An extended newsboy problem with shortage-level constraints

It is only an order quantity which is decided for a spot selling time in the classical newsboy problem. Both the purchase timing and the time-variant variance of the demand are neglected. When the vendor gives a price discount for early purchase, the buyer purchases the quantities of price discount at the cost of forecast bias. The buyer has to forecast the demand early before purchase ahead of schedule, but it may increase the variance of the forecasted demand, which is a forecast bias. This time-variant variance of the demand is embedded into the model. When the average shortage-level is specified to an upper limit, both the purchase timing and the order quantity are simultaneously considered as the decision variables of the extended newsboy problem in this paper. The resultant outcomes could apply to some cases in the futures contracts.