1. Denise Sakai Troxell (2000) Solving Nonlinear Optimization Problems with Excel Solver for Microsoft Excel 2000

2. Denise Sakai Troxell (2000) A Nonlinear Optimization Problem a simple Inventory Model* A manufacturer would like to produce 98,000 units of a certain product in a year, in lots of a fixed size. The fixed setup cost per lot is $500 and the production cost per unit is $5. The average inventory during a year is half of the lot size and the average annual inventory carrying cost per unit is $0.50. What is the fixed lot size that minimizes the balance between production and inventory carrying costs? * from Mathematics with Applications in Management and Economics by Gordon Prichett and John Saber, Richard D. Irwin, Inc., 7 th ed., 1994.

3. Denise Sakai Troxell (2000) Formulate the Problem obtain the function to be optimized number of lots f(x) = 98,000 (500) + 98,000 (5) + x (0.50) x 2 The objective is to find the lot size x, where 0 x 98,000, that minimizes the balance between production and inventory carrying costs given by the function fixed cost per lot Fixed Cost

4. Denise Sakai Troxell (2000) Formulate the Problem obtain the function to be optimized f(x) = 98,000 (500) + 98,000 (5) + x (0.50) x 2 Fixed Cost production cost per unit Variable Cost The objective is to find the lot size x, where 0 x 98,000, that minimizes the balance between production and inventory carrying costs given by the function

5. Denise Sakai Troxell (2000) Formulate the Problem obtain the function to be optimized f(x) = 98,000 (500) + 98,000 (5) + x (0.50) x 2 Fixed Cost Variable Cost Production Cost The objective is to find the lot size x, where 0 x 98,000, that minimizes the balance between production and inventory carrying costs given by the function

6. Denise Sakai Troxell (2000) Formulate the Problem obtain the function to be optimized f(x) = 98,000 (500) + 98,000 (5) + x (0.50) x 2 Production Cost average carrying cost per unit average number of units in inventory Carrying Cost The objective is to find the lot size x, where 0 x 98,000, that minimizes the balance between production and inventory carrying costs given by the function

7. Denise Sakai Troxell (2000) Formulate the Problem obtain the function to be optimized f(x) = 98,000 (500) + 98,000 (5) + x (0.50) x 2 Production Cost Carrying Cost The objective is to find the lot size x, where 0 x 98,000, that minimizes the balance between production and inventory carrying costs given by the function NOTE: We assume that x can be noninteger

8. Denise Sakai Troxell (2000) Preparing the Worksheet for Solver start with a blank sheet

9. Denise Sakai Troxell (2000) Preparing the Worksheet for Solver enter labels Enter labels in cells A1:B1

10. Denise Sakai Troxell (2000) Preparing the Worksheet for Solver enter labels NOTE: These labels are not essential for the use of Solver

11. Denise Sakai Troxell (2000) Preparing the Worksheet for Solver enter the formula of the function to be optimized Variable values in cell A2 Function formula in cell B2 NOTE: These cells will be colored to indicate that they are essential for Solver x f(x) Remember… f(x) = 98,000 (500) + 98,000 (5) + x (0.50) x 2

12. Denise Sakai Troxell (2000) Preparing the Worksheet for Solver enter the formula of the function to be optimized Click on cell B2 Remember… f(x) = 98,000 (500) + 98,000 (5) + x (0.50) x 2

13. Denise Sakai Troxell (2000) Preparing the Worksheet for Solver enter the formula of the function to be optimized Type in the formula =(98000/A2)*500+98000*5+(A2/2)*0.50 Remember… f(x) = 98,000 (500) + 98,000 (5) + x (0.50) x 2

14. Denise Sakai Troxell (2000) Preparing the Worksheet for Solver enter the formula of the function to be optimized Hit Enter Remember… f(x) = 98,000 (500) + 98,000 (5) + x (0.50) x 2 NOTE: The formula in cell B2 is not defined if cell A2 contains the value 0 or it is blank.

15. Denise Sakai Troxell (2000) Preparing the Worksheet for Solver enter the formula of the function to be optimized NOTE: Avoid the error message #DIV/0! in cell B2 by typing in an initial value different from 0 in cell A2

16. Denise Sakai Troxell (2000) Using Solver invoke Solver Click on Tools

17. Denise Sakai Troxell (2000) Using Solver invoke Solver Click on Solver

18. Denise Sakai Troxell (2000) Using Solver invoke Solver

19. Denise Sakai Troxell (2000) Using Solver complete the Solver Parameters dialog box Click on cell B2 NOTE: The cell displayed in the Set Target Cell: box must contain the formula of the function being optimized (minimize cell B2)

20. Denise Sakai Troxell (2000) Using Solver complete the Solver Parameters dialog box Check the Min: circle NOTE: The cell displayed in the Set Target Cell: box must contain the formula of the function being optimized (minimize cell B2)

21. Denise Sakai Troxell (2000) Using Solver complete the Solver Parameters dialog box Click on the By Changing Cells: box

22. Denise Sakai Troxell (2000) Using Solver complete the Solver Parameters dialog box Click on cell A2 NOTE: The cell displayed in the By Changing Cells: box must be the cell containing variable values (cell A2)

23. Denise Sakai Troxell (2000) Using Solver complete the Solver Parameters dialog box Click on Add NOTE: The Subject to the Constraints: box must contain the constraints on the variable values (x 98,000)

24. Denise Sakai Troxell (2000) Using Solver complete the Solver Parameters dialog box Click on cell A2 NOTE: The Subject to the Constraints: box must contain the constraints on the variable values (x 98,000) Click on the Cell Reference: box

25. Denise Sakai Troxell (2000) Using Solver complete the Solver Parameters dialog box NOTE: The Subject to the Constraints: box must contain the constraints on the variable values (x 98,000) Make sure <= is displayed

26. Denise Sakai Troxell (2000) Using Solver complete the Solver Parameters dialog box NOTE: The Subject to the Constraints: box must contain the constraints on the variable values (x 98,000) Click on the Constraint: box and type in 98000

27. Denise Sakai Troxell (2000) Using Solver complete the Solver Parameters dialog box Click on OK NOTE: The Subject to the Constraints: box must contain the constraints on the variable values (x 98,000)

28. Denise Sakai Troxell (2000) Using Solver set the Options Click on Options

29. Denise Sakai Troxell (2000) Using Solver set the Options Check the box Assume Non-Negative NOTE: The formula (in the Target Cell B2) is non- linear on the non-negative variable (x in A2)

30. Denise Sakai Troxell (2000) Using Solver set the Options Accept the remaining default options by clicking on OK NOTE: The formula (in the Target Cell B2) is non- linear on the non-negative variable (x in A2)

31. Denise Sakai Troxell (2000) Using Solver execute Solver Click on Solve

32. Denise Sakai Troxell (2000) Using Solver read solution A lot size of 14,000 units minimizes the balance between production and inventory carrying costs at $497,000.00. NOTE: Solver uses a method known as GENERALIZED REDUCED GRADIENT

33. Denise Sakai Troxell (2000) Using Solver end execution Click on OK

34. Denise Sakai Troxell (2000) Using Solver end execution

35. Denise Sakai Troxell (2000) Final Comments Solver might not find the solution… Try to enter different initial lot sizes (in cell A2), for example, 1, 49000, or 98000 and execute Solver. How to handle problems with Solver? Click on the button if you want some tips now