نکته: شرایط لازم و کافی برای وجود راه حل بهینه مدل تولید - موجودی یک فروشنده و یک خریدار یکپارچه شده با در نظر گرفتن روند بی اعتباری

Chung (2008) discussed the necessary and sufficient conditions for the existence of the optimal solution of an integrated production-inventory model developed by Huang (2004), which allowed a vendor and a buyer to minimize their expected integrated total cost function with an imperfect production process. The objective of Chung (2008) was to improve the solution procedure presented in Huang (2004). In this study, we identify three errors, irrelevant conditions, and inappropriate examples used in Chung (2008). We provide a reformulation of the model and the correct necessary and sufficient conditions for optimality.

Chung (2008) recently claimed that the solution procedure provided in Huang (2004) was likely to cause misunderstandings. The objective of Chung (2008) was to improve the solution procedure in Huang (2004) by providing the necessary and sufficient conditions for the existence of the optimal solution. In this note, we identify three errors, irrelevant conditions for optimality, and inappropriate examples used in Chung (2008). We reformulate the expected integrated total cost function and provide the correct necessary and sufficient conditions for optimality by using the conventional optimization method. In reformulating the model, we use the following notations: Q lot size per production run D annual demand P production rate, P>D SV setup cost per production run for the vendor SB cost of placing an order for the buyer hV unit stock-holding cost per item per year for the vendor hB unit stock-holding cost per item per year for the buyer n number of shipments per lot from the vendor to the buyer T time interval between successive delivery F transportation cost per shipment Y percentage of defective items, a random variable f(y) probability density function of Y v the vendor's unit warranty cost of defective items x screening rate d unit screening cost EK(Q, n) expected annual integrated total cost