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workforce4.m
function workforce4() % Copyright 2022, Gurobi Optimization, LLC % % Assign workers to shifts; each worker may or may not be available on a % particular day. We use Pareto optimization to solve the model: % first, we minimize the linear sum of the slacks. Then, we constrain % the sum of the slacks, and we minimize a quadratic objective that % tries to balance the workload among the workers. % define data nShifts = 14; nWorkers = 7; nVars = (nShifts + 1) * (nWorkers + 1) + nWorkers + 1; avgShiftIdx = (nShifts+1)*(nWorkers+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'}; 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; 1 1 1 0 1 1 1 1 1 1 1 1 1 1 ]; % Build model model.modelname = 'workforce4'; 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); model.obj = zeros(nVars, 1); for w = 1:nWorkers for s = 1:nShifts 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; model.varnames{w + avgShiftIdx} = sprintf('DiffShifts_%s', Workers{w}); model.ub(w + avgShiftIdx) = inf; model.lb(w + avgShiftIdx) = -inf; end % Initialize average shift variable model.ub((nShifts+1)*(nWorkers+1)) = inf; model.varnames{(nShifts+1)*(nWorkers+1)} = 'AvgShift'; % Initialize total slack variable model.ub(totalSlackIdx) = inf; model.varnames{totalSlackIdx} = 'TotalSlack'; model.obj(totalSlackIdx) = 1; % 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 % Save model gurobi_write(model,'workforce4a_m.lp'); % Optimize params.logfile = 'workforce4_m.log'; result = solveandprint(model, params, Shifts, Workers); if ~strcmp(result.status, 'OPTIMAL') fprintf('Quit now\n'); return; end % Constraint the slack by setting its upper and lower bounds totalSlack = result.x(totalSlackIdx); model.lb(totalSlackIdx) = totalSlack; model.ub(totalSlackIdx) = totalSlack; Rows = nShifts+1+nWorkers; for w = 1:nWorkers model.A(Rows+w, w + nShifts * (nWorkers+1)) = 1; model.A(Rows+w, w + avgShiftIdx) = -1; model.A(Rows+w, avgShiftIdx) = -1; model.A(Rows+1+nWorkers, w + nShifts * (nWorkers+1)) = 1; model.rhs(Rows+w) = 0; model.sense(Rows+w) = '='; model.constrnames{Rows+w} = sprintf('DiffShifts_%s_AvgShift', Workers{w}); end model.A(Rows+1+nWorkers, avgShiftIdx) = -nWorkers; model.rhs(Rows+1+nWorkers) = 0; model.sense(Rows+1+nWorkers) = '='; model.constrnames{Rows+1+nWorkers} = 'AvgShift'; % Objective: minimize the sum of the square of the difference from the % average number of shifts worked model.obj = zeros(nVars, 1); model.Q = sparse(nVars, nVars); for w = 1:nWorkers model.Q(avgShiftIdx + w, avgShiftIdx + w) = 1; end % model is no longer an assignment problem, enforce binary constraints % on shift decision variables. model.vtype(1:(nWorkers * nShifts), 1) = 'B'; model.vtype((nWorkers * nShifts + 1):nVars, 1) = 'C'; % Save modified model gurobi_write(model,'workforce4b_m.lp'); % Optimize 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