Excel Solver for Linear Programming Optimization

1. SOLVING LINEAR PROGRAMS
USING EXCEL
Arpee Arruejo, MIT

2. Agenda
 What is Excel Solver?
 How to add Solver in Excel?
 How to use Solver in Excel?
 Excel Solver Examples
 Excel Solver Algorithms

3. What is Excel Solver?
 Excel Solver belongs to a special set of
commands often referred to as What-if Analysis
Tools. It is primarily purposed for simulation and
optimization of various business and
engineering models.
 The Excel Solver add-in is especially useful for
solving linear programming problems, aka linear
optimization problems, and therefore is
sometimes called a linear programming solver

4. How to add Excel Solver?
 Excel, perform the following steps:
 In Excel 2010, Excel 2013, and Excel 2016,
click File > Options.
In Excel 2007, click the Microsoft Office button,
and then click Excel Options.
 In the Excel Options dialog, click Add-Ins on the
left sidebar, make sure Excel Add-ins is
selected in the Manage box at the bottom of the
window, and click Go.

5. How to use Excel Solver?
 Problem. Supposing, you are the owner of a
beauty salon and you are planning on providing
a new service to your clients. For this, you need
to buy a new equipment that costs $40,000,
which should be paid by instalments within 12
months.
 Goal: Calculate the minimal cost per service that
will let you pay for the new equipment within the
specified timeframe.

6. Model:

7. Formulating in Excel
1. Write the LP out on paper, with all
constraints and the objective function.
2. Decide on cells to represent variables.
3. Enter coefficients of each variable in
each constraint in a block of cells.
4. Compute amount of each constraint
being used by current solution.

8. Formulating in Excel
5. Place inequalities in sheet, so you
remember <=, >=
6. Enter amount of each constraint
7. Enter objective coefficients
8. Calculate value of objective function
9. Make sure you have plenty of labels.
10. Widen columns for readability.

9. How to use…cont’n….
1. Run Excel Solver
 On the Data tab, in the Analysis group, click
the Solver button.
2. Define the problem
 The Solver Parameters window will open where you
have to set up the 3 primary components:
 Objective cell
 Variable cells
 Constraints

10. How to use…cont’n….
Objective
 The Objective cell (Target cell in earlier Excel
versions) is the cell containing a formula that
represents the objective, or goal, of the problem.
The objective can be to maximize, minimize, or
achieve some target value.

11. How to use…cont’n….
In this example, the objective cell is B7, which calculates the payment term
using the formula =B3/(B4*B5) and the result of the formula should be equal
to 12:

12. How to use…cont’n….
Variable cells
 Variable cells (Changing cells or Adjustable cells in
earlier versions) are cells that contain variable data
that can be changed to achieve the objective. Excel
Solver allows specifying up to 200 variable cells.

13. How to use…cont’n….
In this example, we have a couple of cells whose values can be changed:
 Projected clients per month (B4) that should be less than or equal to 50;
and
 Cost per service (B5) that we want Excel Solver to calculate.

14. How to use…cont’n….
Constraints
 The Excel Solver Constrains are restrictions or
limits of the possible solutions to the problem. To
put it differently, constraints are the conditions
that must be met.

15. How to use…cont’n….
To add a constraint(s), do the following:
 Click the Add button right to the "Subject to the Constraints" box.

16. How to use…cont’n….
 In the Constraint window, enter a constraint.
 Click the Add button to add the constraint to the list.
 Continue entering other constraints.
 After you have entered the final constraint, click OK to return to the
main Solver Parameterswindow.

17. How to use…cont’n….
Solve the problem
 After you've configured all the
parameters, click
the Solve button at the bottom of
the Solver Parameters window
(see the screenshot above) and
let the Excel Solver add-in find
the optimal solution for your
problem.
 Depending on the model
complexity, computer memory
and processor speed, it may
take a few seconds, a few
minutes, or even a few hours.
 When Solver has finished
processing, it will display
the Solver Results dialog
window, where you select Keep
the Solver Solution and
click OK:

18. How to use…cont’n….
Solve the problem
 In this example, $66.67 appears in cell B5, which is the minimal cost per
service that will let you pay for the new equipment in 12 months, provided
there are at least 50 clients per month:

19. More Examples
Excel Solver example 1
Problem: You want to minimize the cost of shipping goods from 2 different
warehouses to 4 different customers. Each warehouse has a limited supply
and each customer has a certain demand.
Goal: Minimize the total shipping cost, not exceeding the quantity available at
each warehouse, and meeting the demand of each customer.

20. More Examples.. Cont’n…
Formulating the model
To define our linear programming problem for the Excel Solver, let's answer the 3
main questions:
1. What decisions are to be made? We want to calculate the optimal quantity of goods to
deliver to each customer from each warehouse. These are Variable cells (B7:E8).
2. What are the constraints? The supplies available at each warehouse (I7:I8) cannot be
exceeded, and the quantity ordered by each customer (B10:E10) should be delivered.
These are Constrained cells.
3. What is the goal? The minimal total cost of shipping. And this is
our Objective cell (C12).

21. More Examples.. Cont’n…
 The next thing for you to do is to calculate the total quantity shipped
from each warehouse (G7:G8), and the total goods received by each
customer (B9:E9). You can do this with simple Sum formulas
demonstrated in the below screenshot. Also, insert the
SUMPRODUCT formula in C12 to calculate the total cost of shipping:
our Objective cell (C12).

22. More Examples.. Cont’n…
The last thing left for you to
do is configure the Excel
Solver parameters:
 Objective: Shipping_cost set
to Min
 Variable cells:
Products_shipped
 Constraints: Total_received =
Ordered and Total_shipped
<= Available

23. More Examples.. Cont’n…
Solution
 Click the Solve button at the bottom of the Solver Parameters window, and
you will get your answer. In this example, the Excel Solver add-in calculated
the optimal quantity of goods to deliver to each customer from each
warehouse with the minimal total cost of shipping:

24. Standard Form
Max 7x1 + 5x2
s.t. 4x1 + 3x2 <= 240
2x1 + 1x2 <= 100
x1 >= 0
x2 >= 0
electronics
assembly

25. Constraint
coefficients
Current
value of
variables
Formulating in Excel

26. Amount of
each
constraint
used
by current
solution
Formulating in Excel

27. RHS of constraints,
Inequality signs.
Objective Function
Coefficients
Formulating in Excel
Objective
Function
Value

28. Solving in Excel
All we have so far is a big ‘what if” tool. We
need to tell the LP Solver that this is an
LP that it can solve.
 Choose ‘Solver’ from ‘Tools’ menu

29. Click “Data” then “Solver”

30. If No Solver, Office2010

31. If No Solver, Office 2007

32. Solving in Excel
1. Choose ‘Solver’ from ‘Data’ tab
2. Tell Solver what is the objective function,
and which are variables.
3. Tell Solver to minimize or maximize

33. Solver Dialog Box
Set the Target Sell
Tell to minimize or maximize
Where the variables are

34. Solving in Excel
1. Choose ‘Solver’ from ‘Tools’ menu
2. Tell Solver what is the objective function, and
which are variables.
3. Tell Solver to minimize or maximize
4. Add constraints:
 Click ‘Add’, enter LHS, RHS, choose inequality
 Click ‘Add’ if you need to do more, or click ‘Ok’ if
this is the last one.
5. Add rest of constraints

35. Add Constraint Dialog Box

36. Constraints Added

37. Assuming Linear
6. You have to tell Solver that the model is
Linear. Click ‘options,’ and make sure
the ‘Assume Linear Model’ box is
checked.

38. Assume Linear

39. Assuming Linear
6. You have to tell Solver that the model is
Linear. Click ‘options,’ and make sure the
‘Assume Linear Model’ box is checked.
On this box, checking “assume non-negative”
means you don’t need to actually add the non-
negativity constraints manually.
7. Solve the LP: Click ‘Solve.’ Look at Results.

40. Office
2010

41. Office 2010
Options
 Set maximum time
to look for a solution
 OR maximum # of
iterations
Our problems
should solve quickly

42. Solution is Found
When a solution has been found, this box comes up.
You can choose between keeping the solution and going
back to your original solution.
Highlight the reports that you want to look at.

43. Successful Solution

44. Answer Report
 Gives optimal and initial values of
objective function
 Gives optimal and initial values of
variables
 Tells amount of ‘slack’ between LHS and
RHS of each constraint, tells whether
constraint is binding.

45. Answer Report

46. Excel Solver algorithms
When defining a problem for the Excel Solver, you can choose one of
the following methods in the Select a Solving Method dropdown box:
 GRG Nonlinear. Generalized Reduced Gradient
Nonlinear algorithm is used for problems that are smooth nonlinear,
i.e. in which at least one of the constraints is a smooth nonlinear
function of the decision variables. More details can be found here.
 LP Simplex. The Simplex LP Solving method is based the Simplex
algorithm created by an American mathematical scientist George
Dantzig. It is used for solving so called Linear
Programming problems - mathematical models whose requirements
are characterized by linear relationships, i.e. consist of a single
objective represented by a linear equation that must be maximized
or minimized. For more information, please check out this page.
 Evolutionary. It is used for non-smooth problems, which are the
most difficult type of optimization problems to solve because some
of the functions are non-smooth or even discontinuous, and
therefore it's difficult to determine the direction in which a function is
increasing or decreasing. For more information, please see this
page.