document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198377>Question 1198377</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Assume that your widget manufacturing company has a total annual demand of N widgets per year evenly distributed across the year. Each widget cost $b dollars in material and manufacturing costs to make. Every time you do a production run to make some widgets, you incur a set-up cost of P dollars. Any widgets awaiting sale must be stored and thus incur an average storage fee of c dollars per widget per year. Let x be the size of each production run (i.e. x is the number of widgets per production run).\r<BR>\n" );
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document.write( "Write a cost function C(x) and explain each term in the equation and how it was determined.<BR>\n" );
document.write( "Write down any constraints on the allowable values of x.<BR>\n" );
document.write( "Determine a formula for the value of x that minimizes total annual cost. Show all of your work.<BR>\n" );
document.write( "Prove that your formula actually corresponds to the global minimum cost.<BR>\n" );
document.write( "Write down a formula for the number of production runs per year as a function of x. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848295 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848295\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ### **1. Cost Function \( C(x) \):**\r<BR>\n" );
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document.write( "The total cost \( C(x) \) includes three components:<BR>\n" );
document.write( "1. **Material and manufacturing cost**:<BR>\n" );
document.write( "   - The total number of widgets demanded annually is \( N \).<BR>\n" );
document.write( "   - Each widget costs \( b \) dollars to make.<BR>\n" );
document.write( "   - The **total manufacturing cost** is:<BR>\n" );
document.write( "     \[<BR>\n" );
document.write( "     \text{Manufacturing Cost} = N \cdot b<BR>\n" );
document.write( "     \]\r<BR>\n" );
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document.write( "2. **Setup cost**:<BR>\n" );
document.write( "   - A production run is done every \( x \) widgets.<BR>\n" );
document.write( "   - The total number of production runs per year is \( \frac{N}{x} \) (since \( N \) widgets are needed annually, and \( x \) widgets are produced per run).<BR>\n" );
document.write( "   - Each production run incurs a setup cost \( P \).<BR>\n" );
document.write( "   - The **total setup cost** is:<BR>\n" );
document.write( "     \[<BR>\n" );
document.write( "     \text{Setup Cost} = \frac{N}{x} \cdot P<BR>\n" );
document.write( "     \]\r<BR>\n" );
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document.write( "3. **Storage cost**:<BR>\n" );
document.write( "   - Widgets are produced in batches of \( x \) and consumed uniformly over the year.<BR>\n" );
document.write( "   - The average number of widgets stored at any time is \( \frac{x}{2} \) (half of the production run, assuming constant consumption).<BR>\n" );
document.write( "   - The storage cost per widget per year is \( c \).<BR>\n" );
document.write( "   - The **total storage cost** is:<BR>\n" );
document.write( "     \[<BR>\n" );
document.write( "     \text{Storage Cost} = \frac{x}{2} \cdot c<BR>\n" );
document.write( "     \]\r<BR>\n" );
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document.write( "Combining these components, the total cost function is:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "C(x) = N \cdot b + \frac{N}{x} \cdot P + \frac{x}{2} \cdot c<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **2. Constraints on \( x \):**\r<BR>\n" );
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document.write( "1. \( x > 0 \): The production run size must be positive.<BR>\n" );
document.write( "2. \( x \leq N \): The production run size cannot exceed the total annual demand.\r<BR>\n" );
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document.write( "Thus, \( x \) must satisfy:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "0 < x \leq N<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **3. Minimizing \( C(x) \):**\r<BR>\n" );
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document.write( "To minimize \( C(x) \), take the derivative of \( C(x) \) with respect to \( x \) and set it to zero.\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "C(x) = N \cdot b + \frac{N}{x} \cdot P + \frac{x}{2} \cdot c<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\frac{dC}{dx} = -\frac{N \cdot P}{x^2} + \frac{c}{2}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Set \( \frac{dC}{dx} = 0 \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "-\frac{N \cdot P}{x^2} + \frac{c}{2} = 0<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\frac{N \cdot P}{x^2} = \frac{c}{2}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "x^2 = \frac{2 \cdot N \cdot P}{c}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "x = \sqrt{\frac{2 \cdot N \cdot P}{c}}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **4. Verifying Minimum:**\r<BR>\n" );
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document.write( "To confirm that this value of \( x \) gives a global minimum, examine the second derivative:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\frac{d^2C}{dx^2} = \frac{2N \cdot P}{x^3}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "For \( x > 0 \), \( \frac{d^2C}{dx^2} > 0 \), indicating that \( C(x) \) is convex and has a global minimum at \( x = \sqrt{\frac{2 \cdot N \cdot P}{c}} \).\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **5. Number of Production Runs per Year:**\r<BR>\n" );
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document.write( "The number of production runs per year is:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\text{Production Runs} = \frac{N}{x}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Substitute \( x = \sqrt{\frac{2 \cdot N \cdot P}{c}} \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\text{Production Runs} = \frac{N}{\sqrt{\frac{2 \cdot N \cdot P}{c}}} = \sqrt{\frac{N \cdot c}{2 \cdot P}}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **Summary of Results:**\r<BR>\n" );
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document.write( "1. **Cost Function**:<BR>\n" );
document.write( "   \[<BR>\n" );
document.write( "   C(x) = N \cdot b + \frac{N \cdot P}{x} + \frac{x \cdot c}{2}<BR>\n" );
document.write( "   \]\r<BR>\n" );
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document.write( "2. **Optimal Production Run Size**:<BR>\n" );
document.write( "   \[<BR>\n" );
document.write( "   x = \sqrt{\frac{2 \cdot N \cdot P}{c}}<BR>\n" );
document.write( "   \]\r<BR>\n" );
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document.write( "3. **Number of Production Runs per Year**:<BR>\n" );
document.write( "   \[<BR>\n" );
document.write( "   \text{Production Runs} = \sqrt{\frac{N \cdot c}{2 \cdot P}}<BR>\n" );
document.write( "   \] </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );