Research on delivery times scheduling with truncated learning effects

Abstract

This paper addresses the single-machine scheduling with general truncated learning effects (GTLE) and delivery times that are past sequence dependent $({\text {DT}}_\textrm{psd})$. The objective is to determine an optimal job schedule such that makespan, sum of the $\beta$th power of job completion times, and total weighted completion time are minimized, where $\beta >0$ is a given constant. We proved that the makespan, the sum of the $\beta$th power of job completion times, and a special case of the total weighted completion time can be solved in polynomial time. For the general case of the total weighted completion time, we propose the heuristic, simulated annealing and branch-and-bound algorithms. Extensive numerical experiments validate the efficiency of the proposed solution algorithms on a set of randomly generated instances. Furthermore, the computational experiment is also conducted to show that the simulated annealing performs effectively and efficiently.

Appendix

Lemma 1

If the parameters $A>0$, $\lambda \le 1$ , $f'(y)\le 0$, $f''(y) \ge 0$, and $m''(y)\le 0$, then $K(y)=\frac{\lambda \max \{f(A+m(y))h(k+1),\theta \}-\max \{f(A)h(k),\theta \}}{y}$ is a non-decreasing function on y; if $m(0)=0$, $h(1)=1$ , $0 \le \theta \le 1$, and $y\ge 1$ , then h(k) is a non-increasing function.

Proof

The proof of this lemma will be discussed in the following three cases:

• (i) If $f(A+m(y))h(k+1)\ge \theta$, then $f(A)h(k)\ge \theta$, it follows that

  $$\begin{aligned} K(y)=\frac{\lambda f(A+m(y))h(k+1)-f(A)h(k)}{y} \end{aligned}$$

  then,

  $$\begin{aligned} K'(y)=\frac{\lambda f'(A+m(y))h(k+1)m'(y)y-\lambda f(A+m(x))h(k+1)+f(A)h(k)}{y^2} \end{aligned}$$

  Let

  $$\begin{aligned} H(y)=\lambda f'(A+m(y))h(k+1)m'(y)y-\lambda f(A+m(y))h(k+1)+f(A)h(k) \end{aligned}$$

  it follows that

  $$\begin{aligned} H'(y)=\lambda yh(k+1)[f''(A+m(y))(m''(y))^2+f'(A+m(y))m''(y)] \end{aligned}$$

  by $f'(y)\le 0, f''(y)\ge 0, m''(y)\le 0$, it follows that $H'(y)\ge 0$, $H(y)\ge H(0)=f(A)(h(k)-\lambda h(k+1))\ge 0$. Thus, $K'(y)\ge 0$, i.e., K(y) is a non-decreasing function on y.

• (ii) If $f(A+m(y))h(k+1)< \theta$, $f(A)h(k)\ge \theta$, then $K(y)=\frac{\lambda \theta -f(A)h(k)}{y}$ is a non-decreasing function on y.

• (iii) If $f(A+m(y))h(k+1)< \theta$, $f(A)h(k)<\theta$, then $K(y)=\frac{(\lambda -1)}{y}$ is a non-decreasing function on y.

$\square$