Example: Linear Programming Modeling and Solving LP Model Graphically This example is adapted from Problems 3&4 at the end of Chapter 2 of the Textbook. The Munchies Cereal Company makes a cereal from several ingredients. Two of the ingredients, oats and rice, provide vitamins A and B. The company wants to know how many ounces of oats and rice it should include in each box of cereal to meet the minimum requirements of 48 milligrams of vitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats contributes 8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice contributes 6 milligrams of A and 2 milligrams of B. Due to taste consideration, the company wants to ensure that the amount of oats does not exceed the amount of rice by 3 ounces in each box of cereal. An ounce of oats costs $0.05, and an ounce of rice costs $0.03. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis. c. What if unit cost of rice is $0.06 per ounce? 1. Define decision variables clearly X: ounces of oats included in each box of cereal Y: ounces of rice included in each box of cereal 2. Formulate objective function MIN Z = 0.05X + 0.03Y 3. Formulate all constraints 8X + 6Y >= 48 (Vitamin A requirement) X + 2Y >= 12 (Vitamin B requirement) X - Y <= 3 (taste consideration) X>=0, Y>=0 Steps 1-3 complete the LP modeling. The following steps solve the problem graphically. 4. Create coordinate system and plot the boundary lines of all constraints (See graph) 5. Identify the feasible region (See the shaded area on graph) (This is the key to solving LP model graphically. Be really careful which area is the feasible region.) 6. Identify ALL corners of the feasible region and calculate coordinate and the resulting value of objective function for each corner Corner I: (X=0, Y=8); resulting value of objective function is 0.05*0+0.03*8=0.24 Corner II: (2.4, 4.8), this comes from simultaneously solving two equations 8X+6Y=48 and X+2Y=12. The resulting value of objective function is 0.05*2.4+0.03*4.8=0.12+0.144=0.264 Corner III: (6, 3), this comes from simultaneously solving two equations X+2Y=12 and X-Y=3. The resulting value of objective function is 0.05*6+0.03*3=0.3+0.09=0.39. 7. Determine optimal solution and optimal value of objective function The optimal solution is (X=0, Y=8). That is, no oats but 8 ounces of rice in each box of cereal. The optimal value of objective function is $0.24 per box. If the cost of rice is $0.06 per ounce, then the resulting values of objective function are: Corner I: 0.05*0 + 0.06*8 = 0.48 Corner II: 0.05*2.4+0.06*4.8 = 0.12 + 0.288 = 0.408 Corner III: 0.05*6 + 0.06*3 = 0.3+0.18 = 0.48. Thus, the optimal solution is to include 2.4 ounces of oats and 4.8 ounces of rice in each box of cereal. The optimal value of objective function is $0.408 per box of cereal