Capacity optimization and competition with cyclical and lead-time-dependent demands

Abstract

Capacity acquisition is often capital- and time-consuming for a business, and capacity investment is often partially or fully irreversible and difficult to change in the short term. Moreover, capacity determines the action space for service/production scheduling and lead-time quotation decisions. The quoted lead-time affects the customer’s perceived service quality. Thus, capacity acquisition level and lead-time quotation affect a firm’s revenue/profit directly or indirectly. In this paper, we investigate a joint optimization problem of capacity acquisition, delivery lead-time quotation and service-production scheduling with cyclical and lead-time-dependent demands. We first explore the structural properties of the optimal schedule given any capacity and lead-time. Then, the piecewise concave relationship between the delay penalty cost and the capacity acquisition level is found. Thereby, an efficient and effective polynomial time algorithm is provided to determine the optimal capacity acquisition level, delivery lead-time quotation and service/production schedule simultaneously. Furthermore, a capacity competition game among multiple firms is addressed. The numerical studies show that capacity equilibrium often exists and converges to a unique solution.

Appendices

Algorithm 1

Apart from the notations defined in Table 1, the following notations are also used in the algorithms.

• X = $(x_{ij})_{T\times T}$ the service schedule matrix;

• $\bar{d}$= average demand of the firm per period;

• Q= the number of orders waiting to be processed in any period;

• $q_{t}$ = the queue length at the end of period t.

Algorithm 2

Algorithm 3

In order to facilitate the description of the algorithms, the following notations are applied.

• $TI_{2\times n}$ = the matrix to record the time intervals with positive queue length;

• $\ell _1$ = TI(k, 1), the first period in the $k_{th}$ time interval;

• $\ell _2$ = TI(k, 2), the last period in the $k_{th}$ time interval;

• $\varOmega _{5 \times T}$ = the matrix to describe the penalty cost function;

• NTI = the start period of a new time interval;

• KP= the intercept of penalty cost function P(L, C);

• SP = the slope of penalty cost function P(L, C);

• $C_{BP}$ = capacity break points.