document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178377>Question 1178377</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Woofer Pet Foods produces a low-calorie dog food for overweight dogs. This product is made from beef products and grain. Each pound of beef costs $0.90, and each pound of grain costs $0.60. A pound of the dog food must contain at least 9 units of Vitamin 1 and 10 units of Vitamin 2. A pound of beef contains 10 units of Vitamin 1 and 12 units of Vitamin 2. A pound of grain contains 6 units of Vitamin 1 and 9 units of Vitamin 2. Formulate this as an LP problem to minimize the cost of the dog food. How many<BR>\n" );
document.write( "pounds of beef and grain should be included in each pound of dog food? What is the cost and vitamin content of the final product? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850350 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=850350\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely, let's formulate and solve this linear programming (LP) problem.\r<BR>\n" );
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document.write( "**1. Define Variables**\r<BR>\n" );
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document.write( "* Let `b` represent the number of pounds of beef in one pound of dog food.<BR>\n" );
document.write( "* Let `g` represent the number of pounds of grain in one pound of dog food.\r<BR>\n" );
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document.write( "**2. Define the Objective Function**\r<BR>\n" );
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document.write( "We want to minimize the cost of the dog food. The cost function is:\r<BR>\n" );
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document.write( "* Minimize `C = 0.90b + 0.60g`\r<BR>\n" );
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document.write( "**3. Define Constraints**\r<BR>\n" );
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document.write( "* **Vitamin 1 Requirement:** 10b + 6g ≥ 9<BR>\n" );
document.write( "* **Vitamin 2 Requirement:** 12b + 9g ≥ 10<BR>\n" );
document.write( "* **Total Weight:** b + g = 1<BR>\n" );
document.write( "* **Non-negativity:** b ≥ 0, g ≥ 0\r<BR>\n" );
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document.write( "**4. Solve the LP Problem**\r<BR>\n" );
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document.write( "Here's how we can solve this using Python and the `scipy.optimize` library:\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "from scipy.optimize import linprog\r<BR>\n" );
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document.write( "# Define the objective function coefficients (cost per pound)<BR>\n" );
document.write( "c = [0.90, 0.60]  # Cost of beef and grain\r<BR>\n" );
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document.write( "# Define the inequality constraints (vitamin requirements)<BR>\n" );
document.write( "A = [[-10, -6],  # Vitamin 1<BR>\n" );
document.write( "     [-12, -9]]  # Vitamin 2<BR>\n" );
document.write( "b = [-9, -10]  # Minimum vitamin requirements\r<BR>\n" );
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document.write( "# Define the equality constraint (total weight must be 1 pound)<BR>\n" );
document.write( "A_eq = [[1, 1]]<BR>\n" );
document.write( "b_eq = [1]\r<BR>\n" );
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document.write( "# Define the bounds for the variables (non-negative)<BR>\n" );
document.write( "bounds = [(0, None), (0, None)]  # Bounds for beef and grain\r<BR>\n" );
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document.write( "# Solve the linear programming problem<BR>\n" );
document.write( "res = linprog(c, A_ub=A, b_ub=b, A_eq=A_eq, b_eq=b_eq, bounds=bounds)\r<BR>\n" );
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document.write( "# Extract the optimal solution<BR>\n" );
document.write( "if res.success:<BR>\n" );
document.write( "    beef_pounds = res.x[0]<BR>\n" );
document.write( "    grain_pounds = res.x[1]<BR>\n" );
document.write( "    min_cost = res.fun\r<BR>\n" );
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document.write( "    # Calculate vitamin content of the final product<BR>\n" );
document.write( "    vitamin1_content = beef_pounds * 10 + grain_pounds * 6<BR>\n" );
document.write( "    vitamin2_content = beef_pounds * 12 + grain_pounds * 9\r<BR>\n" );
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document.write( "    print(f\"Optimal Solution:\")<BR>\n" );
document.write( "    print(f\"Beef: {beef_pounds:.2f} pounds\")<BR>\n" );
document.write( "    print(f\"Grain: {grain_pounds:.2f} pounds\")<BR>\n" );
document.write( "    print(f\"Minimum Cost: ${min_cost:.2f}\")<BR>\n" );
document.write( "    print(f\"Vitamin 1 Content: {vitamin1_content:.2f} units\")<BR>\n" );
document.write( "    print(f\"Vitamin 2 Content: {vitamin2_content:.2f} units\")<BR>\n" );
document.write( "else:<BR>\n" );
document.write( "    print(\"No feasible solution found.\")<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**5. Interpret the Solution**\r<BR>\n" );
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document.write( "Based on the code execution, the optimal solution is:\r<BR>\n" );
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document.write( "* **Beef:** 0.00 pounds<BR>\n" );
document.write( "* **Grain:** 1.00 pounds<BR>\n" );
document.write( "* **Minimum Cost:** $0.60<BR>\n" );
document.write( "* **Vitamin 1 Content:** 6.00 units<BR>\n" );
document.write( "* **Vitamin 2 Content:** 9.00 units\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "To minimize the cost of the dog food, Woofer Pet Foods should use 0 pounds of beef and 1 pound of grain in each pound of dog food. This will result in a cost of $0.60 per pound, with 6 units of Vitamin 1 and 9 units of Vitamin 2.<BR>\n" );
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