1 Outline  multi-period stochastic demand  base-stock policy  convexity

2 Properties of Convex Functions  let f and f i be convex functions  cf: convex for c  0 and concave for c  0  linear function: both convex and concave  f+c and f  c: convex  sum of convex functions: convex  f 1 (x) convex in x and f 2 (y) convex in y: f(x, y) = f 1 (x) + f 2 (y) convex in (x, y)  a random variable D: E[f(x+D)] convex  f convex, g increasing convex: the composite function g  f convex  f  convex: sup  f  convex  g(x, y) convex in its domain C = {(x, y)| x  X, y  Y(x)}, a convex set, for a convex set X; Y(x) an non-empty set; f(x) > -∞: f(x) = inf {y  Y(x)} g(x, y) a convex function

3 Illustration of the Last Property  Conditions:  g(x, y) convex in its domain C  C = {(x, y)| x  X, y  Y(x)}, a convex set  X a convex set  Y(x) an non-empty set  f(x) > -∞  Then f(x) = inf {y  Y(x)} g(x, y) a convex function  Try: g(x, y) = x 2 +y 2 for -5  x, y  5. What is f(x)?

4 Two-Period Problem: Base Stock Policy

5 General Idea of Solving a Two-Period Base-Stock Problem  D i : the random demand of period i; i.i.d.  x (  ) : inventory on hand at period (  ) before ordering  y (  ) : inventory on hand at period (  ) after ordering  x (  ), y (  ) : real numbers; X (  ), Y (  ) : random variables D1D1 x1x1 D2D2 X 2 = y 1  D 1 y1y1 Y2Y2 discounted factor , if applicable

6 General Idea of Solving a Two-Period Base-Stock Problem  problem: to solve  need to calculate  need to have the solution of for every real number x 2 D2D2 D1D1 x1x1 y1y1 X 2 = y 1  D 1 Y2Y2

7 General Idea of Solving a Two-Period Base-Stock Problem  convexity  optimality of base-stock policy  convexity of f 2  convex  convexity  convex in y 1 D2D2 D1D1 x1x1 y1y1 X 2 = y 1  D 1 Y2Y2

8 Multi-Period Problem: Base Stock Policy

9 Problem Setting  N-period problem with backlogs for unsatisfied demands and inventory carrying over for excess inventory  cost terms  no fixed cost, K = 0  cost of an item: c per unit  inventory holding cost: h per unit  inventory backlogging cost:  per unit  assumption:  > (1  )c and h+(1  )c > 0 (which imply h+   0)  terminal cost v T (x) for inventory level x at the end of period N  : discount factor

10 General Approach  FP: functional property of cost-to-go function f n of period n  SP: structural property of inventory policy S n of period n period N period N-1 period N-2 period 2period 1 … FP of f N SP of S N FP of f N-1 SP of S N-1 FP of f N-2 SP of S N-2 FP of f 2 SP of S 2 FP of f 1 SP of S 1 … attainment preservation

11 Necessary and Sufficient Condition for the Optimality of the Base Stock Policy in a Single-Period Problem  H(y): expected total cost for the period for ordering y units  the necessary and sufficient condition for the optimality of the base stock policy: the global minimum y * of H(y) being the right most minimum y H(y)H(y) H(y)H(y) y y H(y)H(y)  problem with the right-most-global-minimum property: attaining (i.e., implying optimal base stock policy) but not preserving (i.e., f n being right-most-global-minimum does not necessarily lead to f n-1 having the same property)

12 f n with right most global minimum What is Needed? optimality of base- stock policy in period n f n with right most global minimum plus an additional property optimality of base-stock policy in period n f n-1 with all the desirable properties additional property: convexity

13 Properties of Convex Functions  let f and f i be convex functions  cf: convex for c  0 and concave for c  0  linear function: both convex and concave  f+c and f  c: convex  sum of convex functions: convex  f 1 (x) convex in x and f 2 (y) convex in y: f(x, y) = f 1 (x) + f 2 (y) convex in (x, y)  a random variable D: E[f(x+D)] convex  f convex, g increasing convex: the composite function g  f convex  f  convex: sup  f  convex  g(x, y) convex in its domain C = {(x, y)| x ∈ X, y  Y(x)}, a convex set, for a convex set X; Y(x) an non-empty set; f(x) > -∞: f(x) = inf {y  Y(x)} g(x, y) a convex function

14 Illustration of the Last Property  Conditions:  g(x, y) convex in its domain C  C = {(x, y)| x  X, y  Y(x)}, a convex set  X a convex set  Y(x) an non-empty set  f(x) > -∞  Then f(x) = inf {y  Y(x)} g(x, y) a convex function  Try: g(x, y) = x 2 +y 2 for -5  x, y  5. What is f(x)?

15 Period N  G N (y): a convex function in y if v T being convex  minimum inventory on hand y * found, e.g., by differentiating G N (y)  if x < y *, order (y *  x); otherwise order nothing

16 Period N-1  f N (x): a convex function of x  f N-1 (x): in the given form  G N-1 (y): a convex function of y  implication: base stock policy for period N-1

17 Example Example  two-period problem backlog system with v T (x) = 0  cost terms  unit purchasing cost, c = $1  unit inventory holding cost, h = $3/unit  unit shortage cost,  = $2/unit  demands of the periods, D i ~ i.i.d. uniform[0, 100]  initial inventory on hand = 10 units  how to order to minimize the expected total cost

18 A Special Case with Explicit Base Stock Level  single period with v T (x) =  cx  objective function:  c(y  x) + hE(y  D) + +  E(D  y) + +  E(v T (y  D))  c(1  )y + hE(y  D) + +  E(D  y) + +  c   cx  optimal:

19 A Special Case with Explicit Base Stock Level  f t+1 : convex and with derivative  c  G t (x)=cx+hE(x  D) + +  E(D  x) + +  E(f t+1 (x  D))  same optimal as before:  problem: derivative of f N   c for all x  fortunately good enough to have derivative  c for x  S, i.e., if v T (x) =  cx, all order-up-to-level are the same

20 Mid-Term Results  mean: 39.57; standard deviation:  6| 9  5| 6  4| 3  3| 2  2|0 8 9