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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

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2 Single-Period Problem

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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

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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

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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?

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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

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

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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

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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

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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

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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

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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?

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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] +

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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] +

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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

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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 * = ?

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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]

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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

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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

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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

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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

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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

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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.

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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]

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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

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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) + =

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

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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 * )

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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 *

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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

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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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