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Personnel requirement planning

Description
The requirements for construction workers at a construction site during a period of six months are known. Transfers from other sites to this one are possible on the first day of every month and at the end of every month workers may leave to other sites. Transfer, understaffing, and overstaffing incur known costs per month per post. Overtime work is limited to 25 of the hours worked normally. The monthly arrivals and departures are limited. Three workers are already present on site at the beginning of the planning period and that three workers need to remain on-site at the end. Which are the number of arrivals and departures every month to minimize the total cost?

Further explanation of this example: 'Applications of optimization with Xpress-MP', Section 14.6 'Planning the personnel at a construction site'

persplangr.zip[download all files]

Source Files

Data Files





persplan_graph.mos

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

   file persplan.mos
   `````````````````
   TYPE:         Personnel planning
   DIFFICULTY:   2
   FEATURES:     simple MIP problem, formulation of balance constraints
                 using inline 'if'
   DESCRIPTION:  The requirements for construction workers at a construction 
                 site during a period of six months are known. Transfers 
                 from other sites to this one are possible on the first day 
                 of every month and at the end of every month workers may 
                 leave to other sites. Transfer, understaffing, and 
                 overstaffing incur known costs per month per post.
                 Overtime work is limited to 25% of the hours worked 
                 normally. The monthly arrivals and departures are limited. 
                 Three workers are already present on site at the beginning 
                 of the planning period and that three workers need to 
                 remain on-site at the end. Which are the number of arrivals 
                 and departures every month to minimize the total cost?     
   FURTHER INFO: `Applications of optimization with Xpress-MP', 
                 Section 14.6 `Planning the personnel at a construction site'
   
   (c) 2008 Fair Isaac Corporation
       author: S. Heipcke, 2002, rev. Sep. 2017
*******************************************************!)

model "Construction site personnel"
 uses "mmxprs", "mmsvg"

 declarations
  FIRST = 3; LAST = 8
  MONTHS = FIRST..LAST                 ! Set of time periods (months)
  
  CARR, CLEAVE: integer                ! Cost per arrival/departure
  COVER, CUNDER: integer               ! Cost of over-/understaffing
  NSTART, NFINAL: integer              ! No. of workers at begin/end of plan
  REQ: array(MONTHS) of integer        ! Requirement of workers per month
  
  onsite: array(MONTHS) of mpvar       ! Workers on site
  arrive,leave: array(MONTHS) of mpvar ! Workers arriving/leaving
  over,under: array(MONTHS) of mpvar   ! Over-/understaffing
 end-declarations

 initializations from 'persplan.dat'
  CARR CLEAVE COVER CUNDER NSTART NFINAL REQ
 end-initializations
 
! Objective: total cost
 Cost:= sum(m in MONTHS) (CARR*arrive(m) + CLEAVE*leave(m) +
                          COVER*over(m) + CUNDER*under(m))

! Satisfy monthly need of workers
 forall(m in MONTHS) Demand(m):= onsite(m) - over(m) + under(m) = REQ(m) 

! Balances
 forall(m in MONTHS)
  Balance(m):= 
   onsite(m) = if(m>FIRST, onsite(m-1) - leave(m-1), NSTART) + arrive(m)
 BalanceFinal:= NFINAL = onsite(LAST) - leave(LAST)
  
! Limits on departures, understaffing, arrivals; integrality constraints
 forall(m in MONTHS) do
  LimitLeave(m):= leave(m) <= 1/3*onsite(m)
  LimitUnder(m):= under(m) <= 1/4*onsite(m)
  arrive(m) <= 3
  arrive(m) is_integer; leave(m) is_integer; onsite(m) is_integer
  under(m) is_integer; over(m) is_integer  
 end-do
 
! Solve the problem
 minimize(Cost)

! Solution printing
 declarations
  NAMES: array(MONTHS) of string       ! Names of months
 end-declarations
 
 initializations from 'persplan.dat'
  NAMES
 end-initializations
 
 writeln("Total cost: ", getobjval)
 write("Month     ")
 forall(m in MONTHS) write(NAMES(m)," ") 
 write("\nOn site ")
 forall(m in MONTHS) write(strfmt(getsol(onsite(m)),4)) 
 write("\nArrive  ")
 forall(m in MONTHS) write(strfmt(getsol(arrive(m)),4)) 
 write("\nLeave   ")
 forall(m in MONTHS) write(strfmt(getsol(leave(m)),4)) 
 write("\nOverst. ")
 forall(m in MONTHS) write(strfmt(getsol(over(m)),4)) 
 write("\nUnderst.")
 forall(m in MONTHS) write(strfmt(getsol(under(m)),4))
 writeln
 
! Solution drawing
 svgsetgraphviewbox(FIRST, 0, LAST-FIRST+1, max(m in MONTHS) REQ(m)+1)
 svgsetgraphscale(5)
 svgsetgraphlabels("Time periods", "Number of staff")

 svgaddgroup("Req", "Staff requirement", SVG_RED)
 svgsetstyle(SVG_STROKEOPACITY, 0.75)
 svgaddline(sum(m in FIRST..LAST) [m, REQ(m)])
 svgaddgroup("Avail","Actual staff level", SVG_GOLD)
 svgsetstyle(SVG_STROKEOPACITY, 0.75)
 svgaddline(sum(m in FIRST..LAST) [m, round(getsol(onsite(m)))])

 svgsave("persplan.svg")
 svgrefresh
 svgwaitclose("Close browser window to terminate model execution.", 1)
end-model

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