There is class of problems known as Product Mix problems.

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1 Professor Shoemaker There is class of problems known as Product Mix problems. Suppose you need to make two kinds of widgets and you have only so much material/time/money to make them. How many of each widget should I make? i.e. what should be the mix of widgets? 1

2 Mix problems are an example of optimization problems. That is: to find the optimal solution given the limitations you have to live with. In our example the objective is to make as many widgets as possible To Maximize the number of Widgets produced 2

3 Mix problems always have constraints that limit the choices you can make to solve the problem. Constraints are just the facts of life Some Examples You have only 5,000 pounds of Nickel available You have only 8,000 pounds of Copper available You must make at least 50 small widgets You must make at least 75 large widgets Constraints limit the objective 3

4 Optimization problems always have variables that are adjusted in an attempt to achieve the objective. In our example, the quantity of each type of widget will be varied in an effort to make the maximum number of widgets. The three important aspects of an optimization problem: An Objective to minimize or maximize Constraints that limit your choices Variables values that are changed in an attempt to achieve the objective 4

5 17-Solver.xlsx Two types of Widgets to make Amount of metals each Widget needs Amount of each metal needed to make that quantity of Widgets If we make 2 Small and 3 Large this is the total pounds of Nickel and Copper we ll need 5

6 How many of each type we ll make - Variables Constraints Total Widgets produced. The Objective. We want to maximize this number. The Constraints limit the values in the plan at the top 6

7 You must produce at least 50 small and 75 large widgets. These will have to be at least 50 and 75 when the solution is complete. You have at most 5,000 pounds of Nickel and 8,000 pounds of Copper available These cannot exceed 5,000 and 8,000 when the solution is complete. 7

8 You can t produce 1/3 of a widget. You have to produce them in whole number integer amounts. These will have to be integers when the solution is complete. It s not available by default. You may have to add it 8

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10 First link the Objective and Variable Cells Try running the Solve now. What happens? Adding a Constraint. Click on Add 10

11 Constraint: Limit the Nickel to 5,000 pounds Also add the constraint for Copper With just the metal constraints in, try running the solver. What happens? 11

12 The Solver found a solution, but didn t make any of the Large model widgets. Why? Add the two constraints to require some of each model of widget Also add the Integer constraint 12

13 The complete setup Go ahead and Solve it Then select Answer before you click on Ok The solution has us making 208 small and 75 large widgets. 283 is the maximum number of widgets we can make given the constraints 13

14 Whenever a solver achieves an optimal solution one or more of the constraints prevented it from going further. In this case, one or more constraints prevented the solution from making more widgets Two of the constraints prevented us from making any more widgets. Which ones? The requirement to make at least 75 large widgets limited the total Try changing the Minimum quantity of Large constraint from 75 to 25 and solve again. How many more widgets can you then make? 14

15 The 5,000 pound limit on Nickel also prevented us from making more widgets We used up 4,995 of the available 5,000 pounds of Nickel. With only 5 pounds left over, we couldn t make one more of either widget. When a constraint limits further optimization the constraint is called binding. If a constraints binds but there s still a little left over the amount left is called the slack. The Answer Report shows the constraints 15

16 The Nickel isn t shown as binding since there was some slack, But in fact it did bind the solution since there wasn t enough Nickel to make one more widget Your boss: How can we make more widgets? Answer: Get me more Nickel or let me make fewer Large widgets. How many widgets can you make if I get you a additional 1,000 pounds of Nickel? Then what binds the solution? 16

17 The three important aspects of an optimization problem: An Objective to minimize or maximize Constraints that limit your choices Variables values that are changed in an attempt to achieve the objective The Solver feature lets you determine the optimal solution It shows you which constraints limited further optimization It lets you answer What If questions 17