A note on a single-shift days-off scheduling problem with sequence-dependent labor costs

Abstract

Elshafei and Alfares have developed a dynamic programming (DP) algorithm for minimizing the total sequence-dependent cost in a constrained single-shift days-off scheduling problem. In general, however, their proposed DP algorithm may not obtain an optimal solution unlike their claim; in their highlighted security personnel scheduling example, we show how to improve their alleged optimal schedule just by inspection. The purpose of this paper is to describe better solution approaches to their challenging sequence-dependent workforce scheduling problem. We first explain why their DP algorithm encounters difficulties in finding an optimal solution with emphasis on the importance of proper state description for DP. We then describe how to correct their DP procedure with minimal effort. After that, we confirm such a better scheduling result by solving an associated mathematical programming problem with the Gurobi Optimizer, a widely recognized mathematical programming solver.

Appendix: a backward procedure

In Sect.4.2, we have shown a forward dynamic programming (DP) procedure with Eq. (25); for comparison, we present its backward DP version using the same notations defined therein; it appears to be more natural than the preceding forward counterpart.

Backward DP formulation:

• ① Definition of an optimal value function:

• $S_t(x_{t-L},x_{t-L+1},...,x_{t-2},x_{t-1})=$ minimum cost to go, starting day t to end, when $x_{t-k}$, $k \!=\! 1,...,L$, are the assignments made over the last L days.

• ② Recurrence relation:

  $$\begin{aligned} \begin{array}{l} S_t \big (x_{t-L},x_{t-L+1},...,x_{t-2},x_{t-1} \big )\quad \\ \qquad = \!\! \begin{array}{c} \\ \text{ minimum } \\ {{{x_{t} \!=\! v^k_t \! \in \! J_v(t)}}} \end{array} \! \! \Big \{ \! {\displaystyle Z_{t}\big (x_{t-L},x_{t-L+1}, ...,x_{t-1}, x_{t} \big ) } \\ + ~~ S_{t+1}\big (x_{t-L+1},...,x_{t-1},x_{t}\big ) \Big \} \\ \end{array} \end{aligned}$$

  (A1)

  where the decision is to choose a best v-pattern in (12) to be recorded by the policy function $\pi _t(x_{t-L},...,x_{t-1})$.

• ③ Boundary conditions:

  $$\begin{aligned} S_{T+1} \big ( x_{{\tiny {T \!-\! L \!+\! 1}}}, x_{{\tiny {T \!-\! L \!+\! 2}}}, ..., x_{{\tiny {T \!-\! 1}}}, x_{{\tiny {T}}} \big ) = 0. \end{aligned}$$

  (A2)

• ④ Answer is given by $S_1(\text{-,-, } \text{... } \text{,- })$ with no past history.

In Eq. (A1), $Z_{t}(x_{t-L},x_{t-L+1}, ...,x_{t-1}, x_{t})$ is the stage cost for the daily assignment on day t as defined for Eq. (25).