b-1 operations management linear programming module b - part 2

B-1

Operations Operations ManagementManagement

Linear ProgrammingLinear ProgrammingModule B - Part 2Module B - Part 2

B-2

Problem B.23Problem B.231. Gross Distributors packages and distributes industrial

supplies. A standard shipment can be packaged in a class A container, a class K container, or a class T container. The profit from using each type of container is: $8 for each class A container, $6 for each class K container, and $14 for each class T container. The amount of packing material required by each A, K and T container is 2, 1 and 3 lbs., respectively. The amount of packing time required by each A, K, and T container is 2, 6, and 4 hours, respectively. There is 120 lbs of packing material available each week. Six packers must be employed full time (40 hours per week each). Determine how many containers to pack each week.

B-3

Problem B.23Problem B.23

Container ProfitPacking

material (lbs.)Packing

time (hrs.)A $8 2

1

Amount available

K T

$6 $14 3

264

120 =240

B-4

Problem B.23Problem B.23

Maximize: 8xA + 6xK + 14xT

2xA + xK + 3xT 120 (lbs.) 2xA + 6xK + 4xT = 240 (hours)

xA, xK, xT 0

xi = Number of class i containers to pack each week. i=A, K, T

B-5

Linear Programming SolutionsLinear Programming Solutions Unique Optimal Solution. Multiple Optimal Solutions. Infeasible (no solution).

x + y 800

x 1000

x, y 0

Unbounded (infinite solution).Maximize 3x + 2y

x + y 1000

B-6

Computer SolutionsComputer Solutions

Optimal values of decision variables and objective function.

Sensitivity information for objective function coefficients.

Sensitivity information for RHS (right-hand side) of constraints and shadow price.

B-7


Enter data from formulation in Excel. 1 row for the coefficients of objective. 1 row for coefficients & RHS of each constraint. 1 final row for solution (decision variable) values.

Select Solver from the Tools Menu.

B-8

Computer Solutions - SpreadsheetComputer Solutions - Spreadsheet

B-9


B-10


B-11

Computer Solutions - SolverComputer Solutions - Solver

B-12

Computer Solutions - SolverComputer Solutions - Solver

B-13

Computer Solutions - Solver ParametersComputer Solutions - Solver Parameters

B-14


Set Target Cell: to value of objective function. E3

Equal To: Max or Min By Changing Cells: = Sol’n values (decision

variable values). B7:D7

Subject to the Constraints: Click Add to add each constraint: LHS =, , RHS

B-15

Computer Solutions - Adding ConstraintsComputer Solutions - Adding Constraints

Cell Reference: LHS location Select sign : <=, =, >= Constraint: RHS location

B-16

Computer Solutions - Adding ConstraintsComputer Solutions - Adding Constraints

1st constraint. Click Add. Repeat for second constraint.

B-17


Click Options to set up Solver for LP.

B-18

Computer Solutions - Solver OptionsComputer Solutions - Solver Options

Check ‘on’ Assume Linear Model and Assume Non-Negative.

B-19


Click Solve to find the optimal solution.

B-20

Computer Solutions - Solver ResultsComputer Solutions - Solver Results

B-21

Computer Solutions - Optimal SolutionComputer Solutions - Optimal Solution

Optimal solution is to use: 0 A containers

17.14 K containers

34.29 T containers

Maximum profit is $583 per week. Actually $582.857… in Excel values are rounded.

B-22


Optimal solution is to use: 0 class A containers. 17.14 class K containers. 34.29 class T containers.

Maximum profit is $582.857 per week.

Select Answer and Sensitivity Reports and click OK. New pages appear in Excel.

B-23

Computer Solution - Answer ReportComputer Solution - Answer ReportMicrosoft Excel 8.0e Answer ReportWorksheet: [probb.23.xls]Sheet1Report Created: 1/31/01 9:53:27 PM

Target Cell (Max)Cell Name Original Value Final Value

$E$3 Objective LHS 28 582.8571429

Adjustable CellsCell Name Original Value Final Value

$B$7 Sol'n values A cont. 1 0$C$7 Sol'n values K cont. 1 17.14285714$D$7 Sol'n values T cont 1 34.28571429

ConstraintsCell Name Cell Value Formula Status Slack

$E$4 lbs. LHS 120 $E$4<=$F$4 Binding 0$E$5 hours LHS 240 $E$5=$F$5 Binding 0

B-24

Sensitivity AnalysisSensitivity Analysis

Projects how much a solution will change if there are changes in variables or input data.

Shadow price (dual) - Value of one additional unit of a resource.

B-25

Computer Solution - Sensitivity ReportComputer Solution - Sensitivity ReportMicrosoft Excel 8.0e Sensitivity ReportWorksheet: [probb.23.xls]Sheet1Report Created: 1/31/01 9:53:27 PM

Adjustable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$B$7 Sol'n values A cont. 0 -1.142857143 8 1.142857143 1E+30$C$7 Sol'n values K cont. 17.14285714 0 6 8 1E+30$D$7 Sol'n values T cont 34.28571429 0 14 1E+30 1.6

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$E$4 lbs. LHS 120 4.285714286 120 60 80$E$5 hours LHS 240 0.285714286 240 480 80

B-26




Optimal solution: 0 class A containers 17.14285… class K containers 34.28571… class T containers Profit = 0(8) + 17.14285(6) + 34.28571(14) = $582.857

B-27






B-28

Sensitivity for Objective CoefficientsSensitivity for Objective Coefficients

As long as coefficients are in range indicated, then current solution is still optimal, but profit may change!

Current solution is optimal as long as:

Coefficient of xA is between -infinity and 9.142857

Coefficient of xK is between -infinity and 14

Coefficient of xT is between 12.4 and infinity

Objective Allowable AllowableCoefficient Increase Decrease

8 1.142857143 1E+306 8 1E+30

14 1E+30 1.6

B-29


If profit for class K container was 12 (not 6), what is optimal solution?


8 1.142857143 1E+306 8 1E+30

14 1E+30 1.6

B-30



xA=0, xK=17.14, xT=34.29 (same as before)

profit = 685.71 (more than before!)


8 1.142857143 1E+306 8 1E+30

14 1E+30 1.6

B-31




8 1.142857143 1E+306 8 1E+30

14 1E+30 1.6

B-32



Different! Resolve problem to get solution.


8 1.142857143 1E+306 8 1E+30

14 1E+30 1.6

B-33






B-34

Sensitivity for RHS valuesSensitivity for RHS values

Shadow price is change in objective value for each unit change in RHS as long as change in RHS is within range.

Each additional lb. of packing material will increase profit by $4.2857... for up to 60 additional lbs.

Each additional hour of packing time will increase profit by $0.2857... for up to 480 additional hours.

Shadow Constraint Allowable AllowablePrice R.H. Side Increase Decrease

4.285714286 120 60 800.285714286 240 480 80

B-35


Suppose you can buy 50 more lbs. of packing material for $250. Should you buy it?


4.285714286 120 60 800.285714286 240 480 80

B-36


Suppose you can buy 50 more lbs. of packing material for $250. Should you buy it?

NO. $250 for 50 lbs. is $5 per lb.

Profit increase is only $4.2857 per lb.


4.285714286 120 60 800.285714286 240 480 80

B-37


How much would you pay for 50 more lbs. of packing material?


4.285714286 120 60 800.285714286 240 480 80

B-38


How much would you pay for 50 more lbs. of packing material?

$214.28

50 lbs. $4.2857/lb. = $214.2857...


4.285714286 120 60 800.285714286 240 480 80

B-39


If change in RHS is outside range (from allowable increase or decrease), then we can not tell how the objective value will change.


4.285714286 120 60 800.285714286 240 480 80

B-40

Extensions of Linear ProgrammingExtensions of Linear Programming

Integer programming (IP): Some or all variables are restricted to integer values. Allows “if…then” constraints.

Much harder to solve (more computer time).

Nonlinear programming: Some constraints or objective are nonlinear functions. Allows wider range of situations to be modeled.

Much harder to solve (more computer time).

B-41

Integer ProgrammingInteger Programming

1 if we build a factory in St. Louis0 otherwise.

1x {2x { 1 if we build a factory in Chicago

0 otherwise.

We will build one factory in Chicago or St. Louis.

x1 + x2 1

We will build one factory in either Chicago or St. Louis.

x1 + x2 = 1

If we build in Chicago, then we will not build in St. Louis.

x2 1 - x1

B-42

You are creating an investment portfolio from 4 investment options: stocks, real estate, T-bills (Treasury-bills), and cash. Stocks have an annual rate of return of 12% and a risk measure of 5. Real estate has an annual rate of return of 10% and a risk measure of 8. T-bills have an annual rate of return of 5% and a risk measure of 1. Cash has an annual rate of return of 0% and a risk measure of 0. The average risk of the portfolio can not exceed 5. At least 15% of the portfolio must be in cash. Formulate an LP to maximize the annual rate of return of the portfolio.

Harder Formulation ExampleHarder Formulation Example

B-43

A business operates 24 hours a day and employees work 8 hour shifts. Shifts may begin at midnight, 4 am, 8 am, noon, 4 pm or 8 pm. The number of employees needed in each 4 hour period of the day to serve demand is in the table below. Formulate an LP to minimize the number of employees to satisfy the demand.

Another Formulation ExampleAnother Formulation Example

Midnight- 4 am

4 am - 8 am

3 6

8 am - noon

13

Noon - 4 pm 15

4 pm - 8 pm

12

8 pm -midnight 9

b-1 operations management linear programming module b - part 2

Documents

rhs slide

hours x

class t containers

rhs location slide

multiple optimal solutions

y x y

t containers maximum

type of container