Inventory models with ramp type demand rate, partial backlogging and Weibull deterioration rate

Recently, Skouri et al. (2009) proposed two inventory models with general ramp-type demand rate, Weibull deterioration rate, and partial backlogging of unsatisfied demand: (a) Model 1 was starting with no shortages, and Model 2 was starting with shortages. They derived the optimal solutions for both models. Then they ran 2 numerical examples, and concluded that ”the total cost for the model starting with shortages (i.e., Model 2) is less than the total cost for the model starting with no shortages (i.e., Model 1). This observation agrees with known, results ∗2000 Mathematics Subject Classification. Primary 90B05. †Corresponding author. 62 Jinn-Tsair Teng, Chun-Tao Chang, and Donald Chan from literature concerning the finite horizon inventory models.” In this note, we will provide two counterexamples to show that the total cost for Model 2 is not always cheaper than the total cost for Model 1. In fact, the inventory models here involve in various demand rate, backlogging rate, and deterioration rate with respect to time. Consequently, it seems not to have simple conditions to determine which model is cheaper to operate than the other.