Using Solver to solve a minimization LP + interpretation of output BSAD 30 Dave Novak Source: Anderson et al., 2013 Quantitative Methods for Business 12.

Using Solver to solve a minimization LP+ interpretation of output

BSAD 30

Dave Novak

Source: Anderson et al., 2013 Quantitative Methods for Business 12th edition – some slides are directly from J. Loucks © 2013 Cengage Learning

Overview

Focus on section 7.5 Solving the minimization LP using MS

Solver and interpreting output Surplus variables Standard formulation of LP

Setting up the problem

Go to Excel choose the “Data” tab and then click on Solver

The Solver Screen pops up and we will need to reference the cells for our problem:Set Objective refers to the OF in B4Click the Min Radio button as this is a

minimization problem

Setting up the problem

By Changing Variable Cells refers to our two variables (cells B1 and B2)

Subject to the Constraints: refers to our constraints for the problem. We will add each one separately. Select “Add” in the Add Constraint Menu choose cell B8 for the left-hand-side “Cell Reference” for constraint #1 and choose cell C8 for the right-hand-side “Cell Reference” for constraint #1

Setting up the problem

Make sure the Make Unconstrained Variables non-negative check box is checked (this ensures that X1 and X2 are non-negative). We cannot produce a negative number of either decision variable

And using the Pull-down menu make sure that Select Solving Method is Simplex LP

Select SOLVE

Setting up the problem

The cells in your spreadsheet model should change and cell B1 should now = 3.2, cell B2 should now = 0.8, and cell B4 should = 17.6

Using the Solver Results Menu that pops up on your screen highlight both “Answer” and “Sensitivity” on the right-hand-side under Reports and click OK

Minimization example from last class We solved this graphically

ObjectiveFunction

“Regular”Constraints

Non-negativity Constraints

Min 5x1 + 2x2

s.t. #1) 2x1 + 5x2 > 10

#2) 4x1 - x2 > 12

#3) x1 + x2 > 4

x1, x2 > 0

Minimization problem

#2) 4x1 - x2 > 12#2) 4x1 - x2 > 12

#1) 2x1 + 5x2 > 10#1) 2x1 + 5x2 > 10

x1x1

Feasible Region6

5

4

3

2

1

#3) x1 + x2 > 4#3) x1 + x2 > 4

1 2 3 4 5 6

x2x2

Objective functionObjective function

Setting up the problem

Download, save, and then open the Excel template file from the URL on the course schedule

X1 (# of units of product 1)X2 (# of units of product 2)

Objective Function (Maximize Profit) 0

Constraints LHS RHSST:1) Constraint#1 0 102) Constraint #2 0 123) Constraint #3 0 4

Answer reportObjective Cell (Min)

Cell Name Original Value Final Value$B$4 Objective Function (Maximize Profit) 0 17.6

Variable CellsCell Name Original Value Final Value Integer

$B$1 X1 (# of units of product 1) 0 3.2 Contin$B$2 X2 (# of units of product 2) 0 0.8 Contin

ConstraintsCell Name Cell Value Formula Status Slack

$B$10 3) Constraint #3 LHS 4 $B$10>=$C$10 Binding 0$B$8 1) Constraint#1 LHS 10.4 $B$8>=$C$8 Not Binding 0.4$B$9 2) Constraint #2 LHS 12 $B$9>=$C$9 Binding 0

(16/5, 4/5)

Answer Report

Excel Solver uses the term “slack” when referring to any non-binding ≥ or ≤ constraintsNon-binding constraints do not restrict the

feasible region at the optimal solution point

Slack and surplus variables

Recall that slack and surplus variables are added to the standard form LP whenever there is an inequality

Less-than or equal to constraints ≤ require a SLACK variable that is added to the LHS of the constraint so that we can set the LHS = the RHS

Slack and surplus variables

Greater-than or equal to constraints ≥ require a SURPLUS variable that is subtracted from the LHS of the constraint so that we can set the LHS = the RHS

Equality constraints do not require either a slack or surplus variable in standard form

Slack and surplus variables

Surplus variables describe EXCESS quantity or the amount over the RHS of the constraint that is being used (associated with ≥)It is feasible to use less

Slack variables describe IDLE resources or the amount of the RHS of the constraint that is not being used (associated with ≤)It is feasible to use more

Answer Report

Constraint #1 is a non-binding constraint and has “surplus” = 0.4 (although Excel uses the term slack for both ≥ and ≤)

Constraint #1 does not directly affect the optimal solution – it is a redundant, non-binding constraint

Because this is a minimization problem, and the constraints are ≥, this lets us know that at the optimal point (3.2, 0.8) we exceed the RHS of constraint #1 by 0.4 units

Magnified view of extreme points

#2) 4x1 - x2 > 12#2) 4x1 - x2 > 12

#1) 2x1 + 5x2 > 10#1) 2x1 + 5x2 > 10

#3) x1 + x2 > 4#3) x1 + x2 > 4

Optimal A (3.2, 0.8)This point is restricted or bound by constraints #2 and #3, but NOT by constraint #1

B (3.33, 0.67)

Surplus in non-binding constraint #1at optimal point

Answer Report

Binding constraints let us know that all the resources associated with those specific constraints are used (there is no excess and there is no shortage)

The constraint “binds” the problem, and all resources are used – there is no slack or surplus

Standard form of the LP

Standard form of the LP involves adding slack or surplus variables to the mathematical model so the problem can be solved using strict equalities For each ≥ constraint, there should be a

“surplus” variable (denoted by Si)

Surplus variables have a coefficient of 0 in the OF

Surplus variables have a coefficient of 1 in the constraints

Standard form of the LP

Reformulate our problem

Standard form of the LP

Reformulate our problem

Slack or surplus? (I just randomly

created this, so don’t solve it)

ObjectiveFunction

“Regular”Constraints

Non-negativity Constraints

Min 10x1 + 25x2

s.t. #1) 2x1 + 5x2 > 15

#2) 4x1 - x2 > 20

#3) x1 + x2 ≤ 7

#4) 2x1 = 15

x1, x2 > 0

Standard Form?

Standard form of the LP

You could think of slack/surplus variables as “placeholder” variables that hold the numeric difference between the LHS and RHS of any inequality

In an equality constraint, the RHS MUST EQUAL the LHS, so there are no slack/surplus variables

Summary

Solver setup for an LP Answer report summary

Define surplus variable in context of any minimization problem

Formulate Standard Form LP