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Outline. single-period stochastic demand without fixed ordering cost base-stock policy minimal expected cost maximal expected profit ( s , S ) policy: minimal expected cost. Single-Period Problem. Common Features. stochastic demands how many items to order

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  1. Outline • single-period stochastic demand without fixed ordering cost • base-stock policy • minimal expected cost • maximal expected profit • (s, S) policy: minimal expected cost 1

  2. Single-Period Problem 2

  3. Common Features • stochastic demands • how many items to order • too many: costly and with many leftovers • obsolescence • too few: opportunity loss • real-life problems • multi-product • multi-period or continuous-time • starting with a single-product, single-period problem 3

  4. Useful Policies in Inventory Control • any simple way to control inventory? yes, some simple rule-type policies • base-stock policy • base-stock levelS • an order of S-p units for an inventory position p units on review • (s, S) policy • reorder point s • an order of S-p units only for an inventory position ps on review • special case: (S-1, S) policy for continuous review under unit demands • (R, Q) policy • R in (R, Q)  sin (s, S) • all replenishment orders being of Q units • (R, nQ)policy • a variant of (R, Q) to replenish multiple batches of Q units to make inventory position above R 4

  5. How Good Are the Policies? • how good are the policies? • would they be optimal in some sense? • what are the conditions to make them optimal? • would the conditions for single- and multi-period problems be different? • would the conditions change with the setup cost? 5

  6. Answers • basically yes to the questions • single-period problems • base-stock policy being optimal among all policies: zero setup cost under convexity (and certain other assumptions) • (s,S) policy being optimal among all policies: positive setup cost under K-convexity (and certain other assumptions) • multi-period problems • similar conditions as the single-period problems 6

  7. Single-Period Problem • Optimality of the Base-Stock Policy • Minimizing Expected Cost 7

  8. I1 I3 I2 Ideas of the Optimality of the Based-Stock Policy for Single-Period Problems • let H(q) be the expected cost of the period if the period starts with q items (ignoring the variable unit cost for the time being) • let Ij be the initial inventory on hand • how to order? H(q) q 8

  9. H(q) H(q) H(q) I1 I2 I2 I3 I2 I3 I1 I4 I3 I1 q q q Ideas of the Optimality of the Based-Stock Policy for Single-Period Problems • let H(q) be the expected cost of the period if the period starts with q items (ignoring the variable unit cost for the time being) • let Ij be the initial inventory on hand • how to order? 9

  10. H(q) H(q) H(q) I1 I2 I2 I3 I2 I3 I1 I4 I3 I1 q q q Ideas of the Optimality of the Based-Stock Policy for Single-Period Problems • unique minimum point q* of H(q) the base-stock policy being optimal • order nothing if initial inventory I0 q*; else order q*I0 • (ignoring the variable unit cost for the time being) 10

  11. Newsvendor Models: Optimality of the Base-Stock Policy • a single-period stochastic inventory problem • order lead time = 0 • cost items • c = the cost to buy each unit • h = the holding cost/unit left over at the end of the period •  = the shortage cost/unit of unsatisfied demand • D = the demand of the period, a continuous r.v. ~ F • v(q) = the expected cost of the period for ordering q items • objective: minimizing the total cost of the period 11

  12. Newsvendor Models: Optimality of the Base-Stock Policy • cost: variable ordering cost, holding cost, and shortage cost • ordering q units • variable ordering cost = cq • how about holding cost and shortage cost? 12

  13. Newsvendor Models: Optimality of the Base-Stock Policy • first assume that no inventory on hand • inventory on hand after ordering = q • expected holding cost • when D is a constant d • holding cost = 0, if d q • holding cost = h(qd), if d< q • holding cost = hmax(0, qd) = h(qd)+ • in general when D is random, expected holding cost = hE[q-D]+ 13

  14. Newsvendor Models: Optimality of the Base-Stock Policy • expected shortage cost • when D is a constant d • shortage cost = 0, if q d • shortage cost = (dq), if q< d • shortage cost = max(0, dq) = (dq)+ • in general when D is random, expected holding cost = E[D-q]+ 14

  15. v(q) Newsvendor Models: Optimality of the Base-Stock Policy • the problem: • min v(q) = cq + hE[q-D]+ + E[D-q]+ • how to order? what is the shape of v(q)? complicated optimal ordering policy if the shape of v(q) is complicated q 15

  16. Example 6.1.1. • let us play around with a concrete example • standard Newsvendor problem • c = $1/unit • h = $3/unit •  = $2/unit • D ~ uniform[0, 100] • q* = ? 16

  17. Example 6.1.1. • D ~ uniform[0, 100] • min v(q) = cq + hE[q-D]+ + E[D-q]+ = q + 3E[q-D]+ + 2E[D-q]+ 17

  18. Newsvendor Models: Optimality of the Base-Stock Policy • for I0q* • the expected total cost to order y = v(I0+y) c (I0+y)+ cy = v(I0+y) c I0 • the expected total cost not to order = v(I0) c I0 • optimal not to order because v(q) is increasing for q q* • for I0<q* • the expected total cost of ordering q*I0 = v(q*)cq*+c(q*I0) = v(q*)cI0 • the expected total cost ordering y = v(I0+y)c(I0+y)+cy = v(I0+y)cI0 • optimal to order q*I0 because v(I0+y)  v(q*) for y q*I0 • ordering policy: base-stock type • for I0 < q*, cost of the buying option = v(q*)cI0; cost of the non-buying option = v(I0)cI0 18

  19. Newsvendor Models: Optimality of the Base-Stock Policy • min v(q) = cq + hE[q-D]+ + E[D-q]+ • would v(q) be always a convex function? Yes • optimal policy: always a base-stock policy [d-q]+ [q-d]+ d d 19

  20. Newsvendor Models: Optimality of the Base-Stock Policy • can we differentiate v(q)? • v(q) = cq + hE[q-D]+ + E[D-q]+ • first-order condition: • c + hF(q) - Fc(q) = 0 20

  21. Example 6.1.1. • q* = F1((c)/(+h)) • c = $1/unit, h = $3/unit,  = $2/unit  q* = F1(0.2) • F = uniform[0, 100]  q* = 20 • another way to show the result 21

  22. Newsvendor Models: Optimality of the Base-Stock Policy • a similar problem as before, except • unit selling price = p • objective: maximizing the total profit of the period • skipping shortage and holding costs for simplicity • r(q) = the expected profit if ordering q pieces • max r(q) = pE[min(D, q)] – cq • r(q) concave 22

  23. Newsvendor Models: Optimality of the Base-Stock Policy • r(q) = pE[min(D, q)] – cq • min (D, q) = D + q – max(D, q) = D – [D–q]+ • let  = E(D) • r(q) = p  pE(Dq)+ cq • r(q) concave • first-order condition gives Example 6.1.2. 23

  24. Example 6.1.2. • revenue: p = $5/unit • cost terms • unit purchasing cost, c = $1 • unit inv. holding cost, h = $3/unit (inv. holding cost in example) • unit shortage cost,  = $2/unit (shortage in example) • the demand of the period, D ~ uniform[0, 100] 24

  25. Example 6.1.2. • expected total profit • r(q) = pE[min(D, q)] – cq – hE(q–D)+– E(D–q)+ • min (D, q) = D + q – max(D, q) = D – [D–q]+ •  = E(D) • r(q) = p[ – E(D–q)+] – cq – hE(q–D)+– E(D–q)+ • r(q) = p – cq – (p+)E(D–q)+ – hE(q–D)+ • concave function • r(q) = – c + (p+)Fc(q) – hF(q) • q* = Fc(0.6) = 60 25

  26. Example 6.1.2. • deriving from sketch • r(q) = p – cq – (p+)E(D–q)+ – hE(q–D)+ • D ~ unif[0, 100]  E(D–q)+ = E(q–D)+ = 26

  27. Single-Period Problem • Optimality of the (s, S) Policy • Minimizing Expected Cost 27

  28. Fixed-Cost Models: Optimality of the (s, S) Policy • a similar problem, with fixed cost K per order • the sum of the variable cost in ordering, the expected inventory holding cost, and the expected shortage cost: • v(q) = cq + hE[q D]+ + E[Dq]+ • S*: the value of q that minimizes v(q) • s* < S* such that v(q*) = K+v(S*) 28

  29. s* K v(s*) v(q) v(S*) S* q Figure 1. The Definition of s* and S* Fixed-Cost Models: Optimality of the (s, S) Policy • for I0>S*: no point to order • ordering y: v(I0+y)+K+cy • not ordering: v(I0) • for sI0S*: do not order • not ordering: v(I0)cI0 • ordering y: v(I0+y)c(I0+y)+K+cy = v(I0+y)+KcI0 • for I0s: orderup to S* • not ordering: v(I0)cI0 • ordering y: v(I0+y)c(I0+y)+K+cy = v(I0+y)+KcI0 • optimal policy: (s, S) policy can break I0 S* into two cases, I0 < s and s  I0  S* 29

  30. Example 6.2.1. • cost terms • fixed ordering cost, K = $5 per order • unit purchasing cost, c = $1 • unit inventory holding cost, h = $3/unit • unit shortage cost,  = $2/unit • the demand of the period, D ~ uniform[0, 100] • the expected total cost for ordering q items 30

  31. Example 6.2.1. • v'(S*) = 0 S* = 20 • v(S*) = 90 • s* = the value of q < S* such that v(s*) = v(S*) • s* = 5.8579 • optimal policy • if on hand stock x < 5.8579, order S-x • otherwise do nothing 31

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