The newsboy model Notation-the newsboy model A single product is to be ordered at the beginning of a period and can be used only to satisfy the demand during that period. Assume that all relevant costs can be determined on the basis of ending inventory.Define: co-Cost per unit of positive inventory remaining at the end of the period (overage cost); c-Cost per unit of unsatisfied demand,which can be thought as a cost per unit of negative ending inventory (underage cost). Assume that the demand D is a continuous nonnegative random variable with density functionfx)and cumulative distribution function F(x). The decision variable O is the number of units to be purchased at the beginning of the period. The goal is to determine O to minimize the expected costs incurred at the end of the period
The newsboy model Notation-the newsboy model A single product is to be ordered at the beginning of a period and can be used only to satisfy the demand during that period. • Assume that all relevant costs can be determined on the basis of ending inventory. Define: c 0=Cost per unit of positive inventory remaining at the end of the period (overage cost); c u=Cost per unit of unsatisfied demand, which can be thought as a cost per unit of negative ending inventory (underage cost). • Assume that the demand D is a continuous nonnegative random variable with density function f(x) and cumulative distribution function F(x). • The decision variable Q is the number of units to be purchased at the beginning of the period. • The goal is to determine Q to minimize the expected costs incurred at the end of the period
The newsboy model A general outline for analyzing most stochastic inventory problems is as follows: 1.Develop an expression for cost incurred as a function of both the random variable D and the decision variable Q 2.Determine the expected value of this expression with respect to the density function or probability function of demand. 3.Determine the value of Q such that the expected cost function is minimized. Development of Cost Function Define G(Q,D)as the total overage and underage cost incurred at the end of the period when Q units are ordered at the start of the period and D is the demand. Q-D is the demand units left at the end of the period as long as Q>D; If Q<D,then Q-D is negative and the number of units remaining on hand at the end of the period is 0
The newsboy model A general outline for analyzing most stochastic inventory problems is as follows: 1. Develop an expression for cost incurred as a function of both the random variable D and the decision variable Q. 2. Determine the expected value of this expression with respect to the density function or probability function of demand. 3. Determine the value of Q such that the expected cost function is minimized. Development of Cost Function • Define G(Q, D) as the total overage and underage cost incurred at the end of the period when Q units are ordered at the start of the period and D is the demand. • Q-D is the demand units left at the end of the period as long as Q D; • If Q<D, then Q-D is negative and the number of units remaining on hand at the end of the period is 0
不8 The newsboy model Q-DfQ≥D, max -D,0)= 0 fQ≤D. mto-w-&-08 fD≤ ● max(Q-D,0)represents the units left at the end of the period. ● max (D-Q,0)indicates the excess demand over supply,or unsatisfied demand. G(Q,D)=comax{Q-D,0)+cmax{D-Q,0) The expected cost function is defined as: G(Q)=E(GQ,D)) G(O)=co"max{O-x.O)f(x)dx+c.max{x-Q.O)f(x)dx -c(Q-x)f(x)dx+c.J(x-O)f(x)dx
The newsboy model • max{Q-D, 0} represents the units left at the end of the period. • max {D-Q, 0} indicates the excess demand over supply, or unsatisfied demand. , max { , 0} 0 . Q D if Q D Q D if Q D , max { , 0} 0 . D Q if D Q D Q if D Q G(Q, D)=c 0max{Q-D, 0}+c umax{D-Q, 0} The expected cost function is defined as: G(Q)=E(G{Q, D)) 0 0 0 0 0 ( ) max{ , 0} ( ) max{ , 0} ( ) ( ) () ( ) () u Q u Q G Q c Q x f x dx c x Q f x dx c Q x f x dx c x Q f x dx
The newsboy model Determining the Optimal Policy Determine the value of Q that minimizes the expected cost G(Q). 0-6fegfk G(Q)is convex such do that Q(Q)has minimal =cF(Q)-c(1-F(Q) value G())0forallz0 do2 010-6-20 100 200 300 400 dG(Q) =cF(0)-c(1-F(0) do 1Q=0 Since the slope is =-c,<0,(since F(O)=0) negative at Q-0,G(Q) is decreasing at Q-0
The newsboy model • Determining the Optimal Policy Determine the value of Q that minimizes the expected cost G(Q). G(Q) is convex such that Q(Q) has minimal value 0 0 0 ( ) () () ( ) (1 ( )) Q u Q u dG Q c f x dx c f x dx dQ cFQ c FQ 2 2 0 ( ) ( )() 0 0 u dGQ c c f Q for all Q dQ 0 0 ( ) (0) (1 (0)) 0, (since (0) 0) u Q u dG Q cF c F dQ c F Since the slope is negative at Q=0, G(Q) is decreasing at Q=0
The newsboy model 13 dG(Q) =cF(Q)-c(1-F(Q)=0 0 do 98 7 b 4 2 0 Optimal solution,Q",such 10-6-20 100 200 300 400 Q→ that cF(Q')-c(1-F(Q)》=0 Fig5-3 Expected Cost Function for Newsboy Model or F(Q")=c,l(co+c) The critical ratio. The critical ratio is strictly between 0 and 1,meaning that for a continuous demand,this equation is always solvable
The newsboy model Optimal solution, Q *, such that 0 ( ) ( ) (1 ( )) 0 u dG Q cFQ c FQ dQ Fig5-3 Expected Cost Function for Newsboy Model * * 0 * 0 ( ) (1 ( )) 0 ( ) /( ) u u u cFQ c FQ or FQ c c c The critical ratio. The critical ratio is strictly between 0 and 1, meaning that for a continuous demand, this equation is always solvable