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

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### 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: deliveries in different markets

- 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: deliveries in different marketsWhat 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.

The first order optimum condition is: deliveries in different markets

.

The second order condition of a unique maximum is satisfied since

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 programming (LP and QP) modelsForest 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.

Figure programming (LP and QP) models

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

. programming (LP and QP) models

Now, we will compare two strategies, A and B. programming (LP and QP) models

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.

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

We can simplify the expression this way:

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

Let us investigate the difference between the expected present values (costs) of strategies A and B!

- Observation 1. present values (costs) of strategies A and B!
- 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. present values (costs) of strategies A and B!

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:

Case 1. local supply), is sufficiently high in relation to the marginal storage cost, then it is optimal to increase the stock level.

Parameters:

n = 2

p = ½

The Binomial distribution of exogenous wood deliveries local supply), is sufficiently high in relation to the marginal storage cost, then it is optimal to increase the stock level.

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).

Case 2. local supply), is sufficiently high in relation to the marginal storage cost, then it is optimal to increase the stock level.

Parameters:

n = 6

p = ½

Case 3. local supply), is sufficiently high in relation to the marginal storage cost, then it is optimal to increase the stock level.

Parameters:

n = 6

p = ¼

The optimization problem at a general level local supply), is sufficiently high in relation to the marginal storage cost, then it is optimal to increase the stock 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 local supply), is sufficiently high in relation to the marginal storage cost, then it is optimal to increase the stock level.

s.t.

The mill local supply), is sufficiently high in relation to the marginal storage cost, then it is optimal to increase the stock level.

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 deliveries marginal profit in the mill, margprof, is defined as the product price minus variable production costs other than the raw material costs.

The 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: marginal profit in the mill, margprof, is defined as the product price minus variable production costs other than the raw material costs.

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:

The constraints: marginal profit in the mill, margprof, is defined as the product price minus variable production costs other than the raw material costs.

@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: marginal profit in the mill, margprof, is defined as the product price minus variable production costs other than the raw material costs.

d (=

Optimally controlled stochastic stock path under marginal profit in the mill, margprof, is defined as the product price minus variable production costs other than the raw material costs. monopsony or perfect raw material market when the entering stock level state is 0.

Optimally controlled stochastic stock path under marginal profit in the mill, margprof, is defined as the product price minus variable production costs other than the raw material costs. monopsony or perfect raw material market when the entering stock level state is 1.

Optimally controlled stochastic stock path under marginal profit in the mill, margprof, is defined as the product price minus variable production costs other than the raw material costs. monopsonywhen the entering stock level state is 6.

Optimally controlled stochastic stock path under marginal profit in the mill, margprof, is defined as the product price minus variable production costs other than the raw material costs. perfect raw material market when the entering stock level state is 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;

Conclusions stock states under optimal control and monopsony (with state constraints)

- 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 stock states under optimal control and monopsony (with state constraints)

- 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

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