Planning the Personnel at a Construction Site

Contents

Problem description

Construction workers who erect the metal skeleton of skyscrapers are called steel erectors. The following table lists the requirements for steel erectors at a construction site during a period of six months. Transfers from other sites to this one are possible on the first day of every month and cost $100 per person. At the end of every month workers may leave to other sites at a transfer cost of $160 per person. It is estimated that understaffing as well as overstaffing cost $200 per month per post (in the case of unoccupied posts the missing hours have to be filled through overtime work).

Monthly requirement for steel erectors

+---+---+---+---+---+---+
|Mar|Apr|May|Jun|Jul|Aug|
+---+---+---+---+---+---+
| 4 | 6 | 7 | 4 | 6 | 2 |
+---+---+---+---+---+---+

Overtime work is limited to 25% of the hours worked normally. Every month, at most three workers may arrive at the site. The departure to other sites is limited by agreements with labor unions to 1/3 of the total personnel of the month. We suppose that three steel erectors are already present on site at the end of February, that nobody leaves at the end of February and that three workers need to remain on-site at the end of August. Which are the number of arrivals and departures every month to minimize the total cost?

Variables

transferin                 Cost to transfer in
transferout                Cost to transfer out
staffingdevcost            Cost for over or under employment
overtimemax                Maximum overtime
maxtransferin              Maximum amount to transfer in
maxtransferout             Maximum amount to transfer out
startstaff                 Starting staff
endstaff                   Staff required at the end of the period
demands                    Staff required each month

Reference

Applications of optimization... Gueret, Prins, Seveaux

% Marcus Edvall, Tomlab Optimization Inc, E-mail: tomlab@tomopt.com
% Copyright (c) 2005-2009 by Tomlab Optimization Inc., $Release: 7.2.0$
% Written Oct 7, 2005.   Last modified Apr 8, 2009.

Problem setup

transferin      = 100;
transferout     = 160;
staffingdevcost = 200;

overtimemax     = 0.25;
maxtransferin   = 3;
maxtransferout  = 1/3;

startstaff      = 3;
endstaff        = 3;

demands         = [4 6 7 4 6 2]';

m = length(demands);
onsite = tom('onsite',m,1,'int');
arrive = tom('arrive',m,1,'int');
leave  = tom('leave',m,1,'int');
over   = tom('over',m,1,'int');
under  = tom('under',m,1,'int');

% Variables are binary
bnds = {onsite >= 0, arrive >= 0, leave >= 0, over >= 0, under >= 0};
bnds = {bnds, arrive <= maxtransferin};

% Start constraint
con1 = {onsite(1) == startstaff + arrive(1)};

% Final constraint
con2 = {endstaff == onsite(end) - leave(end)};

% Intermediate constraints
con3 = {onsite(2:end) == onsite(1:end-1) - ...
    leave(1:end-1) + arrive(2:end)};

% Precense constraints
con4 = {onsite - over + under == demands};

% Overtime constraints
con5 = {under <= onsite*overtimemax};

% Leaving constraints
con6 = {leave <= onsite*maxtransferout};

% Objective
objective = transferin*sum(arrive) + transferout*sum(leave) + ...
    staffingdevcost*sum(over+under);
constraints = {bnds, con1, con2, con3, con4, con5, con6};
options = struct;
options.solver = 'cplex';
options.name   = 'Plan Personnel at a Constr Site';
sol = ezsolve(objective,constraints,[],options);

PriLev = 1;
if PriLev > 0
    months = ['Mar'; 'Apr'; 'May'; 'Jun'; 'Jul'; 'Aug'];
    for m = 1:size(months,1),
        disp(['-- ' months(m,:) ' --' ])
        disp(['  ' num2str(sol.onsite(m)) ' worker(s) on site'])
        disp(['  ' num2str(sol.arrive(m)) ' worker(s) have arrived'])
        disp(['  ' num2str(sol.leave(m)) ' worker(s) will leave'])
        disp(['  ' num2str(sol.over(m)) ' worker(s) too many'])
        disp(['  ' num2str(sol.under(m)) ' worker(s) too few'])
    end
end

% MODIFICATION LOG
%
% 051205 med   Created
% 060118 per   Added documentation
% 060126 per   Moved disp to end
% 090325 med   Converted to tomSym

Problem type appears to be: mip
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Plan Personnel at a Constr Site  f_k    1780.000000000000000000
                                                f(x_0)      0.000000000000000000

Solver: CPLEX.  EXIT=0.  INFORM=101.
CPLEX Branch-and-Cut MIP solver
Optimal integer solution found

FuncEv    6 
-- Mar --
  4 worker(s) on site
  1 worker(s) have arrived
  0 worker(s) will leave
  0 worker(s) too many
  0 worker(s) too few
-- Apr --
  6 worker(s) on site
  2 worker(s) have arrived
  0 worker(s) will leave
  0 worker(s) too many
  0 worker(s) too few
-- May --
  6 worker(s) on site
  0 worker(s) have arrived
  0 worker(s) will leave
  0 worker(s) too many
  1 worker(s) too few
-- Jun --
  6 worker(s) on site
  0 worker(s) have arrived
  0 worker(s) will leave
  2 worker(s) too many
  0 worker(s) too few
-- Jul --
  6 worker(s) on site
  0 worker(s) have arrived
  2 worker(s) will leave
  0 worker(s) too many
  0 worker(s) too few
-- Aug --
  4 worker(s) on site
  0 worker(s) have arrived
  1 worker(s) will leave
  2 worker(s) too many
  0 worker(s) too few