Chapter 10

Wood Inventory Management in Paper Industry

G. S. R. Murthy, A. L. N. Murthy and Katta G. Murty

1 Brief History of Paper Making

Paper making is considered one of the four great inventions of China, and the paper
making process is traced to Cai Lun, a court eunuch during the Hun Dynasty in
105 CE (Current Era). However, paper was being used for wrapping and padding
in China since second century BCE (Before Current Era), as toilet paper since sixth
century CE (an Arab traveler to China in 851 CE wrote about the strange practice
of the Chinese use of toilet paper “ . . . the Chinese do not wash themselves with
water when they have done their morning necessities, but only wipe themselves with
paper!”), and for government issued paper currency by around 1000 CE.
Thick paper-like material produced from the pith of the Cyperus Papyrus plant
by lamination, called papyrus, was used even much earlier in Ancient Egypt.
The Chinese process for making paper in those days consisted of soaking and
pounding rags (recycled fibres from used textiles (of hemp, linen, or cotton)) and
some plant fibres into a watery pulp, then pouring this pulp onto a woven bamboo
screen. The layer of pulp that formed on the screen was then lifted off carefully, and
as it dried, it matted into a sheet of paper.
Use of paper spread slowly outside China to other parts of East Asia. In those days,
it was thin and translucent, hence written only on one side. The paper making tech-
nology traveled from China to Japan with Buddhist priests around 610 CE, and the
Japanese started making paper themselves using fibres of mulberry trees called bast.

The online version of this chapter (doi: 10.1007/978-1-4939-1007-6_10) contains supplementary

material, which is available to authorized users.

G. S. R. Murthy () · A. L. N. Murthy · K. G. Murty

SQC & OR Unit, Indian Statistical Institute, Street No.8, Habsiguda,
500 007 Hyderabad, India
e-mail: murthygsr@gmail.com
A. L. N. Murthy
e-mail: simhaaln@rediffmail.com
K. G. Murty
e-mail: murty@umich.edu

K. G. Murty (ed.), Case Studies in Operations Research, 199

International Series in Operations Research & Management Science 212,
DOI 10.1007/978-1-4939-1007-6_10, © Springer Science+Business Media New York 2015
200 G. S. R. Murthy et al.

With the defeat of China in the battle of Talas in Kyrgyzstan in 751 CE by the
Arabs, the secret of paper making was obtained from Chinese prisoners of war by
the Islamic world, and they set up a paper mill in Samarkand. Also they refined the
manufacturing process, and made paper making into a major industry by setting up
plants in Baghdad, Damascus and other places; and introduced books, bookbinding,
and bookshops. They started wrapping goods sold to customers in markets with paper
by twelfth century CE.
Soon paper making was diffused across all of the Islamic world and Europe. Paper
manufacturing reached India in thirteenth century CE through Arab merchants, and
it quickly replaced traditional writing materials there.
Archeological evidence indicates that in the USA, similar bark-paper writing
material (parchment made by boiling and pounding the inner bark of trees until the
material becomes suitable for art and writing) was in use among Mayas and other
Mesoamerican cultures earlier than fifth century CE.
In Europe, paper making started in the then Islamic part of the Iberian peninsula in
tenth century CE, and slowly spread to Italy, France, and Germany by 1400 CE, and
was mechanized by the use of waterpower. Its rapid expansion was truly enhanced
by the invention of the printing press and the beginning of the printing revolution
in fifteenth century CE. However, due to their noise and smell, paper mills were
allowed to be erected only outside the city boundaries in those days.
Until mid-nineteenth century CE, the most common fibre source for paper produc-
tion was recycled fibre from rags. A process of removing printing inks from recycled
paper (called de-inking now-a-days) was developed at the end of the eighteenth cen-
tury. Also around that time, continuous paper making machines were introduced.
Observing American wasps chewing up wood fiber and making their nests from it,
people got the idea that pulp from wood may make an ideal raw material to make
paper. Soon experiments began for using wood as a fibre source for paper making.
By mid-nineteenth century CE, a machine for extracting fibres from wood by
pulping it (instead of rags) was invented; and the technique for bleaching the pulp
(so that the paper produced will be white) was developed. This started a new era in
paper making. By the end of the nineteenth century, all paper making machines were
using wood instead of rags to make paper.
By 1900 CE, with the invention of the practical fountain pen, and mass-produced
pencil, and the introduction of the rotary printing press, wood-based paper and the
newspapers, schoolbooks, fiction books, magazines, etc. printed on paper caused a
major transformation of the human society.
For more details see articles in references [1, 2, 3, 4].

2 Brief Description of the Paper Manufacturing Process

We will describe the paper manufacturing process using wood as the basic raw
material (see Fig. 10.1). Wood of conifers is generally preferred because it consists
of longer fibres than the wood of deciduous trees (like eucalyptus, subabul, beech,
10 Wood Inventory Management in Paper Industry 201

Fig. 10.1 From Forest to Finished Products: The Paper Making Process (Details at: http://www.
internationalpaper.com/documents/EN/Sustainability/PaperProcess.pdf.)

etc.), as longer fibres form a firmer fibrous web leading to firmer paper on the paper
machine. Most paper manufacturing companies in India, including the large ones,
use debarked logs of eucalyptus, subabul, casurina trees as these are locally grown
abundantly. The company where this project was carried out uses 1,300,000 t of wood
202 G. S. R. Murthy et al.

logs of eucalyptus, subabul, casurina (the percentage of these species in the mix is
approximately 60, 30, 10 %, respectively) annually; and some imported wood pulp,
to manufacture writing and printing paper, and paperboard (used for manufacturing
packing boxes for packing a variety of consumer goods like toothpaste tubes etc.).
About 70 % of their wood pulp requirement is met by pulp made at the company with
the wood supply mentioned above, and 30 % met by imported wood pulp in the form
of dry sheets. Paperboard is manufactured on the paper machine by gluing together
three sheets of paper made simultaneously.
The first major operation in paper making is the preparation of pulp from which
paper is made. Paper mills can make the pulp from wood themselves (these mills
are called integrated mills), or they can buy pulp in the form of dry pressed sheets
(these sheets are broken down into pulp with water as they enter the manufacturing
process).
Wood consists of mostly cellulose, lignin (dark colored chemical compounds that
hold the cellulose fibres in wood together), and minor amounts of resins, tanins, and
mineral material. There are two basic ways of pulping wood logs. In mechanical
pulping, wood logs are processed into pulp by grinding them against a quickly
rotating stone disc with water, resulting in mechanical pulp (MP); this process yields
95 % pulp, but it leaves all the lignin in the pulp. The mechanical pulping process
damages the fibres in wood, with the result that paper made from it is not strong; and
since all the lignin is left in MP, paper made from it becomes yellow very quickly.
In chemical pulping, logs are processed into wood chips of about 3” size in the
chipping process at the beginning of paper making. Wood chips of various species
of wood in the raw material mix are mixed together, and this mixture then enters the
chemical cooking process. Here, the wood chips are cooked in a chemical solution
which converts lignin into a water-soluble substance that is then washed out, leaving
a pulp that consists of mostly cellulose fibre. So the yield of chemical pulping is
approximately 50 % of the input wood, but the fibres in it are clean and undamaged.
Spent fluids from chemical cooking are processed to recover the chemicals for reuse.
The pulp at this stage is brown in color, it goes to the bleaching process next.
Bleaching is a continuation of the chemical cooking process, it also removes the
remaining lignin in the pulp. Bleaching is a complex process consisting of several
chemical process steps with washing of the pulp between the various treatments.
Then, the pulp enters the refining process, where it goes through a fine blending
of the fibre ends for a close-knit connection between individual fibres (for increasing
the strength of paper produced), addition of water, fillers, sizing materials, dyes, and
additives (like kaolin, china clay, chalk, titanium dioxide etc. to make paper more
opaque, softer, and flexible). Then, the watery pulp enters the paper machine.
Once pulp has been prepared, the next operation in making paper for writing
and printing, is forming it into a sheet. This transforms the diluted pulp into a fine
uniform laminate on the paper machine consisting of a head box, and a forming
machine. Pulp arrives at the head box through round tubing, but it transfers it onto
the paper forming wire mesh in the form of a thin wide and uniform wet sheet.
Pulp exits the head box through a series of equispaced nozzles spread across the
width-line of the paper sheet to be formed; pulp comes out of each nozzle as a jet
10 Wood Inventory Management in Paper Industry 203

spray onto a continuously moving paper forming wire mesh (normally made of either
polyethylene or polyurethane). The paper sheet is consolidated during this journey;
water passing freely through the wire mesh drops down by gravity from it first, and
is sucked from it by vacuum suction towards the end of the forming section.
On leaving the forming section, the paper sheet has 20 % consistency (i.e., it is
80 % water), and enters the further drying process called wet pressing. In this process,
the paper sheet is passed between rolls in contact with felt which turns around the
rolls of the presses absorbing water from the sheet.
When the paper sheet comes out of wet pressing, its water content is about 60 %.
Next it enters the drying operation, in which more water is eliminated from it through
the application of heat by passing through heated air, through contact over huge
steam-heated cylinders, and through infrared drying. At the end of this operation, the
water content of the paper sheet is reduced to its target of approximately 5 %. This
drying is one of the costliest operations in paper manufacturing.
Then, the paper sheet enters the coating operation, where a starch based emulsion
consisting of special coatings is applied to the paper sheet in two stages called as
pre-coat and a final-coat, to give it smoothness and shine necessary so that inks can
easily be applied on it for writing, and printing. The composition of the coating
emulsion is tailored to the specific properties required for the final sheet of paper,
such as water resistance, gloss, opacity, smoothness, and whiteness.
Some papers that require high surface finish, now pass through an operation known
as calendering whose main purpose is to improve the gloss and printing properties
of paper.
After the calendering operation, the paper sheet is rolled up in the form of reels
known as jumbo rolls. Customers require paper either in the form of reels of specified
sizes, or in the form of sheets of standard sizes. Depending on these requirements,
in the finishing section, jumbo rolls are either rewound into reels as required by the
customer, or cut into sheets and packed into packets for delivery to customers.

2.1 Process for Making Paper Board

Above we discussed the process of making writing and printing paper on a paper
machine in which paper is manufactured as a single sheet. The process for making
paper board is the same as above until the paper machine stage. The paper machine
for making paper board consists of three head boxes arranged vertically one below
the other, each with its own separate forming machine and forming wire mesh,
forming simultaneously three sheets of paper one below the other vertically. Each of
these paper sheets goes through the same operations as above simultaneously until
the calendering operation, after which the three sheets are glued together to form a
single sheet of paper board, which is rolled up at the end to form a jumbo roll of
paperboard.
204 G. S. R. Murthy et al.

3 Wood Inventory Management in a Paper Mill

In this chapter, we will discuss wood inventory management and optimization, based
on a consulting project carried out by the Indian Statistical Institute at a leading paper
manufacturing company in India. The first process at the company is the chipping
process, and the mixture of wood of different species fed into the chippers is called
wood mixture, and the proportions of different species in the wood mixture is referred
to as furnish. The Management develops furnish specifications in each planning
horizon based upon the yield forecasts of the various species in that year, cost/ton of
each species, and other considerations.
When furnish is specified, we will refer to the specified proportions of various
wood species in the wood mixture as the furnish constraints. Out of different species,
subabul and eucalyptus account for about 90 % of the wood mixture. At present rates,
the cost of these two woods in the annual consumption approximately accounts for
about 10 % of the total annual turnover of the company.
Wood quantities are measured in tons. The quantity of wood that the company
can procure daily depends on the availabilities of wood, and trucks to haul it; both of
which tend to be low during the summer months of March to August. This seasonal
variation in the supply of wood over the year is the reason for the company main-
taining an inventory of wood in a storage yard to avoid the possibilities of shortages
and stock outs.
Wood is stored in stacks which are maintained species-wise in the storage yard.
Stored wood loses moisture over time, and is also subject to degradation by insect
damage, hence its weight decreases. For these reasons, the company forms stacks to
facilitate withdrawal by the first in first out (FIFO) principle, which it follows as far
as possible; and the company imposes an upper bound on the shelf-life for wood in
storage, that varies by species between 3 and 6 months.
Each day procured wood logs arrive at the company in trucks. Some of these
trucks are sent to the production facility as they arrive, to unload the wood in them
directly into the chippers. Wood fed into the chippers like this is called direct feed.
The remaining incoming trucks are sent to the wood storage yard for storing the
wood in them. Each day, besides the direct feed, some wood is also drawn from the
storage yard to feed the chippers, stored wood fed like this is referred to as indirect
feed. Thus, each day the feed into the chippers consists of direct and indirect feeds.
The planning horizon is a financial year, April 1st to March 31st. The production
manager of the pulp mill specifies the daily requirements of wood for the entire
year. From this and the specified furnish, the daily requirements are aggregated into
monthly requirements of wood for each wood species. The company uses month as
the planning unit. The raw materials manager uses these monthly requirements for
wood, and the proportion of direct and indirect feed patterns, to work out a month-
wise procurement plan for the year, and accordingly fixes deals with suppliers for
wood.
10 Wood Inventory Management in Paper Industry 205

3.1 Decision Making Problem Considered

The decision making problem that the management faces is that of deciding on
(1) how much wood (to be specified species wise) to procure each month, and (2)
how much of it to be used as direct feed (again to be specified species wise) so that
various constraints are satisfied and the overall cost is minimized.
Since subabul and eucalyptus constitute 90 % of input wood mixture, we will
confine the discussion in this chapter to just these two species only for simplicity.
Other species in the wood mixture can be handled in a similar way. The focus of
this chapter is to determine an optimum plan for procuring wood monthly, feeding it
into the chippers, and storing it in the storage yard. The mathematical model for this
decision making problem is given in detail in the next section. Section 5 presents the
results of application of the model to sample data, and Sect. 6 will summarize the
chapter.

4 Optimization Model for Wood Inventory Management

For the decision making problem in question, we shall consider two years—the
current year and the previous year. Thus, we have 24 months in all. These are
denoted by the set M = {1, 2, . . . , 24}. Here, 1 represents April of previous year,
2 represents May of the previous year, and so on.
The planning can be done (revised) at the beginning of any month of the current
year comprising months represented by 13 to 24. Let j0 denote the first month of the
planning period. That is, we are planning for the period month j0 of the current year
to rest of the months in the current year.
The decisions involved are how much of wood from each species to be purchased
in each month and how much of it to be fed directly into the production as direct
feed. Since the company follows FIFO policy, the decision on the indirect feed from
the yard depends only on the furnish. If FIFO is to be built into the model, then
the model becomes nonlinear and complex. Instead we can relax this constraint in
the model but the solution obtained will make it feasible to implement FIFO in the
practical execution. This aspect will be clear from the model we are building.
In our model, we define our decision variables as the quantities to be distributed
from each month’s stock/purchases to the production months without violating the
shelf life constraints. Let Ls and Le denote the shelf lives (in months) of subabul
and eucalyptus, respectively. This means that subabul procured in month i can be
used up to month i + Ls but not in month i + Ls + 1 or later. Similarly eucalyptus
procured in month i can be used up to month i + Le but not in month i + Le + 1
or later. Let u = max(1, j0 − Ls ) and v = max(1, j0 − Le ). Let t = min(u, v).
Thus, the months to be considered for the model are t, t + 1, . . . , 24. Define the sets
P = {j0 , j0 + 1, . . . , 24}, S = {(i, j ) : j ≥ i ≥ u, j ∈ P and j − i ≤ Ls } and
E = {(i, j ) : j ≥ i ≥ v, j ∈ P and j − i ≤ Le }.
The decision variables are defined in the following subsection.
206 G. S. R. Murthy et al.

4.1 Decision Variables

The decision variables are the quantities of wood distributed from each month’s
stock/availabilities to future months.
1. For each (i, j ) ∈ S define sij = quantity of subabul taken from month i and to be
used for production of month j .
2. For each (i, j ) ∈ E, eij = quantity of eucalyptus taken from month i and to be
used for production of month j .
3. It is possible that we may end up in an infeasible situation. The infeasibility occurs
when there is a shortage of wood (subabul and eucalyptus). For this reason, we
need to define the artificial variables only for the production months, of course,
two sets of variables—one for subabul and the other for eucalyptus. Define
(a) xj = quantity of subabul that will be made available at the beginning of month
j ∈ P for month j ’s production/opening inventory requirement,
(b) yj = quantity of eucalyptus that will be made available at the beginning of
month j ∈ P for month j ’s production/opening inventory requirement,

4.2 Symbols Used for Model Parameters (Data Elements)

We shall now describe the model parameters.

1. Availabilities.
(a) asi = quantity of subabul available from month i, u ≤ i ≤ 24. For i < j0 , asi
is a known quantity; and for i ≥ j0 , asi is a forecasted availability.
(b) aei = quantity of eucalyptus available from month i, v ≤ i ≤ 24. For i < j0 ,
aei is a known quantity; and for i ≥ j0 , aei is a forecasted availability.
2. Material Costs.
(a) csi = cost of subabul/ton purchased in month i, i ∈ M.
(b) cei = cost of eucalyptus/ton purchased in month i, i ∈ M.
3. Handling Costs. There are two types of handling—direct and indirect. Indirect
handling cost is much higher than the direct handling cost. Since the wood loses
its weight over time, the weight of wood also varies over time and hence the
calculation of cost of wood handled indirectly (that is first storing it in yard and
then feeding it to production later) also varies. For simplicity, it has been agreed
to take a reasonable rate, cost per ton, as indirect cost that will be proportional to
the weight at the time of storage. The present rates per ton of wood are Rs. 10 for
direct handling and Rs. 110 for indirect handling. Let
(a) cd = cost per ton for direct handling.
(b) ch = cost per ton for indirect handling.
4. Moisture Losses. Moisture losses are specified separately for subabul and euca-
lyptus. It is assumed that one ton of subabul (eucalyptus) purchased in month i
would weigh lijs (lije ) tons in month j .
10 Wood Inventory Management in Paper Industry 207

5. Opening Inventories. For each production month j , there is an opening inventory

requirement. This is specified for subabul and eucalyptus separately. Accordingly,
there must be at least qsj (qej ) tons of subabul (eucalyptus) at the beginning of
month j , j ∈ P .
6. Direct Feed Percentages. The direct feed refers to wood that is used directly from
the purchases, and the indirect feed is the wood fed from the stocks. One of the
requirements is that a certain percentage of the wood (by current weight in tons)
fed to production should be the direct feed. For production month j , the direct
feed percentage should be at least dj .
7. Production Requirement. At the time of planning, the management will specify
the total quantity of wood (in our case, the total quantity of subabul and eucalyptus
put together) required for each of the production months in the planning horizon.
We shall denote this by rj tons for the j th month, j ∈ P .
8. Furnish Limits. If rj is the production requirement for the month j , then the
percentage of subabul in that is called the furnish percentage. Two limits are
specified for each month—fj and Fj . The subabul percentage in rj should lie
between fj and Fj .

4.3 Constraints

There are a number of constraints in this optimization model. These are listed below.
1. Availability Constraints. As noted earlier asi s and aei s denote stocks/forecasts
of subabul and eucalyptus available from month i. Consider month i; the total
quantity of subabul drawn from this month is given by j :(i,j )∈S sij , i ≥ u, and
this should be at most asi . A similar expression holds for eucalyptus. Therefore,
the availability constraints are:
 
sij ≤ asi , for i ≥ u, and eij ≤ aei , for i ≥ v. (10.1)
j :(i,j )∈S j :(i,j )∈E

2. Shelf Life Constraints. The shelf life constraints have been built in the model by
defining the decision variables (sij s and eij s) appropriately.
3. Production Requirement Constraints. Consider month j ∈ P and let rj be the
quantity of wood required for this month. The quantities of subabul and eucalyptus
supplied to month j from month i are sij and eij . Due to moisture losses, sij
would weight sij lijs and eij would weight eij lije . Taking the artificial variables into
consideration, the production constraints are

xj + yj + sij lijs + eij lije ≥ rj , for all j ∈ P . (10.2)

4. Opening Inventory Constraints. Consider a month j ∈ P . Let us compute the

opening inventory of subabul for this month. Let i be such that i < j0 and
j −1
(i, j ) ∈ S. Note that k=j0 sik is the quantity of subabul drawn for the months
208 G. S. R. Murthy et al.
j −1
j0 , j0 + 1, . . . , j − 1. Therefore, the balance asi − k=j0 sik will remain at the
beginning of month j from the stocks of month i, and its weight at the beginning
j −1
of month j is (asi − k=j0 sik ) × lijs . Note that for (i, j ) to be in S, i must be at
j0 −1 j −1
least j − Ls . Thus, i=j −Ls ((asi − k=j0 sik ) × lijs ) is the amount of inventory
of subabul from months u, u + 1, . . . , j0 − 1. Now consider a month i such that
i+Ls
i ≥ j0 and (i, j ) ∈ S. Note that lijs k=j sik is the weight of subabul purchased
j −1 i+Ls
in month i and available at the beginning of month j . Thus, i=j0 lijs k=j sik
is the total quantity of subabul that is available as the opening inventory from the
purchases planned for the months j0 , j0 + 1, . . . , j − 1. Therefore, the total
inventory of subabul at the beginning of month j ∈ P is given by
⎛ ⎞
j0 −1 j −1 j −1
   s
i+L
Is (j ) = xj + lijs ⎝asi − sik ⎠ + lijs sik .
i=j −Ls k=j0 i=j0 k=j

Similarly, the opening inventory of eucalyptus for month j is given by

⎛ ⎞
j0 −1 j −1 j −1
   e
i+L
e ⎝ ⎠
Ie (j ) = yj + lij aei − eik + e
lij eik .
i=j −Le k=j0 i=j0 k=j

Recall that qsj and qej are the opening inventory requirements of subabul and
eucalyptus, respectively for the month j ∈ P . Therefore, we have the following
opening inventory constraints:
⎛ ⎞
j0 −1 j −1 j −1
   s
i+L
xj + lijs ⎝asi − sik ⎠ + lijs sik ≥ qsj for j ∈ P , (10.3)
i=j −Ls k=j0 i=j0 k=j

⎛ ⎞
j0 −1 j −1 j −1
   e
i+L
yj + lije ⎝aei − eik ⎠ + lije eik ≥ qej for j ∈ P . (10.4)
i=j −Le k=j0 i=j0 k=j

5. Direct Feed Constraints. For j ∈ P , recall that dj denotes the percentage of

direct feed. As sjj and ejj denote the direct feeds of subabul and eucalyptus,
respectively for month j , we must have

sjj + ejj ≥ dj rj /100, j ∈ P . (10.5)

6. Furnish Constraints. For j ∈ P , recall that fj and Fj denote the furnish limits
of subabul in month j with production requirement rj . Then, the furnish con-
straint states that the subabul content in rj must be at least fj rj /100 and at most
Fj rj /100. Therefore, we have
 
lijs sij ≥ fj rj /100 and lijs sij ≤ Fj rj /100. (10.6)
i:(i,j )∈S i:(i,j )∈S
10 Wood Inventory Management in Paper Industry 209

4.4 Objective Function

The objective function comprises three components, namely, the material cost, the
cost of handling and the cost of inventory. Since we are solving the problem with
artificial variables, the artificial variables (the shortages) are priced very high. For
this reason, we can omit the inventory and handling costs on these variables (in a
sense, these are included in the cost itself).
1. Material Cost. The cost of the material is associated with the old stocks (asi
and aei for i < j0 ) and new purchases (sij s, eij s for i ≥ j0 ). Note that if
a part of the old stock is not used, then it is rational to add the cost of this
unused material to the objective function. Therefore, the cost of material can be
taken as the cost of used material plus the cost of unused material. From the
definition of the decision variables, the unused  materialarises only from the old
stocks. Cost of used material is given by csi sij +  cei eij . Cost of unused
material is given by i<j0 (a si − s )c
ij si for subabul and
 − eij )cei for
i<j0 (aei 
eucalyptus.
 But c
 si ij s + (a
i<j0  si − s )c
ij si = a c
 i<j0 si si + (i≥j0 csi sij
and  cei eij + i<j0 (aei − eij )c = i<j0 aei cei + i≥j0 cei eij . As i<j0 asi csi
and i<j0 aei cei are constants, these two terms can be dropped from the objective
function. Take θ , a large positive number, as the cost of artificial variables xj and
yj . Then, the material cost component of the objective function is given by
  
CM = csi sij + cei eij + θ (xj + yj ). (10.7)
i≥j0 i≥j0 j ≥j0

2. Cost of Handling. The total quantity of wood handled is given by (sij + eij ) of
which (sjj +ejj ) is the quantity of direct handling. Again we are not considering
the artificial variables which is unnecessary. Therefore, handling cost is given by
 
CH = ch (sij + eij ) + (cd − ch ) (sjj + ejj ). (10.8)

3. Cost of Inventory. We take the cost of holding as the interest on cost of wood
held in inventory. Let r denote the rate of interest per annum. Subabul purchased
at a cost csi per ton in month i and used in month j is held in inventory for
j − i months. Therefore, the interest on this is given by sij × csi × (j − i) ×
r
1200
. But the quantity of ith month, i < j0 , not used in any of the production
months
 is  held in stock for (24 − i) months. The holding cost of this is equal to
csi asi − 24 k=j0 sij (24 − i)r/1200. Therefore, the total holding cost is given
by
⎧ ⎛ ⎞ ⎫
r ⎨ ⎬
j0 −1
 24
Hs = sij × csi × (j − i) × + csi ⎝asi − sij ⎠ (24 − i) .
1200 ⎩(i,j )∈S i=u k=j
⎭
0

(10.9)
210 G. S. R. Murthy et al.

Similarly, the holding cost of eucalyptus is given by

⎧ ⎛ ⎞ ⎫
r ⎨  ⎬
j0 −1
 24
He = eij × cei × (j − i) × + cei ⎝aei − eij ⎠ (24 − i) .
1200 ⎩(i,j )∈E i=v k=j
⎭
0

(10.10)
Therefore, the total cost of inventory is given by CI = Hs + He .

4.5 Complete Mathematical Formulation

The complete mathematical formulation for the decision making problem is given
by: Minimize CM + CH + Hs + He subject to constraints (1) to (6) mentioned above.

5 Application to Sample Data

We shall now apply the optimization model to a sample data from the company. A
decision support system (DSS), a software package, was developed for the company
which takes all the inputs for solving the problem and produces the required outputs.
With the help of this software, optimal solutions can be obtained for the dynamic
situations, that is, at the beginning of any month of the planning horizon. With the
help of this DSS, it was possible to study the effectiveness of using the optimization
model, by applying it to the past data at the end of a planning year (that is, end
of month 24) so that the actual decisions made (based on the subjective decisions
made by the management with their experience and expertise) as well as the actual
availabilities and the forecasts were known at the beginning of each month in the past.
In this section, we shall present the typical scenarios of analysis that need to be
performed on the data. One of them is when you are planning at the beginning of
the planning horizon (Sect. 5.2) and the other when you are at the beginning of the
second month of the planning horizon (Sect. 5.3). The second scenario talks about
dynamic optimization. At the beginning of the second month, we know what has
been planned and what has actually happened. This will give us an opportunity to
examine the difference between the plan and the actual scope for revising the plan
using the model.

5.1 Inputs

The inputs for the decision making problem are: (1) moisture losses, (2) old stocks
and forecasted availabilities, (3) cost elements (material costs, handling costs, rate
of interest), (4) shelf lives, (5) production requirements, and (6) opening inventory
requirements.
10 Wood Inventory Management in Paper Industry 211

1. Moisture Losses. Recall that lijs (lije ) is the residual weight of subabul (eucalyptus)
(that is one ton of subabul (eucalyptus) purchased in month i will weigh lijs tons
in month j ). Based on the past data the values of lijs s and lije s are estimated and
they are given below.
⎧
⎪
⎪ 0.68 if j ≥ i + 3
⎨
0.75 if j = i + 2
lijs = (10.11)
⎪
⎪ 0.79 if j = i + 1
⎩
1.00 if j = i

and
⎧
⎨0.63 if j ≥ i + 2
lije = 0.68 if j = i + 1 (10.12)
⎩
1.00 if j = i

2. Handling Costs, Shelf Lives and Rate of Interest. The cost of direct handling per
ton is Rs. 10 and that of indirect handling is Rs. 105. Shelf life of subabul is
4 months and that of eucalyptus is 6 months. The holding cost is taken as the
interest on inventory, and the rate of interest is taken as r = 15 % per annum.
3. Direct Feed and Furnish Percentages. Direct feed percentage is taken as 25 %
for all months, that is, dj = 25 for j = 13, 14, . . . , 24. The furnish limits are
also taken uniformly throughout the year, that is, fj = 40 and Fj = 65 for
j = 13, 14, . . . , 24.
4. Rest of the Inputs. The other inputs, the availabilities and forecasts, the production
requirements, the opening inventory requirement are given in Table 10.1.
In Table 10.1, data on asi , aei , csi , cei for i ≤ 12 are the actual and known figures,
where as the same for i ≥ 13 are forecasts. The production requirements ri are
planned figures and the actual realizations may be at variance due to various factors.
The opening inventories are also planned but usually less prone to changes.

5.2 Planning at the Start

In this section, we shall find the optimal decisions when we are at the beginning of
the planning horizon, that is, at the beginning of month 13. In this case, our j0 = 13
(see first paragraph of Sect. 4). It may be noted that the old stocks from yard are
given in Table 10.1 from the months October and November of the old year (months
7 and 8) for subabul; as the shelf life is only 4 months, the availabilities of subabul
of these two months will be ignored in the problem. The optimal plan is obtained
using the model of Sect. 4 and the input data as given in the previous section with
j0 = 13. The solution is presented in Table 10.2. The overall cost, the objective
value, is Rs. 1,208,985,562. This solution above is obtained with the constraint that
shelf lives of subabul and eucalyptus as 4 and 6 months, respectively. It may be
noted that the solution does not make use of the wood whose shelf life has expired.
212 G. S. R. Murthy et al.

Table 10.1 Inputs for asi , aei , csi , cei , qsi , qei , and ri
Subabul Eucalyptus
Month (i) asi csi qsi aei cei qei ri
Oct (7) 22,000 2222 – 6000 2300 – –
Nov (8) 23,000 2122 – 17,000 2100 – –
Dec (9) 16,700 2322 – 14,500 1900 – –
Jan (10) 26,745 2000 – 26,254 2300 – –
Feb (11) 15,466 2000 – 14,620 2300 – –
Mar (12) 16,964 2000 – 10,456 2300 – –
Apr (13) 17,283 2100 18,000 11,738 2478 18,000 33,100
May (14) 16,260 2200 18,000 12,478 2455 18,000 33,201
Jun (15) 15,772 2150 18,000 11,249 2485 18,000 33,071
Jul (16) 14,386 2300 18,000 11,784 2475 18,000 30,350
Aug (17) 11,793 2300 18,000 7086 2488 18,000 29,663
Sep (18) 9549 2300 18,000 8374 2389 18,000 29,868
Oct (19) 15,720 2300 18,000 15,614 2423 18,000 30,794
Nov (20) 18,615 2300 18,000 23,985 2395 18,000 27,447
Dec (21) 32,338 2100 18,000 30,515 2475 18,000 38,257
Jan (22) 20,473 2200 18,000 22,023 2497 18,000 39,881
Feb (23) 26,221 2150 18,000 18,486 2507 18,000 31,725
Mar (24) 22,643 2150 18,000 15,575 2519 18,000 31,264
All weights are in tons and costs are in Indian rupees per ton

Supposing the manager wishes to know the answer to the following question: If the
subabul shelf life is restricted to 3 months, is it still possible to get a feasible solution,
and if so, at what cost? Running the DSS software developed for this, the answer to
this question can be obtained at the click of a button. The answer is that there will
be shortages and that there is no feasible solution. The solution obtained using the
artificial variables (xj s and yj s) exhibits that the shortages are as follows: subabul
shortages are 1473 t in August, 5043 t in September and 1901 t in November; and the
eucalyptus shortages are 2051 t in the month of September. This information would
help the manager in revising his plans.

5.3 Planning After the First Month

We shall now look at a scenario at the end of April (month number 13). Now the
planning is required for the rest of the months. Recall that the forecasts of availabil-
ities for subabul and eucalyptus for the month of April were 17,283 and 11,738 t,
respectively. The optimal solution at the beginning of April suggested to procure
these amounts (see Table 10.2). Imagine, due to lack of sufficient quantity of euca-
lyptus, only 10,550 t of eucalyptus could be procured. Also, the actual production
quantity consumed is 27,226 t of total wood. Based on the actual consumptions and
procurements, the position of material availabilities and forecasts are given in Ta-
ble 10.3. The solution for this scenario is shown in Table 10.4. The problem has a
feasible solution and the total cost, the objective value (for the rest of the period), is
Rs. 1,108,098,387.
10 Wood Inventory Management in Paper Industry 213

Table 10.2 Solution for the decision making problem at the beginning of the planning horizon
(April)
From (i) To (j ) (lijs ) sij From (i) To (j ) (lije ) eij
9 13 0.68 16,700 7 13 0.63 6000
10 13 0.68 6152 8 14 0.63 17,000
10 14 0.68 20,593 9 15 0.63 14,500
11 15 0.68 15,466 10 14 0.63 278
12 16 0.68 16,964 10 15 0.63 8159
13 13 1.00 5922 10 16 0.63 17,816
13 17 0.68 11,361 11 17 0.63 14,620
14 14 1.00 7529 12 18 0.63 10,456
14 18 0.68 8730 13 13 1.00 7818
15 15 1.00 8290 13 19 0.63 321
15 19 0.68 7482 14 14 1.00 771
16 16 1.00 565 14 20 0.63 11,706
16 20 0.68 13,822 15 20 0.63 6057
17 17 1.00 5644 15 21 0.63 5191
17 21 0.68 6148 16 16 1.00 7023
18 18 1.00 9549 16 21 0.63 596
19 19 1.00 10,215 16 22 0.63 4165
19 23 0.68 5505 17 17 1.00 7086
20 20 1.00 6459 18 18 1.00 7796
20 24 0.68 12,156 18 24 0.63 578
21 21 1.00 20,672 19 19 1.00 15,272
21 24 0.68 7898 19 24 0.63 342
22 22 1.00 20,473 20 20 1.00 403
23 23 1.00 16,865 20 24 0.63 23,582
23 24 0.79 5433 21 21 1.00 9756
22 22 1.00 16,793
22 24 0.63 3826
23 23 1.00 11,104
23 24 0.68 311
24 24 1.00 7816
Note: Only non-zero sij s and eij s in the solution are presented

6 Summary

Wood inventory management is a crucial problem for pulp mills and paper industry.
The problem involves making decisions on procurement of wood of different species,
quantities to be stored and to be fed directly into production. The constraints include
maintaining direct feed proportions, ensuring shelf lives, furnish percentage, monthly
opening inventory requirements, production requirements and wood availabilities.
Experience-based decisions are usually far from optimality, and optimization models
are indispensable for getting optimal solutions. This chapter presented the problem
as a linear programming problem. Examples and solutions of sample scenarios are
presented. The company where this problem was solved was provided with a DSS, a
software tool, to solve the problem as and when required. This will facilitate smooth
planning and analysis of different scenarios.
214 G. S. R. Murthy et al.

Table 10.3 Inputs for asi , aei , csi , cei , qsi , qei , and ri
Subabul Eucalyptus
Month (i) asi csi qsi aei cei qei ri
Oct. (7) 22,000 2222 − 0 2300 − −
Nov. (8) 23,000 2122 − 14,000 2100 − −
Dec. (9) 0 2322 − 14,500 1900 − −
Jan. (10) 23,904 2000 − 26,254 2300 − −
Feb. (11) 15,466 2000 − 14,620 2300 − −
Mar. (12) 16,964 2000 − 10,456 2300 − −
Apr. (13) 12,920 2100 18,000 7850 2478 18,000 27,226
May (14) 16,260 2200 18,000 12,478 2455 18,000 33,201
Jun. (15) 15,772 2150 18,000 11,249 2485 18,000 33,071
Jul. (16) 14,386 2300 18,000 11,784 2475 18,000 30,350
Aug. (17) 11,793 2300 18,000 7086 2488 18,000 29,663
Sep. (18) 9549 2300 18,000 8374 2389 18,000 29,868
Oct. (19) 15,720 2300 18,000 15,614 2423 18,000 30,794
Nov. (20) 18,615 2300 18,000 23,985 2395 18,000 27,447
Dec. (21) 32,338 2100 18,000 30,515 2475 18,000 38,257
Jan. (22) 20,473 2200 18,000 22,023 2497 18,000 39,881
Feb. (23) 26,221 2150 18,000 18,486 2507 18,000 31,725
Mar. (24) 22,643 2150 18,000 15,575 2519 18,000 31,264
All weights are in tons and costs are in Indian rupees per ton

Exercises

1. For more versatile values of lij s, design a two-way table for capturing the inputs.
2. Explain why if first in first out (FIFO) is to be considered in the model, it will
become a nonlinear programming problem. Argue that the model proposed in
this chapter, the linear programming model, does not necessarily give an optimal
solution to the problem with FIFO. Also, explain how the solution obtained by
the model proposed in this chapter can still be adjusted to implement FIFO.
3. Consider the solution presented in Table 10.2. Note that the solution presents
the original values. Verify that the production requirements are actually met by
this solution. Also compute the direct and indirect feeds for each month from the
solution presented in Table 10.2. Compute the furnish met in each month for the
solution provided in Table 10.2.
4. Consider the following requirement. The management wants to ensure that at the
end of the planning horizon, there should be 15,000 t of subabul that is 3 months
old, 22,000 t of subabul that is 2 months old and 17,000 t of subabul that is 1
month old. Explain how you can ensure this using the proposed model.
5. A Rolling Horizon approach for modeling this problem: This is a simpler and
more direct approach for this problem. In this approach, for constructing the
model for the problem, the “year” does not play any role. The model depends
only on the current month, for which we need to determine how much subabul,
eucalyptus wood to buy in the current month for direct feed this month into the
10 Wood Inventory Management in Paper Industry 215

Table 10.4 Solution for the decision making problem at the beginning of the planning horizon
(May)
From (i) To (j ) (lijs ) sij From (i) To (j ) (lije ) eij
10 14 0.68 23,904 8 14 0.63 13,680
11 15 0.68 15,466 9 15 0.63 14,500
12 16 0.68 16,964 10 15 0.63 8194
13 17 0.68 12,920 10 16 0.63 17,816
14 14 1.00 5270 11 17 0.63 14,620
14 18 0.68 10,990 12 18 0.63 10,456
15 15 1.00 8268 13 19 0.63 7026
15 19 0.68 7504 14 14 1.00 3030
16 16 1.00 565 14 20 0.63 9447
16 20 0.68 13,822 15 20 0.63 4235
17 17 1.00 4581 15 21 0.63 2138
17 21 0.68 7212 16 16 1.00 7023
18 18 1.00 9549 16 21 0.63 4761
19 19 1.00 11,129 17 17 1.00 7086
19 23 0.68 4591 18 18 1.00 6255
20 20 1.00 6507 18 24 0.63 2120
20 24 0.68 12,109 19 19 1.00 10,133
21 21 1.00 19,946 19 24 0.63 5481
21 24 0.68 8768 20 20 1.00 2918
22 22 1.00 20,473 20 24 0.63 21,066
23 23 1.00 17,488 21 21 1.00 9058
23 24 0.79 4726 22 22 1.00 19,408
23 23 1.00 11,104
24 24 1.00 7816
Note: Only non-zero sij s and eij s in the solution are presented

paper making process, and for storing in the inventory and using to feed in each
of future months.
Let L = maximum{Ls , Le }, the maximum of the shelf lives of subabul and eu-
calyptus woods. Consider the current month to be month 0. Then, the planning
horizon H = {j : 0 ≤ j ≤ L}. The decision variables to be determined for this
current month are: s0j , e0j = tons of subabul, eucalyptus wood to purchase in the
market for feeding into the paper making process in month j for each j ∈ H .
As time moves forward from month 0 to the next, 0 drops out and L + 1 gets
added to the planning horizon. and a corresponding model has to be solved in that
month 1 in this approach. That is why this type of approach is known as the rolling
horizon approach. The model for the problem to be solved in month 1 is similar
to that in month 0 with appropriate changes for the passage of time; the model
with these changes has to be solved each month. Consider the current month 0.
In this month, we already have the solutions obtained for the models in earlier
months, from which we can easily compute the stock levels of both subabul and
eucalyptus purchased in earlier months for feeding into the paper making process
in month 0. Assuming that all other data are as given above, construct the model
to be solved in the current month 0; and describe how this model can be used by
the company for its decision making over time.
216 G. S. R. Murthy et al.

References

1. Paper. http://en.wikipedia.org/wiki/paper. Accessed 30 June 2014.

2. History of paper. http://en.wikipedia.org/wiki/History_of_paper. Accessed 30 June 2014.
3. Sappi: Inspired by life. www.sappi.com. Accessed 30 June 2014.
4. Toraspapel. www.torraspapel.com. Accessed 30 June 2014.