A column generation approach for stochastic optimization problems

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Wang, Yong Min

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Understanding how uncertainty effects the dynamics and behavior of an organization is a critical aspect of system design. Models and methods that take uncertainty into account can lead to significant reductions in cost. This dissertation investigates the use of stochastic optimization models for the following applications: (1) a generalized assignment problem (GAP) with uncertain resource capacity and unknown processing times and (2) a shift planning and scheduling problem (SPSP) with unknown demand that arises in the construction of the workforce at US Postal Service mail processing and distribution centers. In the models, the first stage decisions are made before the uncertainty is revealed. For the GAP, the first stage decisions correspond to an assignment of jobs to agents. Penalties are incurred when the assignments do not permit all demand to be satisfied. For the SPSP, the number of full-time and part-time employees, as well as the number of full-time shifts by type, must be specified before the demand is known. In the second stage, feasibility is addressed by allocating overtime and calling in temporary workers to handle spikes in the mail volume. This dissertation consists of three parts: (i) the development and analysis of stochastic integer programming models for the GAP and the SPSP, (ii) the estimation of the demand distributions from historical data for the Dallas processing and distribution center, and (iii) the development of column generation algorithms to solve the associated stochastic integer programs. In the first part, the stochastic integer programming models are formulated for the two applications and the value of the stochastic solutions is presented. The second part begins with an analysis of the weekly demand and a test of hypothesis concerning the existence of an end-of-month effect, i.e., that the week at the end of the month might have larger volume. The hypothesis is rejected. After removing outliers, it is shown that the demand is approximated well by a normal distribution. In the last part of the dissertation, a branch-and-price algorithm is developed and used to solve the two applications. Experimental results demonstrate the algorithm’s effectiveness.