graphics circuit boards, USB memory stick, hard disk drives. Each of these technical products requires time, in minutes, on three types of electronic testing equipment as shown in the following table:
</p>
<center>
<table>
<tbody>
<tr>
<th scope="col" style="text-align: center; border: 1px solid black; padding: 10px;"/>
<th scope="col" style="text-align: center; border: 1px solid black; padding: 10px;">Camera</th>
<th scope="col" style="text-align: center; border: 1px solid black; padding: 10px;">Circuit Board</th>
<th scope="col" style="text-align: center; border: 1px solid black; padding: 10px;">Memory Stick</th>
<th scope="col" style="text-align: center; border: 1px solid black; padding: 10px;">Hard Drive</th>
</tr>
<tr>
<th scope="row" style="text-align: center; border: 1px solid black; padding: 10px;">Test device 1</th>
<td style="text-align: center; border: 1px solid black; padding: 10px;">7</td>
<td style="text-align: center; border: 1px solid black; padding: 10px;">12</td>
<td style="text-align: center; border: 1px solid black; padding: 10px;">6</td>
<td style="text-align: center; border: 1px solid black; padding: 10px;">18</td>
</tr>
<tr>
<th scope="row" style="text-align: center; border: 1px solid black; padding: 10px;">Test device 2</th>
<td style="text-align: center; border: 1px solid black; padding: 10px;">2</td>
<td style="text-align: center; border: 1px solid black; padding: 10px;">3</td>
<td style="text-align: center; border: 1px solid black; padding: 10px;">2</td>
<td style="text-align: center; border: 1px solid black; padding: 10px;">15</td>
</tr>
<tr>
<th scope="row" style="text-align: center; border: 1px solid black; padding: 10px;">Test device 3</th>
<td style="text-align: center; border: 1px solid black; padding: 10px;">5</td>
<td style="text-align: center; border: 1px solid black; padding: 10px;">3</td>
<td style="text-align: center; border: 1px solid black; padding: 10px;">2</td>
<td style="text-align: center; border: 1px solid black; padding: 10px;">9</td>
</tr>
</tbody>
<caption>Table 1: Time Requirement</caption>
</table>
</center>
<p>The first two test devices are available 100 hours per week. The third (device 3) requires more preventive maintenance and may be used only 80 hours each week. Watts Electronics believes that it cannot sell more than 1000, 700, 800, and 600 units of each device, respectively. Thus, it does not want to produce more than these units.
In addition, profit per unit is $140 for cameras, $92 graphics circuit boards, $135 USB memory stick, $230 hard disk drives. Watts Electronics wants to maximize its profits. </p>
<p>Below, a solution report and a sensitivity report are given. Using these two exhibits, answer the
following questions. Note that three values, identified by "A"
and "B" are missing in the sensitivity report. Two of
the questions will ask you to compute "A" and "B". The missing
values are not needed for the remaining questions.</p>
<p>
<a class="modal-content" href="/assets/courseware/v1/28f8d1b2d19db1f9220c4f0856f44982/asset-v1:MITx+15.053x+3T2016+type@asset+block/AnswerReport.tiff"> Answer Report</a>
</p>
<p>
<a class="modal-content" href="/assets/courseware/v1/4ffbf2cfe879eaba603a0ef940257e0f/asset-v1:MITx+15.053x+3T2016+type@asset+block/SensitivityReport.tiff"> Sensitivity Report</a>
</p>
<p> Suppose that Watts Electronics is considering to increase the available time of Test Device 2 for the next week.
What would be the increase in the profit if the available time increases to 102 hours?</p>
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PART B
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<p> By how much should the profit for selling one unit of
hard drives, before hard drives could be considered for production in
the optimal production mix?</p>
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PART C
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<p>What would be the increase in the profit if the profits are increased to $145 per unit for cameras and $140 per unit for memory sticks? Round to two decimals.
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PART D
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<p>Watts Electronics is considering the possibility of decreasing the available time of test devices 3 from 80 hours to 30 hours. Which of the following statements is the most accurate? Check one.
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Reducing the number of Test Device 3 hours by 50 will not
affect total profit.
</label>
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<input type="radio" name="input_c116d7c4ba2142848741154cdc48a621_2_1" id="input_c116d7c4ba2142848741154cdc48a621_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c116d7c4ba2142848741154cdc48a621_2_1-choice_1-label" for="input_c116d7c4ba2142848741154cdc48a621_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c116d7c4ba2142848741154cdc48a621_2_1">
Reducing the number of Test Device 3 hours by 50 will decrease
profit by $500.
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Reducing the number of Test Device 3 hours by 50 could
affect total revenue, but we do not have enough information to
compute the magnitude of the change.
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PART E
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<p>Suppose the profit of cameras decreased from 140 to 125.
What would be the new optimal solution? What would be the new
profit to two decimals? (If there is not enough information in the problem to give
the answer, write "-1".)
</p>
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<h3 class="hd hd-3 problem-header" id="090209affaab4e1cb8cefcd794fa711b-problem-title" aria-describedby="block-v1:MITx+15.053x+3T2016+type@problem+block@090209affaab4e1cb8cefcd794fa711b-problem-progress" tabindex="-1">
PART F
</h3>
<div class="problem-progress" id="block-v1:MITx+15.053x+3T2016+type@problem+block@090209affaab4e1cb8cefcd794fa711b-problem-progress"></div>
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<p>Use "pricing out" to compute the missing value B in the sensitivity report. "B"
is the reduced cost of Circuit Boards.
</p>
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\(\)
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PART G
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<p>Suppose that we increased the availability of Device 1 from 6000 to 7000 minutes and
simultaneously decreased the demand for circuit boards from 700 to 500. What would be the "increase"
in the optimal objective value? Use the 100% rule to determine if there is enough information to give an
answer. If there is not enough information, then enter "-1". </p>
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PART A
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<p>Consider the LP:</p>
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<p class="equation">\[\left. \begin{array}{rrcl} \max &amp; -10x_1-4x_2 &amp; &amp; \\
\text {s.t.:} &amp; &amp; &amp; \\
(1)&amp; 2x_1+x_2\leq 4 \\
(2)&amp; x_1+2x_2\leq 4 \\
(3)&amp; x_1 +x _2\geq 1 \\
(4)&amp; x_1,x_2\geq 0 \nonumber \\
\end{array}\right\}\]</p>
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<p>Solve geometrically and also trace the simplex procedure steps graphically. What is the optimal objective value? </p>
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PART B
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<p>Starting with your graphical solution to the previous part, what is the shadow price for the third constraint?</p>
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PART C
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<p>Suppose that the objective function is changed to \(z=-x_1+cx_2\). Graphically determine the values of \(c\) for which the solution found in part (a) remains optimal.
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\(-1 \leq c \leq 0\)
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\(c \leq 0\)
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