document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165403>Question 1165403</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dunder Mifflin supplies paper to corporate clients. An important client needs 1500 reams of paper.<BR>\n" );
document.write( "Dunder Mifflin can procure the paper from several suppliers. The characteristics of the suppliers<BR>\n" );
document.write( "are as follows:<BR>\n" );
document.write( "Cost per ream Delivery time Reams available<BR>\n" );
document.write( "Supplier A 3.50 5 days 200<BR>\n" );
document.write( "Supplier B 2.00 10 days 600<BR>\n" );
document.write( "Supplier C 6.50 15 days 600<BR>\n" );
document.write( "Supplier D 5.00 4 days 200<BR>\n" );
document.write( "Supplier E 4.00 6 days 200<BR>\n" );
document.write( "When procuring the paper, Dunder Mifflin needs to ensure the following conditions are satisfied:<BR>\n" );
document.write( "(a) The client requires that at least 500 of the 1500 reams be delivered within the next 7 days.<BR>\n" );
document.write( "(b) In order to maintain good relations with Supplier C, Dunder Mifflin must order at least 100<BR>\n" );
document.write( "reams from them.<BR>\n" );
document.write( "(c) Because of a rivalry between supplier B and D, Dunder Mifflin must order at least as many<BR>\n" );
document.write( "reams from supplier D as they do from supplier B.<BR>\n" );
document.write( "(d) Each supplier has a limited number of reams of paper available, as shown in the table.<BR>\n" );
document.write( "Formulate a linear optimization model to help Dunder Mifflin procure the paper at the lowest cost.<BR>\n" );
document.write( "(You don’t need to find the optimal solution to the model you formulate.)  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853096 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.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=853096\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a classic linear programming (LP) problem. We need to define the decision variables, the objective function (minimizing cost), and the constraints based on the client and supplier requirements.\r<BR>\n" );
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document.write( "## 1. Decision Variables\r<BR>\n" );
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document.write( "Let $x_i$ be the number of reams of paper ordered from **Supplier $i$**, where $i \in \{A, B, C, D, E\}$.\r<BR>\n" );
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document.write( "* $x_A$: Reams ordered from Supplier A<BR>\n" );
document.write( "* $x_B$: Reams ordered from Supplier B<BR>\n" );
document.write( "* $x_C$: Reams ordered from Supplier C<BR>\n" );
document.write( "* $x_D$: Reams ordered from Supplier D<BR>\n" );
document.write( "* $x_E$: Reams ordered from Supplier E\r<BR>\n" );
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document.write( "Since the number of reams must be non-negative:<BR>\n" );
document.write( "$$x_A, x_B, x_C, x_D, x_E \ge 0$$\r<BR>\n" );
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document.write( "## 2. Objective Function (Minimize Cost)\r<BR>\n" );
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document.write( "The objective is to minimize the total procurement cost. This is the sum of (Cost per Ream $\times$ Reams Ordered) for each supplier.\r<BR>\n" );
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document.write( "| Supplier | Cost per Ream |<BR>\n" );
document.write( "| :---: | :---: |<BR>\n" );
document.write( "| A | 3.50 |<BR>\n" );
document.write( "| B | 2.00 |<BR>\n" );
document.write( "| C | 6.50 |<BR>\n" );
document.write( "| D | 5.00 |<BR>\n" );
document.write( "| E | 4.00 |\r<BR>\n" );
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document.write( "$$\text{Minimize } Z = 3.50x_A + 2.00x_B + 6.50x_C + 5.00x_D + 4.00x_E$$\r<BR>\n" );
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document.write( "## 3. Constraints\r<BR>\n" );
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document.write( "### a) Total Demand Constraint\r<BR>\n" );
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document.write( "The client requires a total of 1500 reams of paper.<BR>\n" );
document.write( "$$x_A + x_B + x_C + x_D + x_E = 1500$$\r<BR>\n" );
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document.write( "### b) Delivery Time Constraint (Within 7 Days)\r<BR>\n" );
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document.write( "The client requires at least 500 reams to be delivered within 7 days. We must identify suppliers with a delivery time $\le 7$ days: Suppliers A (5 days), D (4 days), and E (6 days).\r<BR>\n" );
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document.write( "$$x_A + x_D + x_E \ge 500$$\r<BR>\n" );
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document.write( "### c) Supplier C Relationship Constraint\r<BR>\n" );
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document.write( "Dunder Mifflin must order at least 100 reams from Supplier C.<BR>\n" );
document.write( "$$x_C \ge 100$$\r<BR>\n" );
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document.write( "### d) Rivalry Constraint (Supplier D vs. B)\r<BR>\n" );
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document.write( "The order from Supplier D must be at least as many reams as the order from Supplier B.<BR>\n" );
document.write( "$$x_D \ge x_B$$<BR>\n" );
document.write( "This is typically written as:<BR>\n" );
document.write( "$$x_D - x_B \ge 0$$\r<BR>\n" );
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document.write( "### e) Availability Constraints\r<BR>\n" );
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document.write( "The order from each supplier cannot exceed the reams available.\r<BR>\n" );
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document.write( "| Supplier | Reams Available |<BR>\n" );
document.write( "| :---: | :---: |<BR>\n" );
document.write( "| A | 200 |<BR>\n" );
document.write( "| B | 600 |<BR>\n" );
document.write( "| C | 600 |<BR>\n" );
document.write( "| D | 200 |<BR>\n" );
document.write( "| E | 200 |\r<BR>\n" );
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document.write( "$$x_A \le 200$$<BR>\n" );
document.write( "$$x_B \le 600$$<BR>\n" );
document.write( "$$x_C \le 600$$<BR>\n" );
document.write( "$$x_D \le 200$$<BR>\n" );
document.write( "$$x_E \le 200$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## Linear Optimization Model Summary\r<BR>\n" );
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document.write( "$$\text{Minimize } Z = 3.50x_A + 2.00x_B + 6.50x_C + 5.00x_D + 4.00x_E$$\r<BR>\n" );
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document.write( "$$\text{Subject to:}$$<BR>\n" );
document.write( "1.  $$x_A + x_B + x_C + x_D + x_E = 1500$$ (Total Demand)<BR>\n" );
document.write( "2.  $$x_A + x_D + x_E \ge 500$$ (7-Day Delivery)<BR>\n" );
document.write( "3.  $$x_C \ge 100$$ (Supplier C Minimum)<BR>\n" );
document.write( "4.  $$x_D - x_B \ge 0$$ (Rivalry)<BR>\n" );
document.write( "5.  $$x_A \le 200$$<BR>\n" );
document.write( "6.  $$x_B \le 600$$<BR>\n" );
document.write( "7.  $$x_C \le 600$$<BR>\n" );
document.write( "8.  $$x_D \le 200$$<BR>\n" );
document.write( "9.  $$x_E \le 200$$<BR>\n" );
document.write( "10. $$x_A, x_B, x_C, x_D, x_E \ge 0$$ (Non-negativity) </SPAN>\n" );
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