2024年5月7日

CS代写｜MAST 90014 – Optimisation for Industry

The assignment problem

Data:

• 𝑛 : number of agents, tasks
• 𝑐𝑖 𝑗 : preference for assigning a task 𝑗 to a worker 𝑖

Variables:

• 𝑥𝑖𝑗 equal to one if worker 𝑖 is assigned to task 𝑗. Model:

𝑛𝑛 minimise ÕÕ𝑐𝑖𝑗𝑥𝑖𝑗

𝑖=1 𝑗=1

s.t.

𝑛
Õ𝑥𝑖𝑗 = 1, 𝑖 = 1,…,𝑛,

𝑗=1 𝑛

Õ𝑥𝑖𝑗=1, 𝑗=1,…,𝑛, 𝑖=1

𝑥𝑖𝑗 ∈{0,1}, 𝑖,𝑗=1,…,𝑛.

The set covering problem

• 𝐼: set of regions to be serviced,
• 𝐽: set of candidates for service centres,
• 𝑐 𝑗 : cost of installing the service centre 𝑗 .
• 𝑎𝑖 𝑗 : parameter equal to 1 if centre 𝑗 can serve region 𝑖 and to 0 otherwise.

Variables:

• 𝑥 𝑗 : binary variable equal to 1 if centre 𝑗 is selected and to 0 otherwise. Model:

minimise Õ𝑐𝑗𝑥𝑗 𝑗∈𝐽

s.t.

Õ𝑎𝑖𝑗𝑥𝑗=1, 𝑖∈𝐼, 𝑗∈𝐽

𝑥𝑗∈{0,1}, 𝑗∈𝐽

minimise Õ𝑐𝑗𝑥𝑗 𝑗∈𝐽

s.t.

Õ𝑎𝑖𝑗𝑥𝑗≤1, 𝑖∈𝐼, 𝑗∈𝐽

𝑥𝑗∈{0,1}, 𝑗∈𝐽

minimise Õ𝑐𝑗𝑥𝑗 𝑗∈𝐽

s.t.
Õ𝑎𝑖𝑗𝑥𝑗≥1, 𝑖∈𝐼,

𝑗∈𝐽
𝑥𝑗∈{0,1}, 𝑗∈𝐽