 FICO Xpress Optimization Examples Repository
 FICO Optimization Community FICO Xpress Optimization Home   Project planning with resource constraints

Description
Over the next 6 months we have three projects which can be done. Each of these projects has a profile of manpower requirements over its lifetime, and a benefit which accrues each month when the project has been completed. Our problem is to determine when each project is to start, subject to the constraint that in no month can we try to use more manpower than is available.
• simple MIP problem
• alternative formulation using binary variables or SOS-1
• if-then-else statement
Further explanation of this example: 'Xpress teaching material', Section 2.3 'SOS-1: Manpower planning'; 'Applications of optimization with Xpress-MP', Section 3.4.4 'Special Ordered Sets of type 1'. Similar problem: Section 7.1 'Construction of a stadium'

Source Files

projplan_graph.mos

```(!******************************************************
Mosel Example Problems
======================

file projplan.mos
`````````````````
TYPE:         Planning projects with resource constraints
DIFFICULTY:   3
FEATURES:     MIP problem, alternative formulation with SOS-1,
tricky formulation of resource use profiles and
calculation of benefits, graphical solution representation
DESCRIPTION:  Over the next 6 months we have three projects which can
be done. Each of these projects has a profile of manpower
requirements over its lifetime, and a benefit which accrues
each month when the project has been completed.
Our problem is to determine when each project is to start,
subject to the constraint that in no month can we try to
use more manpower than is available.
FURTHER INFO: `Applications of optimization with Xpress-MP teaching
material', Section 2.3 `SOS-1: Manpower planning';
`Applications of optimization with Xpress-MP',
Section 3.4.4 `Special Ordered Sets of type 1'.
Similar problem:
Section 7.1 `Construction of a stadium'

(c) 2008 Fair Isaac Corporation
author: S. Heipcke, 2001, rev. Sep. 2017
*******************************************************!)

model "Manpower planning"
uses "mmxprs", "mmsvg"

parameters
USESOS = false                   ! Set to true to use formulation with SOS
end-parameters

declarations
NP = 3                           ! Number of projects
PROJ = 1..NP                     ! Projects
NT = 6                           ! Time horizon
TIME = 1..NT                     ! Months to plan for
DUR: array(PROJ) of integer      ! Duration of projects
RESUSE: array(PROJ,TIME) of integer  ! Resource usage per project
RESMAX: array(TIME) of integer   ! Resource available per month
BEN: array(PROJ) of real         ! Benefit per month once project finished

x: array(PROJ,TIME) of mpvar     ! 1 if project p starts in month t, else 0
start: array(PROJ) of mpvar      ! Starting time of project t
end-declarations

initializations from 'projplan.dat'
DUR RESMAX BEN RESUSE
end-initializations

! Objective: Maximize Benefit
MaxBen:=
sum(p in PROJ,t in 1..NT-DUR(p)) (BEN(p)*(NT-t-DUR(p)+1)) * x(p,t)

! Resource availability
forall(t in TIME)
Capacity(t):= sum(p in PROJ,s in 1..t) RESUSE(p,t-s+1)*x(p,s) <= RESMAX(t)

! Each project starts once and only once
forall(p in PROJ) OneStart(p):= sum(t in TIME) x(p,t) = 1

! Connect variables x(p,t) and start(p)
forall(p in PROJ) Link(p):= sum(t in TIME) t*x(p,t) = start(p)

! Finish everything by the end of the planning period
forall(p in PROJ) start(p) <= NT-DUR(p)+1

if not USESOS then                ! Turn variables x into binaries
forall(p in PROJ,t in TIME) x(p,t) is_binary
else                              ! Define SOS-1 sets
forall(p in PROJ) StartSet(p):= sum(t in TIME) t*x(p,t) is_sos1
end-if

! Solve the problem
maximize(MaxBen)

! Solution printing
writeln("Total benefit: ", getobjval)
write("   ")
forall(t in TIME) write(t)
writeln
forall(p in PROJ) do
write(p, ": ")
forall(t in TIME)
write( if(t<getsol(start(p)), " ", if(t<getsol(start(p))+DUR(p), "*", "B")) )
writeln
end-do

! Solution drawing
declarations
ProjGraph: array(PROJ) of string
end-declarations

svgsetgraphviewbox(0,0,NT,max(t in TIME)RESMAX(t)+1)
svgsetgraphscale(10)
svgsetgraphlabels("Time", "Resource usage")

forall(p in PROJ) do
ProjGraph(p):= "P"+p
svgsetstyle(SVG_FILL,SVG_CURRENT)
svgsetstyle(SVG_STROKEWIDTH,0)
end-do
forall(t in TIME) do
u:=0.0
forall(p in PROJ) do
1, getsol(sum(s in 1..t) RESUSE(p,t-s+1)*x(p,s)))
u+=getsol(sum(s in 1..t) RESUSE(p,t-s+1)*x(p,s))
end-do
end-do   