Linear Programming for Assembly Robot Task Scheduling under Cycle Time Limits

Project info

• Field of studyRobotics and Mechatronics
• LevelMaster's thesis (S2)
• Method typeOptimisation
• Estimated build time± 5 working days
• Toolslinprog, optimoptions (Optimization Toolbox)
• PriceUS$25 - US$166

Research goal

Optimise the products per hour with linprog by tuning the robot A work time and the robot B work time within realistic constraints.

Variables

Example result

• 23.011Optimal x
• 11.797Optimal y
• 159.761Optimal value
• 2Binding constraints
• 0.864Shadow price R1
• 85.872Mean utilisation (%)

Linear programming gives a maximum Products per hour of 159.76 at Robot A work time=23.01 and Robot B work time=11.80; 2 constraints are binding, the shadow price of resource 1 is 0.864, and mean resource utilisation is 86%.

Computed from illustrative synthetic data, not from a real study.

About Linear Programming (linprog)

linprog solves linear programming problems: maximising or minimising a linear function under linear constraints. It is used for resource allocation, production mix, and transport or scheduling problems.

Steps

1. 1Define decision variables and the objective
2. 2Turn maximisation into minimisation by multiplying by -1
3. 3Build constraint matrices A, b (and Aeq, beq if any) and bounds lb, ub
4. 4Run linprog
5. 5Check exitflag and active constraints
6. 6Interpret shadow prices from the multipliers

Assumptions to check

• Objective and constraints are linear
• Variables are continuous (use intlinprog if integer)
• Coefficients are treated as fixed and certain

Reading and reporting the results

Report the optimal solution, objective value, and fully used resources. Explain shadow prices to show which resource is most valuable to expand.

Example MATLAB code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

f = -[40; 30];                                       % maximise 40*x1 + 30*x2
A = [2 1; 1 2; 1 0];                                 % machine, labour, market limit
b = [100; 80; 40];
lb = [0; 0];
opts = optimoptions('linprog', 'Display', 'off');
[x, fval, exitflag, output, lambda] = linprog(f, A, b, [], [], lb, [], opts);

robot_a_work_time = x(1);
products_per_hour = -fval;
fprintf('x1 = %.3f, x2 = %.3f, max profit = %.2f (exitflag %d)\n', x(1), x(2), products_per_hour, exitflag);
fprintf('Slack: %s\n', mat2str(b - A*x, 4));
fprintf('Shadow prices: %s\n', mat2str(-lambda.ineqlin', 4));

xs = linspace(0, 60, 200);
figure; hold on;
plot(xs, 100 - 2*xs, xs, (80 - xs)/2);
xline(40, 'g');
plot(x(1), x(2), 'ro', 'MarkerSize', 9);
xlim([0 60]); ylim([0 60]); grid on;
xlabel('x_1'); ylabel('x_2'); legend('Machine', 'Labour', 'Market', 'Optimum');

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

• Payment: 50% down payment before work starts, balance after the deliverables are handed over
• Tax: Prices exclude 11% VAT
• Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
• Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

Frequently asked questions