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Part III: multiple resource types

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(1)

Inventory management

Claudio Arbib

Università dell’Aquila __________

Part III: multiple resource types

(2)

Contents

Multiple stock management

– Different resource types regularly absorbed by the system:

formulation as integer linear programming

– Example of application

(3)

Multiple stock management

• Previous models do not allow us in general to optimize the simultaneous management of distinct stock types

• To cope with this situation one can resort to an

optimization model based on integer linear programming

• Referring to day t, let us indicate as

d it = the demand s it = the stock level

r it = the replenishment g it = the per-unit

inventory cost of

resource type i

(i = 1, …, m)

(4)

Multiple stock management

• Resources are purchased from p different magazines: let R h denote the set of resources purchased from magazine h

• Referring to day t, let us indicate as

d it = the demand s it = the stock level r it = the replenishment g it = the per-unit

inventory cost of resource type i (i = 1, …, m)

Customer

Supplier 1

Supplier p R

1

R

p

(5)

Multiple stock management

c o s t

amount

0 10 20 30 40 50 60

kc

0h

for (k – 1) δ

h

< r < k δ

h

• The delivery cost depends on the delivering magazine: let c 0h be the cost borne to send up to δ h resource units from magazine h

• Referring to day t, let us indicate as

d it = the demand s it = the stock level

r it = the replenishment g it = the per-unit

inventory cost of resource type i (i = 1, …, m)

• Let

x ht = number of deliveries

from magazine h on

day t

(6)

Multiple stock management

• Referring to day t, let us indicate as

d it = the demand s it = the stock level

r it = the replenishment g it = the per-unit

inventory cost of resource type i (i = 1, …, m)

• Let

x ht = number of deliveries from magazine h on dayt

The problem reads:

min Σ c 0h Σ x ht + Σ Σ g it s it

h=1 t∈T i=1 t∈T

s i,t = s i,t–1 + r i,t–1 – d i,t ( for i = 1, …, m, t ∈ T )

Σ r it < δ h x ht

i∈R

h

( for h = 1, …, p , t ∈ T ) x ht , r it , s it > 0 x ht integer

p m

(7)

Example

day 1 2 3 4 5 6 7 8 9 10 11 12

20 20 20 19 19 19 18 17 17 17 17 16 g1

10 12 12 12 12 12 14 14 14 14 15 16 g2

15 14 14 14 12 10 9 9 10 11 12 14 g3

cost

A plant is fed with 3 resource types. A forecast for the 12 days to come refers that their value is going to vary as follows

inventory cost

0 5 10 15 20 25

1 2 3 4 5 6 7 8 9 10 11 12

day

cost Risorsa 1

Risorsa 2 Risorsa 3

(8)

giorno 1 2 3 4 5 6 7 8 9 10 11 12

20 20 20 19 19 19 18 17 17 17 17 16 g1

10 12 12 12 12 12 14 14 14 14 15 16 g2

15 14 14 14 12 10 9 9 10 11 12 14 g3

costo

day 1 2 3 4 5 6 7 8 9 10 11 12

20 20 20 19 19 19 18 17 17 17 17 16 g1

10 12 12 12 12 12 14 14 14 14 15 16 g2

15 14 14 14 12 10 9 9 10 11 12 14 g3

2,0 2,1 2,1 1,8 1,0 1,2 1,6 2,2 0,3 1,0 d1

3,0 3,0 3,0 3,0 4,0 1,0 3,0 d2

2,0 2,1 2,6 4,0 1,0 5,0 d3

cost

absorption

Example

The demand of finite products over time causes the following

absorption of resources 1, 2 and 3

(9)

giorno 0 1 2 3 4 5 6 7 8 9 10 11 12

20 20 20 19 19 19 18 17 17 17 17 16 g1

10 12 12 12 12 12 14 14 14 14 15 16 g2

15 14 14 14 12 10 9 9 10 11 12 14 g3

2,0 2,1 2,1 1,8 1,0 1,2 1,6 2,2 0,3 1,0 a1

3,0 3,0 3,0 3,0 4,0 1,0 3,0 a2

2,0 2,1 2,6 4,0 1,0 5,0 a3

costo

assorbimento

day 0 1 2 3 4 5 6 7 8 9 10 11 12

20 20 20 19 19 19 18 17 17 17 17 16 g1

10 12 12 12 12 12 14 14 14 14 15 16 g2

15 14 14 14 12 10 9 9 10 11 12 14 g3

2,0 2,1 2,1 1,8 1,0 1,2 1,6 2,2 0,3 1,0 d1

3,0 3,0 3,0 3,0 4,0 1,0 3,0 d2

2,0 2,1 2,6 4,0 1,0 5,0 d3

5,0 s1

4,0 s2

2,0 s3

cost

absorption

residual stock

This is the stock level inherited from previous days

Example

The demand of finite products over time causes the following

absorption of resources 1, 2 and 3

(10)

Example

day 0 1 2 3 4 5 6 7 8 9 10 11 12

20 20 20 19 19 19 18 17 17 17 17 16 g1

10 12 12 12 12 12 14 14 14 14 15 16 g2

15 14 14 14 12 10 9 9 10 11 12 14 g3

2,0 2,1 2,1 1,8 1,0 1,2 1,6 2,2 0,3 1,0 d1

3,0 3,0 3,0 3,0 4,0 1,0 3,0 d2

2,0 2,1 2,6 4,0 1,0 5,0 d3

5,0 s1

4,0 s2

2,0 s3

400 c0A 500 c0B cost

absorption

residual stock deliveries

The demand of finite products over time causes the following absorption of resources 1, 2 and 3

Resources 1 and 2 come from magazine A, resource 3 from

magazine B. One delivery from A costs c 0A = 400€, from B

c 0B = 500€

(11)

Example

day 0 1 2 3 4 5 6 7 8 9 10 11 12

20 20 20 19 19 19 18 17 17 17 17 16 g1

10 12 12 12 12 12 14 14 14 14 15 16 g2

15 14 14 14 12 10 9 9 10 11 12 14 g3

2,0 2,1 2,1 1,8 1,0 1,2 1,6 2,2 0,3 1,0 d1

3,0 3,0 3,0 3,0 4,0 1,0 3,0 d2

2,0 2,1 2,6 4,0 1,0 5,0 d3

5,0 s1

4,0 s2

2,0 s3

400 c0A 500 c0B

20 delta1 18 delta2 cost

absorption

residual stock deliveries

capacity

The demand of finite products over time causes the following absorption of resources 1, 2 and 3

In a single delivery one can send up to 20 units (of resource

types 1 and 2) from A and up to 18 (of type 3) from B

(12)

Example

day 0 1 2 3 4 5 6 7 8 9 10 11 12

20 20 20 19 19 19 18 17 17 17 17 16 g1

10 12 12 12 12 12 14 14 14 14 15 16 g2

15 14 14 14 12 10 9 9 10 11 12 14 g3

2,0 2,1 2,1 1,8 1,0 1,2 1,6 2,2 0,3 1,0 d1

3,0 3,0 3,0 3,0 4,0 1,0 3,0 d2

2,0 2,1 2,6 4,0 1,0 5,0 d3

5,0 s1

4,0 s2

2,0 s3

r1 r2 r3

400 c0A 500 c0B

20 delta1 18 delta2 replenishment

deliveries capacity

cost

absorption

residual stock

The decision variables of the model regard, for each day,

• the deliveries from each magazine

• the stock levels and resource amounts sent

The former variables are integer, the latter real, all non-negative

(13)

Example

day 0 1 2 3 4 5 6 7 8 9 10 11 12

20 20 20 19 19 19 18 17 17 17 17 16 g1

10 12 12 12 12 12 14 14 14 14 15 16 g2

15 14 14 14 12 10 9 9 10 11 12 14 g3

2,0 2,1 2,1 1,8 1,0 1,2 1,6 2,2 0,3 1,0 d1

3,0 3,0 3,0 3,0 4,0 1,0 3,0 d2

2,0 2,1 2,6 4,0 1,0 5,0 d3

5,0 s1

4,0 s2

2,0 s3

r1 r2 r3

400 c0A 500 c0B

20 delta1 18 delta2 replenishment

deliveries capacity

cost

absorption

residual stock

The cost is obtained by summing up

• the number of deliveries multiplied by the relevant costs

• the stock levels multiplied (scalar product) by inventory costs

MATR.SOMMA.PRODOTTO(C2:N4;C8:N10) + N$14$*SOMMA(B14:M14) + N$15$*SOMMA(B15:M15) N$15$*SUM(B15:M15)

SUMPRODUCT(C2:N4;C8:N10) + N$14$*SUM(B14:M14) +

(14)

Example

day 0 1 2 3 4 5 6 7 8 9 10 11 12

20 20 20 19 19 19 18 17 17 17 17 16 g1

10 12 12 12 12 12 14 14 14 14 15 16 g2

15 14 14 14 12 10 9 9 10 11 12 14 g3

2,0 2,1 2,1 1,8 1,0 1,2 1,6 2,2 0,3 1,0 d1

3,0 3,0 3,0 3,0 4,0 1,0 3,0 d2

2,0 2,1 2,6 4,0 1,0 5,0 d3

5,0 s1

4,0 s2

2,0 s3

r1 r2 r3

400 c0A 500 c0B

5 0 0 0 0 0 0 0 0 0 0 0

4 0 0 0 0 0 0 0 0 0 0 0

2 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 20 delta1

0 0 0 0 0 0 0 0 0 0 0 0 18 delta2

cost

absorption

residual stock

replenishment deliveries

equilibrium capacity

Once introduced the following constraints

• equilibrium: b

i,t–1

+ r

i,t–1

– b

i,t

= a

i,t

• capacity: δ

h

x

ht

– Σ

Rh

r

it

> 0 one can solve the problem, getting

(15)

Example

day 0 1 2 3 4 5 6 7 8 9 10 11 12

20 20 20 19 19 19 18 17 17 17 17 16 g1

10 12 12 12 12 12 14 14 14 14 15 16 g2

15 14 14 14 12 10 9 9 10 11 12 14 g3

2,0 2,1 2,1 1,8 1,0 1,2 1,6 2,2 0,3 1,0 d1

3,0 3,0 3,0 3,0 4,0 1,0 3,0 d2

2,0 2,1 2,6 4,0 1,0 5,0 d3

5,0 3,0 0,9 2,8 1,0 1,0 0,0 5,1 3,5 1,3 1,3 1,0 0,0 s1 4,0 1,0 1,0 3,0 0,0 0,0 0,0 8,0 4,0 3,0 3,0 3,0 0,0 s2 2,0 2,0 0,0 0,0 0,0 12,6 10,0 10,0 10,0 6,0 5,0 5,0 0,0 s3 0,0 0,0 4,0 0,0 0,0 0,0 6,3 0,0 0,0 0,0 0,0 0,0 r1

0,0 0,0 5,0 0,0 0,0 0,0 11,0 0,0 0,0 0,0 0,0 0,0 r2 0,0 0,0 0,0 0,0 14,7 0,0 0,0 0,0 0,0 0,0 0,0 0,0 r3

0 0 1 0 0 0 1 0 0 0 0 0 400 c0A

0 0 0 0 1 0 0 0 0 0 0 0 500 c0B

2 2 2 2 0 1 1 2 2 0 0 1

3 0 3 3 0 0 3 4 1 0 0 3

0 2 0 0 2 3 0 0 4 1 0 5

0 0 11 0 0 0 3 0 0 0 0 0 20 delta1

0 0 0 0 3 0 0 0 0 0 0 0 18 delta2

replenishment deliveries

equilibrium capacity

cost 2676

absorption

residual stock

Once introduced the following constraints

• equilibrium: b

i,t–1

+ r

i,t–1

– b

i,t

= a

i,t

• capacity: δ

h

x

ht

– Σ

Rh

r

it

> 0 one can solve the problem, getting

(16)

Example

day 0 1 2 3 4 5 6 7 8 9 10 11 12

20 20 20 19 19 19 18 17 17 17 17 16 g1

10 12 12 12 12 12 14 14 14 14 15 16 g2

15 14 14 14 12 10 9 9 10 11 12 14 g3

2,0 2,1 2,1 1,8 1,0 1,2 1,6 2,2 0,3 1,0 d1

3,0 3,0 3,0 3,0 4,0 1,0 3,0 d2

2,0 2,1 2,6 4,0 1,0 5,0 d3

5,0 3,0 0,9 2,8 1,0 1,0 0,0 5,1 3,5 1,3 1,3 1,0 0,0 s1 4,0 1,0 1,0 3,0 0,0 0,0 0,0 8,0 4,0 3,0 3,0 3,0 0,0 s2 2,0 2,0 0,0 0,0 0,0 12,6 10,0 10,0 10,0 6,0 5,0 5,0 0,0 s3 0,0 0,0 4,0 0,0 0,0 0,0 6,3 0,0 0,0 0,0 0,0 0,0 r1

0,0 0,0 5,0 0,0 0,0 0,0 11,0 0,0 0,0 0,0 0,0 0,0 r2 0,0 0,0 0,0 0,0 14,7 0,0 0,0 0,0 0,0 0,0 0,0 0,0 r3

0 0 1 0 0 0 1 0 0 0 0 0 400 c0A

0 0 0 0 1 0 0 0 0 0 0 0 500 c0B

2 2 2 2 0 1 1 2 2 0 0 1

3 0 3 3 0 0 3 4 1 0 0 3

0 2 0 0 2 3 0 0 4 1 0 5

0 0 11 0 0 0 3 0 0 0 0 0 20 delta1

0 0 0 0 3 0 0 0 0 0 0 0 18 delta2

replenishment deliveries

equilibrium capacity

cost 2676

absorption

residual stock

The chart obtained helps understanding daily operation in terms of

resource absorptions, residual stock monitoring and replenishment

(17)

Example

0 5 10 15 20 25

0 1 2 3 4 5 6 7 8 9 10 11

resource 1 resource 2 resource 3

The chart obtained helps understanding daily operation in terms of resource absorption, residual stock monitoring and replenishment

0 4 8 12 16 20

0 1 2 3 4 5 6 7 8 9 10 11

resource 1 resource 2 resource 3

residual stock replenishment

Riferimenti

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