denise sakai troxell (2000) solving nonlinear optimization problems with excel solver for microsoft...

Denise Sakai Troxell (2000)

Solving Nonlinear

Optimization Problems

with Excel Solver

for Microsoft Excel 2000


A Nonlinear Optimization Problema 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., 7th ed., 1994.


Formulate the Problemobtain the function to be optimized

number of lots

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 per

lot

Fixed CostFixed Cost



f(x) = 98,000 (500) + 98,000 (5) + x (0.50)

x 2

Fixed CostFixed Cost

production cost per

unit

production cost per

unit

Variable Cost

Variable Cost






f(x) = 98,000 (500) + 98,000 (5) + x (0.50)

x 2

Fixed CostFixed Cost Variable Cost

Variable Cost

Production CostProduction Cost






f(x) = 98,000 (500) + 98,000 (5) + x (0.50)

x 2


average carrying cost

per unit

average carrying cost

per unit

average number of units in inventory

average number of units in inventory

Carrying CostCarrying Cost






f(x) = 98,000 (500) + 98,000 (5) + x (0.50)

x 2


Carrying CostCarrying Cost




NOTE: We assume that x can be noninteger


Preparing the Worksheet for Solverstart with a blank sheet


Preparing the Worksheet for Solverenter labels

Enter labels in cells A1:B1

Enter labels in cells A1:B1


Preparing the Worksheet for Solverenter labels

NOTE: These labels are not essential for the use of

Solver

NOTE: These labels are not essential for the use of

Solver


Preparing the Worksheet for Solverenter the formula of the function to be optimized

Variable values in cell

A2

Variable values in cell

A2

Function formula in cell

B2

Function formula in cell

B2

NOTE: These cells will be colored to indicate that they are essential for

Solver

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



Click on cell B2Click on cell B2

Remember…

f(x) = 98,000 (500) + 98,000 (5) + x (0.50)

x 2



Type in the formula =(98000/A2)*500+98000*5+(A2/2)*0.50

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



Hit EnterHit 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.

NOTE: The formula in cell B2 is not defined if cell A2

contains the value 0 or it is blank.



NOTE: Avoid the error message #DIV/0! in cell B2 by typing in an initial value different from 0

in cell A2

NOTE: Avoid the error message #DIV/0! in cell B2 by typing in an initial value different from 0

in cell A2


Using Solverinvoke Solver

Click on ToolsClick on Tools



Click on SolverClick on Solver


Using Solvercomplete the Solver Parameters dialog box

Click on cell B2Click 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)


(minimize cell B2)



Check the

Min: circle

Check the

Min: circle


(minimize cell B2)


(minimize cell B2)



Click on the By Changing Cells:

box

Click on the By Changing Cells:

box



Click on cell A2Click on cell A2

NOTE: The cell displayed in the By Changing Cells: box must be the cell containing variable values

(cell A2)

NOTE: The cell displayed in the By Changing Cells: box must be the cell containing variable values

(cell A2)



Click on AddClick on Add

NOTE: The Subject to the Constraints: box must contain the constraints on the variable values (x

98,000)


98,000)



Click on cell A2Click on cell A2


98,000)


98,000)

Click on the Cell Reference: box

Click on the Cell Reference: box




98,000)


98,000)

Make sure <= is displayed

Make sure <= is displayed




98,000)


98,000)

Click on the Constraint: box

and type in 98000

Click on the Constraint: box

and type in 98000



Click on OKClick on OK


98,000)


98,000)


Using Solverset the Options

Click on OptionsClick on Options



Check the box Assume Non-

Negative

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)


in A2)



Accept the remaining default options by clicking

on OK

Accept the remaining default options by clicking

on OK


in A2)


in A2)


Using Solverexecute Solver

Click on SolveClick on Solve


Using Solverread solution

A lot size of 14,000 units

minimizes the balance between production

and inventory carrying costs at $497,000.00.

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

NOTE: Solver uses a method known as GENERALIZED REDUCED

GRADIENT


Using Solverend execution

Click on OKClick on OK


Using Solverend execution


Final CommentsSolver 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

denise sakai troxell (2000) solving nonlinear optimization problems with excel solver for microsoft...

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