Suppose a certain manufacturing company produces connecting rods for 4- and 6-cylinder automobile engines using the same production line. The cost required to set up the production line to produce the 4-cylinder connecting rods is $2,200, and the cost required to set up the production line for the 6-cylinder connecting rods is $3,700. Manufacturing costs are $12 for each 4-cylinder connecting rod and $18 for each 6-cylinder connecting rod. Hawkins makes a decision at the end of each week as to which product will be manufactured the following week. If a production changeover is necessary from one week to the next, the weekend is used to reconfigure the production line. Once the line has been set up, the weekly production capacities are 6,000 6-cylinder connecting rods and 8,000 4-cylinder connecting rods. Let X4 = the number of 4-cylinder connecting rods produced next week x6 = the number of 6-cylinder connecting rods produced next week S4 = 1 if the production line is set up to produce the 4-cylinder connecting rods; 0 if otherwise S6 = 1 if the production line is set up to produce the 6-cylinder connecting rods; 0 if otherwise (a) Using the decision variables x4 and s4, write a constraint that limits next week's production of the 4-cylinder connecting rods to either 0 or 8,000 units. (b) Using the decision variables x and S6, write a constraint that limits next week's production of the 6-cylinder connecting rods to either 0 or 6,000 units. (c) Write a third constraint that, taken with the constraints from parts (a) and (b), limits the production of connecting rods for next week. (d) Write an objective function for minimizing the cost of production for next week. Min
Suppose a certain manufacturing company produces connecting rods for 4- and 6-cylinder automobile engines using the same production line. The cost required to set up the production line to produce the 4-cylinder connecting rods is $2,200, and the cost required to set up the production line for the 6-cylinder connecting rods is $3,700. Manufacturing costs are $12 for each 4-cylinder connecting rod and $18 for each 6-cylinder connecting rod. Hawkins makes a decision at the end of each week as to which product will be manufactured the following week. If a production changeover is necessary from one week to the next, the weekend is used to reconfigure the production line. Once the line has been set up, the weekly production capacities are 6,000 6-cylinder connecting rods and 8,000 4-cylinder connecting rods. Let X4 = the number of 4-cylinder connecting rods produced next week x6 = the number of 6-cylinder connecting rods produced next week S4 = 1 if the production line is set up to produce the 4-cylinder connecting rods; 0 if otherwise S6 = 1 if the production line is set up to produce the 6-cylinder connecting rods; 0 if otherwise (a) Using the decision variables x4 and s4, write a constraint that limits next week's production of the 4-cylinder connecting rods to either 0 or 8,000 units. (b) Using the decision variables x and S6, write a constraint that limits next week's production of the 6-cylinder connecting rods to either 0 or 6,000 units. (c) Write a third constraint that, taken with the constraints from parts (a) and (b), limits the production of connecting rods for next week. (d) Write an objective function for minimizing the cost of production for next week. Min
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:**Title: Optimization of Production for Connecting Rods**
**Introduction:**
Suppose a certain manufacturing company produces connecting rods for 4-cylinder and 6-cylinder automobile engines using the same production line. The cost required to set up the production line to produce the 4-cylinder connecting rods is $2,200, and the cost required to set up the production line for the 6-cylinder connecting rods is $3,700. Manufacturing costs are $12 for each 4-cylinder connecting rod and $18 for each 6-cylinder connecting rod. Hawkins makes a decision at the end of each week as to which product will be manufactured the following week. If a production changeover is necessary from one week to the next, the weekend is used to reconfigure the production line. Once the line has been set up, the weekly production capacities are 6,000 6-cylinder connecting rods and 8,000 4-cylinder connecting rods.
**Variables Defined:**
- \( x_4 \) = the number of 4-cylinder connecting rods produced next week
- \( x_6 \) = the number of 6-cylinder connecting rods produced next week
- \( s_4 \) = 1 if the production line is set up to produce the 4-cylinder connecting rods; 0 if otherwise
- \( s_6 \) = 1 if the production line is set up to produce the 6-cylinder connecting rods; 0 if otherwise
**Constraints:**
(a) **Constraint for 4-cylinder connecting rods:**
Using decision variables \( x_4 \) and \( s_4 \), the constraint limits next week's production of the 4-cylinder connecting rods to either 0 or 8,000 units.
(b) **Constraint for 6-cylinder connecting rods:**
Using decision variables \( x_6 \) and \( s_6 \), the constraint limits next week's production of the 6-cylinder connecting rods to either 0 or 6,000 units.
(c) **Combined production constraint:**
A third constraint is written to limit the production of connecting rods for next week, combining the constraints from parts (a) and (b).
(d) **Objective function:**
Write an objective function for minimizing the cost of production for next week.
Minimization involves both production costs and setup costs associated with each type of connecting rod.
This setup will help in effective decision making to minimize production costs while adhering to constraints on production capacity.
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