workforce4.m
功能Workforce4（）％Copyright 2021，Gurobi优化，L狗万app足彩LC %%分配工人转移;每个工人可能会或可能无法在％特定的日期上使用。我们使用帕累托优化来解决模型：％首先，我们最小化了松弛的线性和。然后，我们限制了休闲级的总和，并且我们最小化了％试图平衡工人之间的工作量的二次目标。％定义数据nshifts = 14;nworkers = 7;nvars =（nshifts + 1）*（nworkers + 1）+ nworkers + 1;avgshiftidx =（nshifts + 1）*（nworkers + 1）;totalslackidx = nvars;转移= {蒙1'; '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
