Sequential Appointment Scheduling Considering Walk-In Patients

Algorithm 1

Step  1. 𝑡₁=0, 𝑄¹₁₀=0, 𝑡𝑒𝑠𝑡𝑠𝑒𝑡=Ø, 𝑛=1;

Step  2. Suppose the n + 1st patient is of type 𝑗∈[1,…,𝐽];

   If 𝑗∈𝑡𝑒𝑠𝑡𝑠𝑒𝑡, reject the patient, go to Step  2;

    Else for [𝑆^𝑛_𝑖−1,𝑆^𝑛_𝑖], 𝑖∈{2,…,𝑛+1}, compute

   𝜏^𝑛+1_𝑖−1=argmin_𝜏∈[𝑆^𝑛_{𝑖−1,𝑆}^𝑛_𝑖]𝐶(Π^𝑛+1_𝑖−1), 𝐶^𝑛+1_𝑖−1=min_𝜏∈[𝑆^𝑛_{𝑖−1,𝑆}^𝑛_𝑖]𝐶(Π^𝑛+1_𝑖−1),

   𝑖̃=argmin_{𝑖∈{2,…,𝑛+1}}𝐶^𝑛+1_𝑖−1, 𝑡_𝑛+1=𝜏_(̃^𝑛+1_𝑖−1);

Step  3. If 𝑃(𝑆^𝑛+1)>𝑃(𝑆^𝑛), accept the patient, 𝑛=𝑛+1,

     𝑡𝑒𝑠𝑡𝑠𝑒𝑡=Ø, go to Step  2;

    Else reject this patient, go to Step  4;

Step  4. If 𝑡𝑒𝑠𝑡𝑠𝑒𝑡≠Ø, 𝑡𝑒𝑠𝑡𝑠𝑒𝑡=[𝑡𝑒𝑠𝑡𝑠𝑒𝑡,𝑗];

     If 𝑡𝑒𝑠𝑡𝑠𝑒𝑡=[1,…,𝐽], stop;

    Else go to Step  5;

Step  5. Scheduling a patient who is sure to come to

    [𝑆^𝑛_𝑖−1,𝑆^𝑛_𝑖], 𝑖∈{2,…,𝑛+1} respectively, compute

    𝐶^𝑛+1_𝑖−1=min_𝜏∈[𝑆^𝑛_{𝑖−1,𝑆}^𝑛_𝑖]𝐶(Π^𝑛+1_𝑖−1), 𝐶(𝑆^𝑛+1)=min_{𝑖∈{2,…,𝑛+1}}𝐶^𝑛+1_𝑖−1.

     If 𝐶(𝑆^𝑛+1)−𝐶(𝑆^𝑛)>𝑟, stop;

      Else 𝑡𝑒𝑠𝑡𝑠𝑒𝑡=[𝑗], go to Step  2.