Newsvendor Problem

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Proposal – Generalizations of Newsvendor Problem

1. Introduction:
The newsvendor model has been used in operations management and applied economics for years to determine optimal ordering quantity under uncertain demand. Perishable goods such as banana and lettuce cannot be carried from one period to another. Managers have to make decision on the inventory level of perishable goods over a very limited period. For example, due to the uncertainty of the demand of the newspaper, the newsboy has to decide how many newspapers to order each day. If he orders too few, he will forgo some profit; if he orders too much, any unsold newspaper left will be worthless at the end of the day. Some nonperishable good like seasonal fashion cloths cannot be sold at the same price when they are carrying from one period to the next. Unsold merchandise at the end of the period is generally sold at a deep discount. In contrast to restocking lettuce or newspaper, which is done daily, this decision would be made once for the selling season. These single-period decision models are called the Newsvendor Problem. The newsvendor problem can be widely applied. For example, a cellular provider like Fido might need to decide its order quantity before the launch of a new phone, knowing that the new product will become obsolete quickly. The retailer is facing the problem of losing potential customers if they order too few and incurring additional inventory holding cost if they order too much. Moreover, the newsvendor model also applies to individual choice problems, such as health care financing and insurance purchasing (Rosenfield 1986, Eeckhoudt et al. 1991). Therefore, newsvendor models are always considered to be a very important mathematical model which can be applied to balance the costs of ordering too few against the costs of ordering too much. 1.1 Methodology

Research (web-based, literature-based): This paper researched web-based and literature-based sources, including the Internet, journals and books in the University of Waterloo Library. These researches provide credibility to our analysis of the different newsvendor models. 2. Single Period Newsvendor Models

2.1 Optimization Model
Assumptions:
(1) Planning is done for a single period.
(2) The probability distribution of demand is known.
(3) Company faces periodic demands that are independent and have the same distribution F(x). (4) Deliveries are made in advance of demand.
2.1.1 Cost Minimization Model
Notation:
Q – The ordering quantity.
Cu – penalty cost (or back order cost) per unit of shortage. Co – inventory holding cost per unit of left over after demand is realized. v – acquisition cost of each unit
p- price of each unit
g - salvage value of each unit
D - demand for the product.
B - penalty for not satisfying demand ($/unit)
2.1.1.1 Cost Minimization Model with Discrete Demand
Cost(Q,D)=CoQ-D, if D<QCuD-Q if D>Q
For discrete demand, the demand distribution is defined by the p.m.f p(D). The expected cost:
E[Cost(Q)]=D=0∞pDCost(Q,D)=CoD=0Q-1p(D)Q-D+CuD=Q∞p(D)D-Q Optimal Ordering Quantity
F(Q)=D=0QpD=CuCu+C0
Therefore, the optimal order quantity for the discrete demand newsvendor problem can be found by selecting the Q* such that F(Q*)=px<Q*=CuCu+C0 Q*=F-1CuCu+C0
2.1.1.2 Cost Minimization Model with Continuous Demand
Cost(Q,D)=CoQ-D, if D<QCuD-Q if D>Q
For continuous demand, the demand distribution is defined by the p.d.f fD. The expected cost:
ECostQ=CuEmaxD-Q,0+CoEmaxQ-D,0 where
E[max(D-Q,0) ]=E(shortage)=0∞max⁡(D-Q,0) fDdD=Q∞D-QfDdD E[max(Q-D,0) ]=E(unit over)=0∞max⁡(Q-D,0) fDdD=0QQ-DfDdD Optimal Ordering Quantity
F(Q)= CuCu+C0
Therefore, the optimal order quantity for the continuous demand newsvendor problem can be found by selecting the Q* such that F(Q*)=px<Q*=CuCu+C0 Q*=F-1CuCu+C0
2.1.2 Utility Max Model
Model Formulation
Dana...
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