Optimal stock and purchase policy with stochastic external deliveries in different markets 12th Symposium for Systems Analysis in Forest Resources, Burlington,

Optimal stock and purchase policy with stochastic external deliveries in different markets

12th Symposium for Systems Analysis in Forest Resources, Burlington, Vermont, USA, September 5-8, 2006

Peter Lohmander

Professor of Forest Management and Economic Optimization, Swedish University of Agricultural Sciences, Faculty of

Forestry, Dept. of Forest Economics, 901 83 Umea, Sweden, http://www.lohmander.com/

Version 060830

Abstract:

• Forest industry companies with mills producing pulp, paper and sawn wood often obtain the roundwood from many different sources. These are different with respect to delays and degrees of variation. Private forest owners deliver pulpwood and timber at a stochastic rate. Imported pulpwood and timber may or may not arrive in large quantities a particular day.

• The losses may be considerable if mill production has to stop. If the stochastic supply falls, you may instantly reduce the stock level or buy more from the local market.

• Large stock level variations are only possible if the average stock is large. Stock holding costs may be considerable. If you control a monopsony and let the amount you buy per period from the local market change over time, this increases the expected cost, since the purchase cost function is strictly convex. In a market with many buyers, the purchase cost function appears almost linear to the individual firm and variations are less expensive.

• Hence, the optimal average stock level is higher if we have a monopsony than if we have a market with many independent buyers.

• The analysis is based on stochastic dynamic programming in Markov chains via linear programming.

Question:What is the optimal way to control a raw material stock in this typical situation:

You have to deliver a constant flow of raw material to a mill. Otherwise, the mill can not run at full capacity utilization, which decreases the revenues very much. Some parts of the deliveries to the raw material stock can not be exactly controlled in the short run. These deliveries are different with respect to delays and degrees of variation.

Imported pulpwood and timber may or may not arrive in large quantities a particular day.

Some private forest owners deliver pulpwood and timber at a stochastic rate. Some parts of the input to the raw material stock can be rather exactly controlled in the short run. The cost of rapidly changing such input may however be considerable. The costs of such changes are typically a function of the properties of the local raw material market.

2 ( / )

Q KC cm h

Q m

0

2

4

6

8

10

12

0 1 2 3 4 5

Time

Sto

ck L

evel

1

2

hC cm Q KmQ

2 02

dC hKmQ

dQ

0; 0; 0K M Q

23

22 0

d CKmQ

dQ

The first order optimum condition is:

The second order condition of a unique maximum is satisfied since

.

Q

2

2

hKmQ

2KmQ

h

The first order optimum condition can be rewritten this way and give us the optimal order quantity as an explicit

function of the parameters:

Deterministic multi period linear (and quadratic) programming (LP and QP) models

In most cases, such models are based on deterministic assumptions. The degree of detail is high but the solutions can not explicitly take

stochastic events into account

Adaptive multi period linear (and quadratic) programming models

Adaptive optimization is necessary. Compare Lohmander (2002) and Lohmander and Olsson (2004).

The round wood sources and properties

Forest industry companies with mills producing pulp, paper and sawn wood often obtain the round wood from many different sources.

These are different with respect to delays and degrees of variation.

Private forest owners deliver pulpwood and timber at a stochastic rate.

Imported pulpwood and timber may or may not arrive in large quantities a particular day.

The losses may be considerable if mill production has to stop.

If the stochastic supply falls, you may instantly reduce the stock level or buy more from the local market.

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12

Quantity

Pri

ce

Figure The local supply (price) function in the monopsony case.

0 1300 10p p

0 1

20 1

( )

c pu

c p p u u

c p u p u

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 2 4 6 8 10 12

Quantity = u

Co

st C(u)

line

(0) (10)2000

2 2

C C

5 1750C

(0) (10)5

2 2

C CC

( ) ( )E C u C E u

.

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12

Quantity

Pri

ce p

0

500

1000

1500

2000

2500

3000

3500

0 2 4 6 8 10 12

Quantity = u

Co

st

C(u)

(0) (10)5

2 2

C CC

( ) ( )E C u C E u

Now, we will compare two strategies, A and B.

In both cases, we have to make sure that the mill can run at full capacity utilization during the

following three periods 0, 1 and 2.

In strategy A, we try to keep the stock level as low as possible.

In strategy B, we buy more wood locally than we instantly need in period 1 in case the entering

stock level in period 1 is higher than zero.

Strategy A

0

2

4

6

8

10

12

0 2 4 6 8 10

Stage

Sto

ck L

evel

Low

High

1 1(6) (2) 4 (6) (2) 4 (6)

2 2AY c c h c c h c

The expected present value of strategy A can be illustrated this way, period by period:

4 (2) 2 (6)AY h c c

We can simplify the expression this way:

Strategy B

0

2

4

6

8

10

12

0 2 4 6 8 10

Stage

Sto

ck L

evel #1

#2

#3

#4

The expected present value of strategy B can be illustrated this way, period by

period:

1 1(6) (3) 4 (6) (1) 5 (2) 4 (5) 1 (6)

2 4BY c c h c c h c h c h c

118 (1) (2) 2 (3) (5) 7 (6)

4BY h c c c c c

Let us investigate the difference between the expected present values (costs) of

strategies A and B!

B AY Y

4 16 4 (2) 8 (6)AY h c c

4 18 (1) (2) 2 (3) (5) 7 (6)BY h c c c c c

4 2 (1) 3 (2) 2 (3) (5) (6)h c c c c c

2 2 2 2 20 14 2 1*1 3*2 2*3 1*5 1*6 1*1 3*2 2*3 1*5 1*6h p p

0 14 2 1 6 6 5 6 1 12 18 25 36h p p

14 2 4h p

1

1

2h p

• Observation 1.

• In case the wood market is perfect, no buyer can affect the market price.

• This means that it is not economically rational to increase the stock level (strategy B). Strategy A is a better alternative.

Observation 2.

In case the wood market is a monopsony, the buyer can affect the market price.

This means that Strategy A or strategy B can be the best choice. It is also possible that

they are equally good.

If the derivative of price with respect to volume (in the local supply), is sufficiently

high in relation to the marginal storage cost, then it is optimal to increase the

stock level.

Strategy B is better than Strategy A if:

1 2

hp

Case 1.Parameters:n = 2p = ½

x1 x2 x1+x2

0 0 0

0 1 1

1 0 1

1 1 2

x1+x2 f(x)

0 0,25

1 0,5

2 0,25

0

0,1

0,2

0,3

0,4

0,5

0,6

0 0,5 1 1,5 2 2,5

x = Total delivery (volume units)

f(x

) =

Pro

ba

bil

ity

The Binomial distribution of exogenous wood deliveries

The theory of the Binomial distribution can be found in many textbooks, such as Anderson et al. (2002). The original work on this distribution was made by Jakob

Bernoulli (1654-1705).

( )( ) (1 )x n xnf x p p

x

( ) .f x the probability that x harvest units are delivered from n sources

n the number of sources

!

!( )!

n n

x x n x

p the probability that a particular harvest unit delivery occurs

Case 2.Parameters:n = 6p = ½

x f(x) x! n! (n-x)! n!/x!/(n-x)!

0 0,015625 1 720 720 1

1 0,09375 1 720 120 6

2 0,234375 2 720 24 15

3 0,3125 6 720 6 20

4 0,234375 24 720 2 15

5 0,09375 120 720 1 6

6 0,015625 720 720 1 1

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0 2 4 6 8

x = Total delivery (volume units)

f(x

) =

Pro

ba

bil

ity

Case 3.Parameters:n = 6p = ¼

X f(x) x! n! (n-x)! n!/x!/(n-x)!

0 0,177979 1 720 720 1

1 0,355957 1 720 120 6

2 0,296631 2 720 24 15

3 0,131836 6 720 6 20

4 0,032959 24 720 2 15

5 0,004395 120 720 1 6

6 0,000244 720 720 1 1

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0 2 4 6 8

x = Total delivery (volume units)

f(x)

= P

rob

abil

ity

The optimization problem at a general level

We want to maximize the expected present value of the profit, all revenues minus costs, over an infinite horizon.

This is solved via stochastic dynamic programming. Compare Howard (1960),

Wagner (1975), Ross (1983) and Winston (2004).

Min ii

w s.t.

, ( )( , ) ,i j i u u U i

j

w j i u w R i u

max 10

1

( )s

i

Z w i

The mill

In each period, the pulp production volume, prod, is constrained by the capacity of the mill and by the amount of raw material (wood) in the stock.

@FOR( s_set(i): @FOR( u_set(j): prod(i,j) = @SMIN( millcap, (s(i)+ u(j)))));

The cost function contains a setup cost, csetup. The marginal profit in the mill, margprof, is defined as the product price minus variable production costs other than the raw material costs.

@FOR( q_set(i): q(i) = i-1);

@FOR( q_set(i)|q(i)#LT#1 : Rev(i) = 0);

@FOR( q_set(i)|q(i)#GE#1 : Rev(i) = -csetup + margprof*q(i));

The stochastic exogenous deliveriesThe exogenous deliveries are assumed to have a probability function of the type defined s “Case 2”.

The probabilities, pdev, of different deviations, dev, from the expected value are defined in this way:

@FOR( b_set(i): dev(i) = i-4 );@for( b_set(i):@free(dev(i)));pdev(1) = 1/64;pdev(2) = 6/64;pdev(3) = 15/64;pdev(4) = 20/64;pdev(5) = 15/64;pdev(6) = 6/64;pdev(7) = 1/64;

The control function:

The company purchase from the local market is the adaptive control, u, in this optimization problem.

The instant cost of the control can be calculated via the price function:

0 1

20 1

( )

c pu

c p p u u

c p u p u

0 1

2

12

2

2

dcp p u

du

d cp

du

The constraints:

@FOR( s_set(i):

@FOR( u_set(j)| prod(i,j)#LE# millcap #AND# (i+u(j)-prod(i,j))#LE#(smax-10):

[w_] w(i) >= Rev(1+prod(i,j)) - c(j) - mcstock*(s(i)+u(j)-prod(i,j)) + d*(pdev(1)*w(i+u(j)-prod(i,j) + 0) + pdev(2)*w(i+u(j)-prod(i,j) + 1) + pdev(3)*w(i+u(j)-prod(i,j) + 2) + pdev(4)*w(i+u(j)-prod(i,j) + 3) + pdev(5)*w(i+u(j)-prod(i,j) + 4) + pdev(6)*w(i+u(j)-prod(i,j) + 5) + pdev(7)*w(i+u(j)-prod(i,j) + 6) )

));

The parameters:

d (=

Parameter Explanation Case “Perfect raw material market”

Case “Monopsony raw material market”

Smax The number of states. 21 21

millcap The mill production capacity. 6 6

Csetup The mill set up cost. 0 0

margprof The “marginal profit” in the mill, the product price minus variable costs other than the raw material costs.

1000 1000

mcstock The marginal cost per stored united and period.

2 2

R Rate of interest (per year). 10% 10%

Dyear Discounting factor per year. 1/(1+r) 1/(1+r)

d Discounting factor per day. dyear^(1/365) dyear^(1/365)

P0 Wood price parameter 0 330 300

p1 Wood price parameter 1 0 10

Entering stock level Optimal control (local purchase) volume in the perfect market case

Optimal control (local purchase) volume in the monopsony case

10 0 2

9 0 2

8 0 2

7 0 2

6 0 3

5 1 3

4 2 3

3 3 3

2 4 4

1 5 5

0 6 6

Optimally controlled stochastic stock path under monopsony or

perfect raw material market when the entering stock level state is 0.

0

2

4

6

8

10

12

14

0 1 2 3 4

Stage

Sto

ck

Le

ve

l

x=0

x=1

x=2

x=3

x=4

x=5

x=6

Optimally controlled stochastic stock path under monopsony or

perfect raw material market when the entering stock level state is 1.

0

2

4

6

8

10

12

14

0 1 2 3 4

Stage

Sto

ck

Le

ve

l

x=0

x=1

x=2

x=3

x=4

x=5

x=6

Optimally controlled stochastic stock path under monopsony

when the entering stock level state is 6.

0

2

4

6

8

10

12

14

16

0 1 2 3 4

Stage

Sto

ck

Le

ve

l

x=0

x=1

x=2

x=3

x=4

x=5

x=6

Optimally controlled stochastic stock path under

perfect raw material market when the entering stock level state is 6.

0

2

4

6

8

10

12

14

0 1 2 3 4

Stage

Sto

ck

Le

ve

l

x=0

x=1

x=2

x=3

x=4

x=5

x=6

Determination of the steady state probabilities of entering stock states under optimal control and monopsony (with state constraints)

• p0 = 1/64*(1*p0 +1*p1 +1*p2 +1*p3);• p1 = 1/64*(6*p0 +6*p1 +6*p2 +6*p3 +1*p4);• p2 = 1/64*(15*p0 +15*p1 +15*p2 +15*p3 +6*p4 +1*p5);• p3 = 1/64*(20*p0 +20*p1 +20*p2 +20*p3 +15*p4 +6*p5 +1*p6 + 1*p7);• p4 = 1/64*(15*p0 +15*p1 +15*p2 +15*p3 +20*p4 +15*p5 +6*p6 +6*p7 +

1*p8);• p5 = 1/64*(6*p0 +6*p1 +6*p2 +6*p3 +15*p4 +20*p5 +15*p6 +15*p7 +6*p8

+1*p9);• p6 = 1/64*(1*p0 +1*p1 +1*p2 +1*p3 +6*p4 +15*p5 +20*p6 +20*p7 +15*p8

+6*p9 +1*p10);• p7 = 1/64*(1*p4 +6*p5 +15*p6 +15*p7 +20*p8 +15*p9 +6*p10);• p8 = 1/64*(1*p5 + 6*p6 + 6*p7 + 15*p8 + 20*p9 +15*p10);• p9 = 1/64*(1*p6 +1*p7 +6*p8 +15*p9 + 20*p10);

• p0 + p1 + p2 + p3 + p4 + p5 + p6 + p7 + p8 + p9 + p10 = 1;

Probability distributions of the optimal entering stock levels

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0 2 4 6 8 10 12

Stock level

Pro

bab

ility

Prob PM

Prob MO

Conclusions• Forest industry companies with mills producing pulp, paper and sawn wood

often obtain the roundwood from many different sources. These are different with respect to delays and degrees of variation. Private forest owners deliver pulpwood and timber at a stochastic rate. Imported pulpwood and timber may or may not arrive in large quantities a particular day.

• The losses may be considerable if mill production has to stop. If the stochastic supply falls, you may instantly reduce the stock level or buy more from the local market.

• Large stock level variations are only possible if the average stock is large. Stock holding costs may be considerable. If you control a monopsony and let the amount you buy per period from the local market change over time, this increases the expected cost, since the purchase cost function is strictly convex. In a market with many buyers, the purchase cost function appears almost linear to the individual firm and variations are less expensive.

• Hence, the optimal average stock level is higher if we have a monopsony than if we have a market with many independent buyers.

• The analysis is based on stochastic dynamic programming in Markov chains via linear programming.

References• Anderson D.R., Sweeney D.J. and Williams T.A., Statistics for Business and

Economics, Thomson, 8th edition, 2002• Baumol, W.J., Economic theory and operations analysis, Prentice Hall, 4 ed.,

1977• Howard, R., Dynamic Programming and Markov Processes, MIT Press, 1960• Lohmander, P., On risk and uncertainty in forest management planning

systems, in: Heikkinen, J., Korhonen, K. T., Siitonen, M., Strandström, M and Tomppo, E. (eds). 2002. Nordic trends in forest inventory, management planning and modelling. Proceedings of SNS Meeting in Solvalla, Finland, April 17-19, 2001. Finnish Forest Research Institute, Research Papers 860, p 155-162, ISBN 951-40-1840-0, ISSN 0385-4283

• Lohmander, P., Olsson, L., Adaptive optimisation in the roundwood supply chain, accepted for publication in SYSTEMS ANALYSIS - MODELLING - SIMULATION and in Olsson Leif 2004, Optimisation of forest road investments and the roundwood supply chain Acta Universitatis agriculturae Sueciae. Silvestria nr 310

• Markland, R.E., Topics in Management Science, Wiley, 3 ed., 1989 • Ross, S., Introduction to Stochastic Dynamic Programming, Academic Press,

1983 • Wagner, H., Principles of Operations Research, 2nd ed., Prentice Hall, 1975• Winston, W., Introduction to Probability Models, Operations Research: Vol. 2,

Thomson, 2004