Try our new documentation site.


workforce5.m


function workforce5()

% Copyright 2023, Gurobi Optimization, LLC
%
% Assign workers to shifts; each worker may or may not be available on a
% particular day. We use multi-objective optimization to solve the model.
% The highest-priority objective minimizes the sum of the slacks
% (i.e., the total number of uncovered shifts). The secondary objective
% minimizes the difference between the maximum and minimum number of
% shifts worked among all workers.  The second optimization is allowed
% to degrade the first objective by up to the smaller value of 10% and 2

% define data
nShifts  = 14;
nWorkers =  8;
nVars    = (nShifts + 1) * (nWorkers + 1) + 2;
minShiftIdx = (nShifts+1)*(nWorkers+1);
maxShiftIdx = minShiftIdx+1;
totalSlackIdx = nVars;

Shifts  = {'Mon1'; 'Tue2'; 'Wed3'; 'Thu4'; 'Fri5'; 'Sat6'; 'Sun7';
    'Mon8'; 'Tue9'; 'Wed10'; 'Thu11'; 'Fri12'; 'Sat13'; 'Sun14'};
Workers = {'Amy'; 'Bob'; 'Cathy'; 'Dan'; 'Ed'; 'Fred'; 'Gu'; 'Tobi'};

shiftRequirements = [3; 2; 4; 4; 5; 6; 5; 2; 2; 3; 4; 6; 7; 5];

availability = [
    0 1 1 0 1 0 1 0 1 1 1 1 1 1;
    1 1 0 0 1 1 0 1 0 0 1 0 1 0;
    0 0 1 1 1 0 1 1 1 1 1 1 1 1;
    0 1 1 0 1 1 0 1 1 1 1 1 1 1;
    1 1 1 1 1 0 1 1 1 0 1 0 1 1;
    1 1 1 0 0 1 0 1 1 0 0 1 1 1;
    0 1 1 1 0 1 1 0 1 1 1 0 1 1;
    1 1 1 0 1 1 1 1 1 1 1 1 1 1
    ];

% Build model
model.modelname  = 'workforce5';
model.modelsense = 'min';

% Initialize assignment decision variables:
%    x[w][s] == 1 if worker w is assigned
%    to shift s. Since an assignment model always produces integer
%    solutions, we use continuous variables and solve as an LP.
model.vtype = repmat('C', nVars, 1);
model.lb    = zeros(nVars, 1);
model.ub    = ones(nVars, 1);

for w = 1:nWorkers
    for s = 1:nShifts
        model.vtype(s+(w-1)*nShifts) = 'B';
        model.varnames{s+(w-1)*nShifts} = sprintf('%s.%s', Workers{w}, Shifts{s});
        if availability(w, s) == 0
            model.ub(s+(w-1)*nShifts) = 0;
        end
    end
end

% Initialize shift slack variables
for s = 1:nShifts
    model.varnames{s+nShifts*nWorkers} = sprintf('ShiftSlack_%s', Shifts{s});
    model.ub(s+nShifts*nWorkers) = inf;
end

% Initialize worker slack and diff variables
for w = 1:nWorkers
    model.varnames{w + nShifts * (nWorkers+1)} = sprintf('TotalShifts_%s', Workers{w});
    model.ub(w + nShifts * (nWorkers+1))       = inf;
end

% Initialize min/max shift variables
model.ub(minShiftIdx)       = inf;
model.varnames{minShiftIdx} = 'MinShift';
model.ub(maxShiftIdx)       = inf;
model.varnames{maxShiftIdx} = 'MaxShift';

% Initialize total slack variable
model.ub(totalSlackIdx)       = inf;
model.varnames{totalSlackIdx} = 'TotalSlack';

% Set-up shift-requirements constraints with shift slack
model.sense = repmat('=', nShifts+1+nWorkers, 1);
model.rhs   = [shiftRequirements; zeros(1+nWorkers, 1)];
model.constrnames = Shifts;
model.A = sparse(nShifts+1+nWorkers, nVars);
for s = 1:nShifts
    for w = 1:nWorkers
        model.A(s, s+(w-1)*nShifts) = 1;
    end
    model.A(s, s + nShifts*nWorkers) = 1;
end

% Set TotalSlack equal to the sum of each shift slack
for s = 1:nShifts
    model.A(nShifts+1, s+nShifts*nWorkers) = -1;
end
model.A(nShifts+1, totalSlackIdx) = 1;
model.constrnames{nShifts+1} = 'TotalSlack';

% Set total number of shifts for each worker
for w = 1:nWorkers
    for s = 1:nShifts
        model.A(w + nShifts+1, s+(w-1)*nShifts) = -1;
    end
    model.A(w + nShifts+1, w + nShifts * (nWorkers+1)) = 1;
    model.constrnames{nShifts+1+w} = sprintf('totShifts_%s', Workers{w});
end

% Set minShift / maxShift general constraints
model.genconmin.resvar = minShiftIdx;
model.genconmin.name   = 'MinShift';
model.genconmax.resvar = maxShiftIdx;
model.genconmax.name   = 'MaxShift';
for w = 1:nWorkers
    model.genconmin.vars(w) = w + nShifts * (nWorkers+1);
    model.genconmax.vars(w) = w + nShifts * (nWorkers+1);
end

% Set multiobjective
model.multiobj(1).objn                = zeros(nVars, 1);
model.multiobj(1).objn(totalSlackIdx) = 1;
model.multiobj(1).priority            = 2;
model.multiobj(1).weight              = 1;
model.multiobj(1).abstol              = 2;
model.multiobj(1).reltol              = 0.1;
model.multiobj(1).name                = 'TotalSlack';
model.multiobj(1).con                 = 0.0;
model.multiobj(2).objn                = zeros(nVars, 1);
model.multiobj(2).objn(minShiftIdx)   = -1;
model.multiobj(2).objn(maxShiftIdx)   = 1;
model.multiobj(2).priority            = 1;
model.multiobj(2).weight              = 1;
model.multiobj(2).abstol              = 0;
model.multiobj(2).reltol              = 0;
model.multiobj(2).name                = 'Fairness';
model.multiobj(2).con                 = 0.0;

% Save initial model
gurobi_write(model,'workforce5_m.lp');

% Optimize
params.logfile = 'workforce5_m.log';
result = solveandprint(model, params, Shifts, Workers);
if ~strcmp(result.status, 'OPTIMAL')
    fprintf('Not optimal\n');
end

end

function result = solveandprint(model, params, Shifts, Workers)
% Helper function to solve and display results

nShifts = length(Shifts);
nWorkers = length(Workers);
result = gurobi(model, params);
if strcmp(result.status, 'OPTIMAL')
    fprintf('The optimal objective is %g\n', result.objval);
    fprintf('Schedule:\n');
    for s = 1:nShifts
        fprintf('\t%s:', Shifts{s});
        for w = 1:nWorkers
            if result.x(s+(w-1)*nShifts) > 0.9
                fprintf('%s ', Workers{w});
            end
        end
        fprintf('\n');
    end
    fprintf('Workload:\n');
    for w = 1:nWorkers
        fprintf('\t%s: %g\n', Workers{w}, result.x(w + nShifts * (nWorkers+1)));
    end
else
    fprintf('Optimization finished with status %s\n', result.status);
end
end

Try Gurobi for Free

Choose the evaluation license that fits you best, and start working with our Expert Team for technical guidance and support.

Evaluation License
Get a free, full-featured license of the Gurobi Optimizer to experience the performance, support, benchmarking and tuning services we provide as part of our product offering.
Academic License
Gurobi supports the teaching and use of optimization within academic institutions. We offer free, full-featured copies of Gurobi for use in class, and for research.
Cloud Trial

Request free trial hours, so you can see how quickly and easily a model can be solved on the cloud.

Search