document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168521>Question 1168521</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A retail outlet for calculators sells 700 calculators per year. It costs $2 to store one calculator for a year. To reorder, there is a fixed cost of $5, plus $2.35 for each calculator. How many times per year should the store order calculators, and in what lot size, in order to minimize inventory costs? The store should order ___ calculators ___ times per year to minimize inventory costs.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851563 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=851563\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem using the Economic Order Quantity (EOQ) model.\r<BR>\n" );
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document.write( "**1. Define Variables**\r<BR>\n" );
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document.write( "* D = Annual demand = 700 calculators<BR>\n" );
document.write( "* H = Holding cost per unit per year = $2<BR>\n" );
document.write( "* S = Ordering cost per order = $5 (fixed cost) + 2.35 * Q (where Q is the lot size)\r<BR>\n" );
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document.write( "**2. EOQ Formula**\r<BR>\n" );
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document.write( "The EOQ formula is:\r<BR>\n" );
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document.write( "EOQ = √(2DS / H)\r<BR>\n" );
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document.write( "However, we have an ordering cost that depends on the quantity, which is not the typical EOQ situation.\r<BR>\n" );
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document.write( "**3. Total Inventory Cost**\r<BR>\n" );
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document.write( "Total cost (TC) = Holding cost + Ordering cost\r<BR>\n" );
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document.write( "* Holding cost = (Q/2) * H = (Q/2) * 2 = Q<BR>\n" );
document.write( "* Ordering cost = (D/Q) * S = (700/Q) * (5 + 2.35Q)\r<BR>\n" );
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document.write( "TC = Q + (700/Q) * (5 + 2.35Q)<BR>\n" );
document.write( "TC = Q + 3500/Q + 1645\r<BR>\n" );
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document.write( "To minimize TC, we need to find the derivative of TC with respect to Q and set it to 0.\r<BR>\n" );
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document.write( "**4. Differentiate and Solve for Q**\r<BR>\n" );
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document.write( "* d(TC)/dQ = 1 - 3500/Q² = 0<BR>\n" );
document.write( "* 1 = 3500/Q²<BR>\n" );
document.write( "* Q² = 3500<BR>\n" );
document.write( "* Q = √3500 ≈ 59.16\r<BR>\n" );
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document.write( "Since we must order whole calculators, we consider Q = 59 and Q = 60.\r<BR>\n" );
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document.write( "**5. Calculate Total Cost for Q = 59 and Q = 60**\r<BR>\n" );
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document.write( "* TC(59) = 59 + 3500/59 + 1645 ≈ 59 + 59.32 + 1645 ≈ 1763.32<BR>\n" );
document.write( "* TC(60) = 60 + 3500/60 + 1645 ≈ 60 + 58.33 + 1645 ≈ 1763.33\r<BR>\n" );
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document.write( "Q = 59 minimizes the total cost.\r<BR>\n" );
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document.write( "**6. Calculate Number of Orders Per Year**\r<BR>\n" );
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document.write( "* Number of orders = D / Q = 700 / 59 ≈ 11.86\r<BR>\n" );
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document.write( "Since we can't have a fraction of an order, we round to 12.\r<BR>\n" );
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document.write( "**7. Final Answer**\r<BR>\n" );
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document.write( "The store should order 59 calculators 12 times per year to minimize inventory costs.<BR>\n" );
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