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

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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

2 Single-Period Problem

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

4 Useful Policies in Inventory Control  any simple way to control inventory? yes, some simple rule-type policies  base-stock policy  base-stock level S  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)  s in (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

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?

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

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

8 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 I j be the initial inventory on hand  how to order? I1I1 I3I3 q H(q)H(q) I2I2

9 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 I j be the initial inventory on hand  how to order? I1I1 I3I3 I1I1 I3I3 q H(q)H(q) q H(q)H(q) I2I2 I2I2 I1I1 I2I2 I4I4 q H(q)H(q) I3I3

10 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 I 0  q * ; else order q *  I 0  (ignoring the variable unit cost for the time being) I1I1 I3I3 I1I1 I3I3 q H(q)H(q) q H(q)H(q) I2I2 I2I2 I1I1 I2I2 I4I4 q H(q)H(q) I3I3

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

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?

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] +

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] +

15 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)? q v(q)v(q) complicated optimal ordering policy if the shape of v(q) is complicated

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 * = ?

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

18 Newsvendor Models: Optimality of the Base-Stock Policy  for I 0  q*  the expected total cost to order y = v(I 0 +y)  c (I 0 +y)+ cy = v(I 0 +y)  c I 0  the expected total cost not to order = v(I 0 )  c I 0  optimal not to order because v(q) is increasing for q  q*  for I 0 < q*  the expected total cost of ordering q*  I 0 = v(q*)  cq*+c(q*  I 0 ) = v(q*)  cI 0  the expected total cost ordering y = v(I 0 +y)  c(I 0 +y)+cy = v(I 0 +y)  cI 0  optimal to order q*  I 0 because v(I 0 +y)  v(q*) for y  q*  I 0  ordering policy: base-stock type  for I 0 < q *, cost of the buying option = v(q * )  cI 0 ; cost of the non- buying option = v(I 0 )  cI 0

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 [q-d]+[q-d]+ d [d-q]+[d-q]+ d

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) -  F c (q) = 0

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

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

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.

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]

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+  )F c (q) – hF(q)   q * = F c (0.6) = 60

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) + =

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

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 * )

29 Fixed-Cost Models: Optimality of the (s, S) Policy  for I 0 > S * : no point to order  ordering y: v(I 0 +y)+K+cy  not ordering: v(I 0 )  for s  I 0  S * : do not order  not ordering: v(I 0 )  cI 0  ordering y: v(I 0 +y)  c(I 0 +y)+K+cy = v(I 0 +y)+K  cI 0  for I 0  s: order up to S *  not ordering: v(I 0 )  cI 0  ordering y: v(I 0 +y)  c(I 0 +y)+K+cy = v(I 0 +y)+K  cI 0  optimal policy: (s, S) policy q v(q)v(q) Figure 1. The Definition of s * and S * S*S* v(S*)v(S*) K v(s*)v(s*) s*s* can break I 0  S* into two cases, I 0 < s and s  I 0  S *

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

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

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