1 Outline single-period stochastic demand without fixed ordering cost base-stock policy minimal...

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OutlineOutline

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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Single-Period Problem

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Common Features 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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Useful Policies in Inventory ControlUseful 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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How Good Are the Policies?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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AnswersAnswers

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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Single-Period Problem Optimality of the Base-Stock Policy

Minimizing Expected Cost

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Ideas of the Optimality of the Based-Stock Policy Ideas of the Optimality of the Based-Stock Policy for Single-Period Problemsfor 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?

I1 I3q

H(q)

I2

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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?

I1 I3I1 I3

q

H(q)

q

H(q)

I2I2 I1 I2 I4q

H(q)

I3

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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)

I1 I3I1 I3

q

H(q)

q

H(q)

I2I2 I1 I2 I4q

H(q)

I3

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Newsvendor Models: Newsvendor Models: Optimality of the Base-Stock PolicyOptimality 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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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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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(qd), if d < q holding cost = hmax(0, qd) = h(qd)+

in general when D is random, expected holding cost = hE[q-D]+

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expected shortage cost when D is a constant d

shortage cost = 0, if q d

shortage cost = (dq), if q < d

shortage cost = max(0, dq) = (dq)+

in general when D is random, expected holding cost = E[D-q]+

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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)complicated optimal ordering policy if the

shape of v(q) is complicated

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Example 6.1.1. 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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min v(q) = cq + hE[q-D]+ + E[D-q]+ = q + 3E[q-D]+ + 2E[D-q]+

2 2 23 2(100 )200 200 40

( ) 100q q qv q q q

2

200( ) qE q D

2(100 )200

( ) qE D q

20'( ) 1 0qv q * 20q

D ~ uniform[0, 100]

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Newsvendor Models: Newsvendor Models: Optimality of the Base-Stock Policy 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

2

40( ) 100qv q q

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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]+

d

[d-q]+

d

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Newsvendor Models: Newsvendor Models: Optimality of the Base-Stock Policy 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

[ ]( )

dE q DF q

dq

[ ]( )cdE D q

F qdq

( )c

F qh

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q* = F1((c)/(+h))

c = $1/unit, h = $3/unit, = $2/unit q* = F1(0.2)

F = uniform[0, 100] q* = 20

another way to show the result

2 23 2(100 )200 200

( ) q qv q q

2

200( ) qE q D

2(100 )200

( ) qE D q

3 2(100 )100 200

'( ) 1 0q qv q * 20q

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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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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(Dq)+ cq

r(q) concave

first-order condition gives ( )p c

F qp

Example 6.1.2.

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Example 6.1.2.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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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

* 1 p c

q Fp h

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deriving from sketch r(q) = p – cq – (p+)E(D–q)+ – hE(q–D)+

D ~ unif[0, 100] E(D–q)+ = E(q–D)+ = 2(100 )

;200

q2

200

q

2 27(100 ) 3( ) 250

200 200

q qr q q

7(100 ) 3'( ) 1

100 100

q qr q

*Setting '( ) 0 gives 60r q q

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Single-Period ProblemSingle-Period Problem

Optimality of the (Optimality of the (ss, , SS) Policy) Policy

Minimizing Expected Cost Minimizing Expected Cost

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Fixed-Cost Models: Fixed-Cost Models: Optimality of the (Optimality of the (ss, , SS) Policy) 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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Fixed-Cost Models: Fixed-Cost Models: Optimality of the (Optimality of the (ss, , SS) Policy) 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)+KcI0

for I0 s: order up to S*

not ordering: v(I0)cI0

ordering y: v(I0+y)c(I0+y)+K+cy = v(I0+y)+KcI0

optimal policy: (s, S) policy

q

v(q)

Figure 1. The Definition of s* and S*

S*

v(S*)K

v(s*)

s*

can break I0 S* into two cases, I0 < s and s I0 S*

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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 items2 2 2(100 ) 40 4000

( )200 200 40

q q q qv q cq h

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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

2 40 4000( )

40

q qv q

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