Xpress Optimization with OMPR

The ompr.xpress package allows the use of the FICO Xpress optimizer to solve mixed integer and linear programming problems with the OMPR package.

Installation

A FICO Xpress installation and xpress package installation are required for ompr.xpress to function. The xpress package is not available on CRAN, but is rather included with downloads of the Xpress solver from FICO for Xpress verions greater than 8.13.

If a commercial licensed installation is not already available to you, a community version can be download from the link below. Community license is limited to problem with rows + columns <= 5000.

FICO® Xpress Community License | FICO

Installation instructions for the xpress package are located inside the directory of the Xpress installation. Instructions from FICO are at the link below:

Installation of the R and Python packages (fico.com)

You can install the development version of ompr.xpress from GitHub with:

# install.packages("devtools")
devtools::install_github("bmdahl1/ompr.xpress")

Example

We can use a simple police dispatch shift scheduling problem:

A police department must employ a specified number of dispatchers for all 4-hour shifts throughout the day. Each dispatcher must work an 8-hour shift, and shifts from 2000 - 0800 earn a higher wage rate.

The goal of the optimization will be to minimize total wage costs while supplying all needed dispatchers.

Shift.Time	Required	Shift Rate
0000 - 0400	6	$45.00
0400 - 0800	7	$45.00
0800 - 1200	13	$35.00
1200 - 1600	13	$35.00
1600 - 2000	9	$35.00
2000 - 0000	6	$45.00

suppressWarnings(suppressMessages(library(ompr.xpress)))
suppressWarnings(suppressMessages(library(ompr)))
suppressWarnings(suppressMessages(library(xpress)))

# Set Shift Parameters
shift_required <-  c(6,7,13,13,9,6)
shift_rate <- c(45, 45, 35, 35, 35, 45)
n_shifts <- 6

# Create OMPR Model
model <- MIPModel() |> 
  add_variable(start_shift[s], type = 'integer', s = 1:n_shifts, lb = 0) |> 
  add_variable(on_shift[s], type = 'continuous', s = 1:n_shifts, lb = 0) |> 
  add_constraint(on_shift[s] == (start_shift[s] + start_shift[s-1]), s = 2:n_shifts) |>
  add_constraint(on_shift[s] == start_shift[s] + start_shift[n_shifts], s = 1) |> 
  add_constraint(on_shift[s] >= shift_required[s], s = 1:n_shifts) |> 
  set_objective(sum_over(on_shift[s]*shift_rate[s], s = 1:n_shifts), sense = 'min')

# Solve with Xpress
model_results <- model |> 
  solve_model(xpress_optimizer())
#> 
#> Final MIP objective                   :                 2150
#> Final MIP bound                       :                 2150
#> Solution time / primaldual integral   :        0s /     0.00%
#> Number of solutions found / nodes     :         3 /        0
#> MIPSTATUS: 6

Shift Time	Shift.Rate	Decision Variables			Required
		Starting	On Shift	Total Hourly Wages
0000 - 0400	45	0	6	270	6
0400 - 0800	45	7	7	630	7
0800 - 1200	35	6	13	1365	13
1200 - 1600	35	9	15	2100	13
1600 - 2000	35	0	9	1575	9
2000 - 0000	45	6	6	1620	6

Sensitivity Analysis

Sensitivity Analysis will be performed on both LP and MIP problems. When the problem is MIP, then all integer variables will be fixed to the solution from the mixed integer solution, and the problem will be rerun as an LP to perform the sensitivity analysis.

Objective Function

Returns upper and lower sensitivity ranges for specified objective function coefficients.
If the objective coefficients are varied within these ranges the current basis remains optimal and the reduced costs remain valid.
(objsa (fico.com))

Objective Function Sensitivity Analysis
Variable	lower	upper
start_shift[1]	-90	100000000000000000000
start_shift[2]	-80	100000000000000000000
start_shift[3]	-70	100000000000000000000
start_shift[4]	-70	100000000000000000000
start_shift[5]	-80	100000000000000000000
start_shift[6]	-90	100000000000000000000
on_shift[1]	-100000000000000000000	100000000000000000000
on_shift[2]	-100000000000000000000	100000000000000000000
on_shift[3]	-100000000000000000000	100000000000000000000
on_shift[4]	-100000000000000000000	100000000000000000000
on_shift[5]	-100000000000000000000	100000000000000000000
on_shift[6]	-100000000000000000000	100000000000000000000

Lower and Upper Bounds

Returns upper and lower sensitivity ranges for specified variables’ lower and upper bounds.
If the bounds are varied within these ranges the current basis remains optimal and feasible.

bndsa (fico.com)

Upper and Lower Bounds Sensitivity Analysis
Variable	lblower	lbupper	ublower	ubupper
start_shift[1]	-0.000001	100000000000000000000	0	100000000000000000000
start_shift[2]	6.999999	100000000000000000000	7	100000000000000000000
start_shift[3]	5.999999	100000000000000000000	6	100000000000000000000
start_shift[4]	8.999999	100000000000000000000	9	100000000000000000000
start_shift[5]	-0.000001	100000000000000000000	0	100000000000000000000
start_shift[6]	5.999999	100000000000000000000	6	100000000000000000000
on_shift[1]	-100000000000000000000.000000	6	6	100000000000000000000
on_shift[2]	-100000000000000000000.000000	7	7	100000000000000000000
on_shift[3]	-100000000000000000000.000000	13	13	100000000000000000000
on_shift[4]	-100000000000000000000.000000	15	15	100000000000000000000
on_shift[5]	-100000000000000000000.000000	9	9	100000000000000000000
on_shift[6]	-100000000000000000000.000000	6	6	100000000000000000000

Right-Hand Side

Returns upper and lower sensitivity ranges for specified right hand side (RHS) function coefficients.
If the RHS coefficients are varied within these ranges the current basis remains optimal and the reduced costs remain valid.

Righ-Hnd Side Sensitivity Analysis
constraint	lower	upper
1	-0.000001	100000000000000000000.000000
2	-0.000001	100000000000000000000.000000
3	-2.000001	100000000000000000000.000000
4	-0.000001	100000000000000000000.000000
5	-0.000001	100000000000000000000.000000
6	-0.000001	100000000000000000000.000000
7	-100000000000000000000.000000	6.000001
8	-100000000000000000000.000000	7.000001
9	-100000000000000000000.000000	13.000001
10	-100000000000000000000.000000	15.000001
11	-100000000000000000000.000000	9.000001
12	-100000000000000000000.000000	6.000001

Row Duals

The dual values for the row constraints automatically generated from the LP optimize function.

getduals (fico.com)

Row Constraint Dual Values
Constraint	Duals
1	45
2	35
3	35
4	35
5	45
6	45
7	0
8	0
9	0
10	0
11	0
12	0

Reduced Costs

The decision variable reduced costs are automatically generated from the LP optimize function.

calcreducedcosts (fico.com)

Decision Variable Reduced Costs
Variable	Reduced Costs
start_shift[1]	90
start_shift[2]	80
start_shift[3]	70
start_shift[4]	70
start_shift[5]	80
start_shift[6]	90
on_shift[1]	0
on_shift[2]	0
on_shift[3]	0
on_shift[4]	0
on_shift[5]	0
on_shift[6]	0