Outline

1. Outline • multi-period stochastic demand • base-stock policy • convexity 1

2. Properties of Convex Functions • let f and fibe 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 • f1(x) convex in x and f2(y) convex in y: f(x, y) = f1(x) + f2(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 2

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) = x2+y2 for -5  x, y  5. What is f(x)? 3

4. Two-Period Problem: Base Stock Policy 4

5. discounted factor , if applicable y1 Y2 X2 = y1 D1 D1 D2 General Idea of Solving a Two-Period Base-Stock Problem • Di: 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 x1 5

6. y1 Y2 x1 X2 = y1 D1 D1 D2 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 x2 6

7. y1 Y2 x1 X2 = y1 D1 D1 D2 General Idea of Solving a Two-Period Base-Stock Problem • convexity  optimality of base-stock policy • convexity of f2 convex • convexity  convex in y1 • convexity  convex in y1 7

8. Multi-Period Problem: Base Stock Policy 8

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 vT(x) for inventory level x at the end of period N : discount factor 9

10. … period 1 period 2 period N-1 period N-2 period N attainment preservation General Approach • FP: functional property of cost-to-go function fn of period n • SP: structural property of inventory policy Sn of period n FP of fN FP of f2 FP of fN-1 FP of f1 FP of fN-2 … SP of SN SP of S2 SP of SN-1 SP of S1 SP of SN-2 10

11. H(y) H(y) y y 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 H(y) y  problem with the right-most-global-minimum property: attaining (i.e., implying optimal base stock policy) but not preserving (i.e., fn being right-most-global-minimum does not necessarily lead to fn-1 having the same property) 11

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

13. Properties of Convex Functions • let f and fibe 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 • f1(x) convex in x and f2(y) convex in y: f(x, y) = f1(x) + f2(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 13

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) = x2+y2 for -5  x, y  5. What is f(x)? 14

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

16. Period N-1 • fN(x): a convex function of x • fN-1(x): in the given form • GN-1(y): a convex function of y • implication: base stock policy for period N-1 16

17. Example 7.3.3 • two-period problem backlog system with vT(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, Di ~ i.i.d. uniform[0, 100] • initial inventory on hand = 10 units • how to order to minimize the expected total cost 17

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

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