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Long term planning of a major national sector using a linear programming model
European Journal of Operational Research 36 (1988) 153-166 North-Holland
153
Case Study
Long term planning of a major national sector using a linear programming model Selahattin KURU
Department of Computer Engineering~ Bogazici University, Bebek, Istanbul, Turkey Abstract: This article discusses a linear programming model for the long term planning of the Turkish iron and steel industry. Iron and steel industry is a complicated industry. It is rich in terms of products and production technologies, and has a large transportation problem associated with it. The model is formulated as a cost minimization problem and is solved in terms of several scenarios, each representing a different strategy for capacity expansion, using a mathematical programming package. Keywords: Sectoral planning, investment, linear programming, iron and steel
1. Introduction
The volume of iron and steel consumption is considered as an indicator of the level of national development. The present level of iron and steel consumption in Turkey is about 100 kg per capita, which is a typical value for a developing country [1]. This value is about 500 for developed countries, and Turkey plans to increase it to 300 in the year 2000. This corresponds to the equivalent of 17 million tons of molten steel consumption a year. The demand for iron and steel products may be met by imports or by increasing the production level, which requires investing for capacity increase of existing plants or building new plants. The subject of this case study is to evaluate different investment alternatives in detail for the planning of the Turkish iron and steel sector up to the year 2000. The work is done for The State Planning Organization of Turkey, which is responsible for developing sectoral and national investment plans.
Received December 1986; revised October 1987
The study of investment involves the determination of the location, the capacity, and the start of the production year for plants to be newly installed, and the capacity increase, and the year of capacity increase for existing plants. Also required is the determination of the production levels of plant units, the flow of raw materials and intermediate and final products within the sector, and the export and import levels.
2. The structure of the sector
The iron and steel sector is a complicated sector. It produces many different products using many different production technologies. Ore mines and consumption centers are distributed over the whole country. Furthermore, the private sector in Turkey has a large number of small plants whose operations are very hard to follow. The sector can be represented as a network as shown in Figure 1, where nodes denote plants, mines, e x p o r t / i m p o r t ports and consumption centers, and arcs denote the flow of material between nodes. As can be seen in the figure, there is
0377-2217/88/$3.50 © 1988, ElsevierSciencePubfishers B.V. (North-Holland)
154
S. Kuru / Long term planning of a major national sector using LP
PORTS K rfw materials
MINES
"I" "I"- \ | |fi I
finol
interm i product s
rawmateriats finalproducts I PLANTS I
ICONSUMPTION I CENTERS
I intermadlate products Figure 1. Network representation of the sector
the flow of raw materials from mines and ports to plants, the flow of final products from plants and ports to consumption centers and from plants to ports, and the flow of intermediate products between plants and ports (both directions) and between plants. The arc denoting the flow of intermediate products between plants also represent the flow of intermadiate products within plants. Note that mines and consumption centers are source and sink nodes, respectively, and plant units are intermediate nodes. A plant unit is actually a conversion point where incoming materials change form. One characteristic of a plant unit is that it may involve incoming flows of more than one type of material, but outgoing flows are always of one single type of material. Plants can be divided into two categories according to the types of final products they produce: long products plants and flat products plants. Long products include sections, rods and bars, and flat products include sheets and plates. For both product groups plants are either integrated plants, consisting of all the major production units, or small plants, consisting of some of the production units. Integrated plants produce essentially all final products of a group starting from iron ore whereas small plants either use an intermadiate product as the input material or produce only a few product types of the product group. There are many variations for small plants such as electric arc furnaces, sponge iron plants, and rolling mills. We will call all these variations non-integrated plants.
The flow diagram of a typical integrated plant producing flat products is given in a simplified form in Figure 2a. The plant may be divided into three main sections: Blast furnaces, steel converters and rolling mills. Blast furnaces convert iron ore into liquid metal. Iron ore may be in the form of lumps, dust, pellet or sinter. Pellet and sinter are pretreated ore. Pellet is produced at an ore mine site. Sinterizing, on the other hand, is performed at the sinter unit of a plant. The other major input to blast furnaces is coke, which is used to reduce iron ore (iron oxide) into metalic iron by a chemical reaction called reduction. Coke is produced from coking coal in the coke works unit of the plant. Most of the liquid metal produced in a plant goes directly into steel converters while the rest is converted into pig iron in the pig iron unit. Steel converters convert liquid metal into molten steel by reducing its carbon content. Lime, which is produced in the lime unit from limestone is used in this process. Another input to steel converters is scrap. Metallurgically, scrap is steel and needs only reshaping for converting it into final products. The price of scrap is low compared to the cost of producing liquid metal from ore, hence making it a desirable raw material. Scrap comes from two sources: Recycling of rolling mills, where it is obtained as a by-product of the shaping process, and from outside. Blast furnaces and steel converters are basically the same for both types of integrated plants. The rolling mill section, which is the section equipped
S. Kuru / Long term planning of a major national sector using LP
c°al~
scrap IliquidI "4 STEEL molten )J BLAST steel FURNACESI I "I CONVERTERS
c°k~ e
jumpore polle
15 5
~
PIG IRON I > "1 UN,T I 'g
dustI ore .... Islnter
Iron
~
plates
_ ~ CASTING SLAB~ (a)
)
L hot slab
rolled
sheets
sheets
c°al~coke lliquid~rap I PLANTI--~I STEEL lumpore )I BLAST metal steel ] ]CONVERTERS ~ FURNACES
Imo,ten
dus~l . . . . ore
Is~nter
--I
PIG UNIT IRON I > mg iroiq ingot
I INGOT CASTING
sectlons
>
rodsandbars
>
(b) Figure 2. Simplified flow diagrams of an integrated plant. (a) Flat products plant; (b) Long products plant
with units for producing specific products, is necessarily different for the two product groups. This section varies considerably even for plants producing products of the same product group. The rolling mills section of a flat products plant typically consists of slab casters, plate mills, and hot and cold rolls. For a long products plant the rolling mills section consists of ingot casting units, section mills, and rods and bars mills. Figure 2b shows the flow diagram of a long products plant in a simplified form. As can be seen in Figure 2 an intermediate product is exported from or im-
ported to the plant, i.e. slab in the case of flat products plants and steel ingot in the case of long products plants. The major unit in a non-integrated plant is the electric arc furnace, which converts scrap into steel. Steel may be in the form of ingot or it may undergo continuous casting for slab production. While ingot is reshaped into sections and bars in rolling mills, slab is converted into plates in hot rolls. The sponge iron technology, which is a relatively new technology, is based on converting pellet into sponge iron, which corresponds to the
156
S. Kuru / Long term planning of a major national sector using LP
mgot I scrap 1
SECTIONS MILLS
] sections )
ELECTRIC FURNACE ARC RODS AND BARS MILLS
(a)
I rods& bars).
ot
scrap
ELECTRIC ARC FURNACE
Imolten | I steel ICASTING ~ UNIT
HOT ROLL
hotrolled ) sheets
(b)
ingot scrap
t
SPONGE IRON
sp°nge ~ODS& iron \II ELECTRICIinn°t/I ARC ~ e~b ! r°dsand bars
UNIT
(c)
IFURNACE ii IMILLS ,,~ ingot
Figure 3. Simplified flow diagrams of a non-integrated plant. (a) Long products plant; (b) Flat products plant; (c) Sponge iron plant
process performed in blast furnaces in integrated plants. Sponge iron then undergoes steel conversion process in an electric furnace. These variations are shown in Figure 3. Iron and steel products are consumed over the whole country. This is especially true for long products since these are mainly consumed by building and construction industries. Flat products, on the other hand, are the main input of the manufacturing industry, thus being consumed mostly at industrial regions. As stated earlier, iron ore and coal are the principal raw materials of the sector. Coking coal is available in one region in Turkey, namely the Zonguldak region. Iron ore, on the other hand, is available in many regions. Ore mines differ greately in terms of production capacity and ore quality. Some ore mines have facilities to convert dust ore into pellet. Ore quality is mainly measured in terms of its tenor, which is the iron content. Tenor is typically around 50%. Size is of secondary importance. Ore lumps are prefered because of ease of transporta-
tion and handling. Dust ore is utilized by converting it into pellet or sinter. The sinterizing process also reduces the sulphur content of the ore. The iron and steel sector introduces a heavy demand on the transportation sector because of the huge amount of material transported within the sector. This includes the transportation of raw materials between mines and plants and between ports and plants, the transportation of intermediate products between plants and between ports and plants, and the transportation of final products between plants and consumption centers, between plants and ports and between ports and consumption centers. The transportation of raw materials within the country is always by rail because the mines are located inland. This means that railroad construction should be completed before a new mine starts production. Railroad reaches a mine region up to a location called a loading station. Mine is transported to the loading station from individual mines of the region by trucks. The transportation of intermediate and final
S. Kuru / Long term planning of a major national sector using LP
products within the country is by rail or by sea, depending on the location of the plant and the consumption center. Export and import of raw materials and intermadiate and final products is by sea. Because of the volume of international trade in the sector, plants are usually built on the coast. Ports utilized by the sector are usually among the largest in a country. For a detailed discussion of both the technological and economical aspects of the iron and steel industry see Manners [2].
3. The approach Long term sectoral planning has been approached in numerous ways. Methodologies include linear [3,4,5], recursive linear [6,7], mixed integer [8], and dynamic programming [9] as well as simulation models [10]. We have adopted linear programming and mixed integer programming methodologies at different stages of the study. We will first briefly discuss how the model evolved into its final form by noting the key issues addressed at those stages and then give the final form of the model in the next section. The study was carried out in cooperation with the experts from The State Planning Organization, from a private engineering and consultancy firm and from the university. The first version of the model was developed in about four months. Many meetings were held during this period trying to determine the details of the model. The experts from the State Planning Organization and the consultancy firm were eager to represent all the fine details of the sector in the model while those from the university tried to keep it small and manageable. Finally the following strategy was adopted: Start with a small linear programming model and expand it into a full size mixed integer programming model gradually. This strategy has two benefits: ease of debugging the model at all stages, and better understanding of the significance of including different phenomena in the model. In the first stage, a linear programming model was constructed. This model ignored the transportation problem completely. In other words, the model assumed that the whole country was a single consumption center, that all of the existing plants formed a single large plant, that all of the ore mines formed a single large ore mine etc. The
157
solution of this model showed that no major inv e s t m e n t - - t o build a new p l a n t - - w a s necessary before the year 1993. The required capacity increase up to the year 1993 could be met by small modernization investments and by eliminating bottlenecks in the existing plants arising from the incompatibilities in the capacities of different units. This is a very important result for two reasons: First, it eliminates the need for 0-1 variables up to the year 1993 for the mixed integer programming model to be constructed at the next stage, and secondly it allows one to decouple the plan periods up to the year 1993. The next stage involved solving mixed integer programming models. The first version of the mixed integer programming model ignored the transportation model completely as the linear programming model. The purpose of this model was to study the types of plants to be built and the times to build them. The model was solved for three levels of demand. It was shown that a new integrated plant was necessary to build in case the demand follows a trajectory at a higher-than-expected level. For the case the demand follows its expected trajectory, on the other hand, a nonintegrated plant could well do. The second version of the mixed integer model included the transportation problem. The purpose of this model was to see whether the location of the new plant was significant. The solution of the model showed that the model was much more sensitive to the time of building the new plant than the location where to build it. Although the second version of the mixed integer programming model represented the final form of the model, it was decided to convert it into a set of linear programming problems for the following reason. The experts from the State Planning Organization did not feel comfortable enough to accept the solution of a mixed integer programming model, but instead, they wanted to see and study in detail several of the linear programming solutions that are considered by the mixed integer programming algorithm (the branch and bound algorithm) while trying to reach the optimum. The linear programming problems actually contained 0-1 variables to represent investment decisions, but their values were fixed externally using special constraints. A given set of values for the integer variables represented an alternative plan for the sector. These alternative plans, which are called
158
S. Kuru / Long term planning of a major national sector using LP
scenarios, are discussed in detail in Section 5. The next section discusses the linear programming formulation of the model.
4. The model
The linear programming model of the Turkish iron and steel industry uses a cost function representing the net present value of all costs associated with the operation of the sector. This includes operating costs, investment cost, transportation cost, imported materials costs and income of exported materials. The investment cost is expressed in terms of the capacity of individual plant units. Although the cost of a plant unit is not a linear function of its capacity, it is approximated by a linear function of its capacity observing that its slope does not vary considerably. The relations governing the behaviour of the sector are basically material balance relations. Speaking in terms of the network representation of the sector in Figure 1, all nodes but those representing plant units are either source or sink nodes. One relation states that the flow from a source node or to a sink node cannot exceed its capacity. A plant unit, which is an intermediate node, actually converts one or more types of material into another type of material. Thus we need two material balance relations for each plant unit, one stating that the flow of material out from a unit cannot exceed its production level, and the other stating that the required amount of each type of input material for a unit is related to the production level of the unit by a conversion factor. Aside from material balance relations there are some technological and policy related constraints imposed on the behaviour of the sector. The fact that the scrap ratio in the feed to a steel converter cannot exceed 20% is such a technological constraint. An example of a policy related constraint is the requirement that there is an upper limit on the amount of final products imported. One point that needs elaboration is the representation of investment alternatives for capacity increase of plant units for new or existing plants. For some units such as rolling mills, the capacity increase takes discrete values while for others such as blast furnaces it is a continuous quantity. By making the following observation discrete values
are approximated as continuous values, thus making it possible to avoid the use of integer variables. Rolling mills of many different capacities are available through vendors. Thus, given the capacity of a rolling mill section, it is always possible to find a combination of mill capacities, which are themselves discrete, whose sum is very close to the continuous value determined by the model. Furthermore, the number of mills required is usually few. We should also note that the error introduced by this approximation is well within the precision limits of such a model. Another point that should be noted is that the model represents only one new plant alternative. This may seem a rather serious restriction but actually it is not. This is because the mixed integer programming model discussed in the previous section showed that at most one new plant was necessary to meet the increasing demand up to the year 2000. The model is represented in terms of several index sets that are used to represent coefficients, parameters, and variables. The following notation is used for indices: period t, T; plant i, u; plant unit type j, e; consumption center n; mine m; port l; raw material type h; final product type k; intermediate product type r. Note that a plant unit is specified by giving the index values of the plant and the unit type. This corresponds to introducing a new set which is the cartesian product of the sets representing plants and unit types. The objective function, which is formulated as a cost function for the sector, is given below. Note that this formulation is used for better readability rather than computational efficiency. Coefficients are factored out of summation signs as much as possible in the formulation used in actual computations.
MinU=~td,(~ij(aijtAijt) +EEE(bmht-q-Cmijht)Fmijht h m ij
(1) (2)
S. Kuru / Long term planning of a major national sector using LP + EEE(bl,~t+Clijht)Flijht h
l
+ EE~_.(btrt+c,ijrt)Ftijr, r
1
(4)
/j
+E E Z k
(3)
ij
1
+
(5)
n
-E~_.E(b,n-c,j,rt)F~jm
(6t
r ij I
--Y'E~_.(b,k,--Ci,,k,)F~,tkt
(7)
k ij l Jr- E E ECijertFijer, r
ij
"1- E E ECijnktFijnkt k
/j
(9)
n
+ ~_, E W,Ttf~TZ,T i
(8)
ue
(10)
T=I t
+ E
E WiTtgijTEiT
(11)
/j T = I
Term (1) of the objective function represents the operating cost of all plant units in terms of their production levels Aij,. The unit production cost a includes all costs (i.e. labor, energy, etc.) associated with producing a unit amount of output material but the cost of input materials. It is assumed that a,j, is independent of A~j,, thus eliminating nonlinearities. This assumption is justified by observing a negligible linearization error. Term (2) represents the cost of raw materials of national origin. The cost is expressed as a linear combination of flow of raw materials Fm,jm from national mines to plants. The coefficients b,~m and c,~om denote the unit price of a particular type of raw material at the mine site and the cost of transporting one unit of raw material from the mine site to the plant, respectively, bmm reflects all the costs associated with producing one unit of raw material at the mine site. The next three terms, terms (3), (4) and (5), represent the cost of imported materials, which include the cost of raw materials and intermediate products imported to plants and final products imported to consumption centers. Each of these terms is expressed in terms of the flow of the raw material under consideration, i.e. Fl~jm, Ftijr , or Ft, k,. Again the coefficients have a unit price
159
Clijht, Clijrt o r Clnkt. The unit prices, which are fixed by the world market, are C I F prices at the nearest national port. The unit transportation cost is the cost of transporting one unit of material from the port to the destination. Terms (6) and (7) represent the negative cost of exported materials, which include intermediate and final products (export of raw materials is not allowed as required by the State Planning Organization). Again, each term is expressed in terms of the flow of the material exported, i.e. Ftjlr t OF F, jlk ,. Coefficients b and c are the same as those for imported materials. Note that the cost of transportation from a plant to a port is positive. Terms (8) and (9) represent the cost of transporting nationally produced intermediate products and final products within the country, expressed in terms of the flow of the respective material Fijuert a n d F,j,~. The last two terms represent the investment costs. Investment costs have a fixed cost component and a variable cost component, which are represented by terms (10) and (11), respectively. The fixed cost component is expressed in terms of the integer variable Ziv which represents the investment decision for plant i at period T and £ r which is the cost of the investment. Integer variables Z,r are treated as continuous variables by the model but they are each forced to take the value 0 or 1 using a constraint. This point will be discussed further later. The cost is distributed over the life cycle of the plant and the portion of f,r charged to period t is expressed in terms of WiT,. w,r t is computed for a plant as the ratio of its nominal production capacity at period t and the sum of its nominal production capacities over its whole life. The alternative to this strategy is to charge the investment cost to a particular period such as the period when production starts or to several periods depending on the cash flow of the investment and to subtract the present worth of the remaining value at the final period. The variable cost component of the investment cost is expressed in terms of the production capacities introduced by the investment. Here gilT is the incremental cost of investment for one unit of increase in the production capacity of a particular plant unit and Eo r is the increase in production capacity of that plant unit. Eij r and Z~T are related to each other through a constraint which will be discussed later. component
S. Kuru / Long term planning of a major national sector using LP
160
Cost terms are reduced to net present cost values through coefficients d. d t is computed using a discount factor, which is taken as the difference between the interest rate and the inflation rate. As stated earlier, material balance relations constitute the major constraints of the linear programming problem. The constraints are given below.
EFiijnkt+EFlnkt~Dnkt ij
forallk, n , t ,
(1)
for all m, h, t,
(2)
l
~Fmijh, <~Gmh, /j
E E Umijhtgmijht "~- E E UlijhtFlijht m h
1 h
-'}- EUieijrtFieijrt "1- EVueijrtFueijrt ie ue
"~- E OlijrtFlijrt -- Aijt >/0
for all 0", t,
(3)
1
Aijt - ( ~n Fijnkt "~- ~l Fijlkt q- ~1Fijlrt
+ ~ F,jiert + ~ F~j~t) <~0 for all /j', t, ie ue (4)
E~j, -yjZ~t >~0
for all/j, t,
(5)
Eij t - YjZit <~0
for all 0, t,
(6)
Z a =
for all i, t,
(7)
for all
(8)
t
Aijt -- E XiTtEijT < Bijt
0", t.
T=I
Constraint (1) is the material balance relation for consumption centers. It states that the demand D, kt of a consumption center for a particular product type should be met in the form of shipments F,j~kt from all plants and imports Fl~kt. Constraint (2) is the material balance relation for mines. It states that the amount of a particular raw material type Fm~/ht shipped from a mine site to all plants cannot exceed its production capacity Gmm for that raw material type. Constraints (3) and (4) are the material balance relations for plant units. Constraint (3) states that a sufficient amount of input material should be supplied to a particular plant unit for it to produce output material of amount equal to its production level Aij,. Depending on the plant unit type input materials may be of more than one
type, and each type may be supplied from more than one source (i.e. it could be ore supplied from mines Fmijht or by import Ftijh t, or different intermediate products supplied from another unit of the same plant F~eijrt, from another plant Fueijn or by import Fti/~t). For each of the input material source there is a conversion factor v which determines the amount of output material to be obtained from one unit amount of the particular input material. Although the constraint is written for a plant unit type producing intermediate product r it is also valid for plant unit types producing final products. Note that this constraint is repeated for every different type of input material for a plant unit type. The second material balance relation for plant units, constraint (4), states that the output of a plant unit, whether it is a final product or an intermediate product, cannot exceed the production level Aiyt of the plant unit. The final products are shipped to consumption centers (F~ynkt) or exported (F, jlkt), while the intermediate products are either used by other units of the same plant (F//ien), or shipped to other plants (Fiy,,ert) or exported (F/fin) . Constraints (5), (6) and (7) are capacity increase constraints and represent investment decisions. Constraint (7) sets the investment decision variable Z~t to 0 or 1. If it is set to 1, then constraints (5) and (6) force E~jt, the capacity introduced by the investment, to take a value between yj and Yj, the minimum and the maximum feasible capacity increases for plant unit type j, respectively. An alternative formulation would be not to introduce constraint (7) and declare Z~t as 0-1 variables. This would convert the linear programming model into a mixed integer programming model. As explained in section 3, the linear programming formulation is prefered for the following reasons: First, it allows the decision maker to see some of the intermediate solutions of the mixed integer programming problem. The decision maker wants to see these solutions to increase his confidence on the choice made by the model. He may even choose to adopt a near optimal solution judging that the near optimal solution is better than the optimal solution because of the things that are not reflected in the model. Secondly, the decision maker has a good understanding of the problem so that he usually performs better to
S. Kuru / Long term planning of a major national sector using LP
reduce the number of linear programming problems to be solved than a typical mixed integer programming strategy such as the branch and bound method. The last constraint, constraint (8), states that the production level A~jt of a plant unit cannot exceed its production capacity. The production capacity of a plant unit at period t is expressed as the sum of two capacity values: Its existing capacity Bij , at this period and the sum of the capacity increases E0~- up to period t. The coefficient x i r t is introduced to reflect the fact that only a portion of the capacity increase E~jT introduced by an investment at period T can be utilized at period t. Although a function of t, the existing capacity B i j t of a plant unit at period t does not include capacity increases introduced by investments. B i j t is usually equal to the capacity available at the first period for all periods, but rarely it changes with t for reasons not reflected in the model. Note that the only constraint causing coupling between constraints is constraint (8). If this constraint did not exist it would be possible to solve the problem separately for each period. This property of the model suggests a natural decomposition scheme for the problem. The cardinalities of the index sets are typically of O(10) and for some sets the cardinality is of O(100). This causes the model to become very large in terms of the number of constraints and the number of variables. The variables F alone quickly exceed thousands in number. It should be noted that the model is for the most general case--i.e, every plant has all types of units and produces all types of final products using all types of raw materials, final products may be shipped to any consumption center or exported via any port, raw materials may be supplied from any mine or imported via any port, etc. Actually many of the variables do vanish when the model is formulated for a specific country, thus reducing the size of the problem considerably. But the model is still too large to debug and to afford the excessive amount of computation time. As stated earlier, a much smaller version of the model was solved at the first stage and the model evolved to its full size gradually. Grouping of ore mines, consumption centers and non-integrated plants are based on the results obtained from the solutions of the models that are used at the early stages. The final form of the model is based on the
161
following index sets: period (base years) intermediate products
final products
plants
plant units consumption centers
raw materials mines
ports
: {1984,1987,1990,1993, 1996 1999}; : {coke, sinter, liquid metal, molten steel, recycle scrap, ingot, hot rolled sheet, sponge iron} ; : {pig iron, plates, hot rolled sheets, cold rolled sheets, sections, rods and bars}; : {Eregli, Karabuk, Iskenderun, Istanbul-1, Izmir-1, a new integrated plant, Istanbul-2, Izmir-2} ; : as in Figures 2 and 3; : (Istanbul, Bursa, Izmir, Ankara, Adana, Kayseri, Samsun, Erzurum, Diyarbakir } ; : {coal, lump ore, dust ore, pellet, scrap}; : {Zonguldak, Divrigi, Hasancelebi, Kangal, Avnik, Yesilhisar, Yahsihan, Akcay } ; : {Eregli, Iskenderun, Istanbul, B a n d i r m a , Izmir, Samsun, Karasu}.
As it is seen each period covers three consecutive years and is represented by a base year, which is the middle year of the period. The first period 1984 is included in the model in order to satisfy the continuity of the model with the near past. Intermediate products, final products and plant units correspond to the flow diagrams given in Figures 2 and 3. Of the three existing integrated plants, Karabuk and Iskenderun produce long products whereas Eregli produces flat products. The existing electric arc furnaces, all of which produce long products in accordance with the flow diagram given in Figure 3a, are groupped as two plants, one in Istanbul and one in Izmir. Investment alternatives considered are as follows. All of the existing plants may undergo capacity expansion at any period. As new plants, one integrated plant and two non-integrated plants
S. Kuru / Long term planning of a major national sector using LP
162
Table 1 Distribution of demand among consumption centers (%)
Table 2 Final product demand (million tons/year)
Consumption center
Pig iron
Plates
Hot rolled sheets
Cold rolled sheets
Sections
Rods & bars
Base year
Pig iron
Plates
Hot rolled sheets
Cold rolled sheets
Sections
Rods & bars
Istanbul Bursa hmir Ankara Adana Kayseri Samsun Erzurum Diyarbakir
40 8 10 30 2 10 . . .
29 2 9 50 4 6 . . .
63 2 8 20 1 6 . . .
54 5 17 15 5 4
25 5 18 20 12 5 4 5 6
23 6 19 21 10 5 7 4 5
1984 1987 1990 1993 1996 1999
0.466 0.542 0.654 0.820 1.036 1.328
0.325 0.418 0.536 0.694 0.904 1.181
0.648 0.834 1.071 1.387 1.812 2.372
0.699 0.886 1.141 1.453 1.893 2.439
0.290 0.423 0.587 0.739 0.943 1.210
1.506 2.084 2.801 3.538 4.506 5.754
. . .
period when it starts production are to be determined by the model. The locations of new non-integrated plants are fixed as Izmir and Istanbul. One of them is to produce long products from scrap as given in Figure 3b and the other is a sponge iron plant as given in Figure 3c~ None of these two plants may start production before the year 1993. The demand for iron and steel products is
are considered. The new integrated plant may produce both flat and long products and may start production at any period not before the year 1996. Five alternative locations are determined for this plant: Samsun, Sivas, Bandirma, Izmir and Karasu. Its location, the capacities of its units, and the
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existing integrated plant a l t e r n a t i v e l o c a t i o n for i n t e g r a t e d
•
existing
non-integrated
o
consumption
• •
ore m i n e coal m i n e port railway
plant
center
Figure 4. Geographical distribution of plants, mines, consumption centers and ports
plant
s. Kuru / Long term planning of a major national sector using LP
assumed to be distributed between consumption centers as in Table 1. As seen from the table, long products are consumed in all 9 consumption centers whereas flat products are consumed in only 6 of them. The demand levels for each final product type are determined by projecting the consumption of the previous years. It is decided to run the model for three different demand levels, normal, maximum and minimum, in order to safeguard the investment decisions for unforeseen changes in the world iron and steel trade. Normal level of demand is given in Table 2. The maximum and the minimum levels are taken as 10% above and below the normal level. The mapping between mine regions and raw material types is as follows: coal : Zonguldak; lump ore: Divrigi, Hasancelebi, Kangal, Yesilhisar, Yahsihan, Akcay; dust ore : Divrigi, Hasancelebi, Avnik, Yesilhisar, Akcay; pellet : Divrigi, Hasancelebi, Avnik; scrap : available at plant site. As it is seen, the number of coking coal mines is reduced to one, lump ore mines to 6, dust ore mines to 5 and pellet plants to 3. This reduces the size of transportation matrix considerably. The transportation matrix is further reduced by mapping plants to ports and consumption centers to ports. A plant or a consumption center at a coastal city uses its own port. This is the case with Eregli, Iskenderun, Istanbul, Izmir, Samsun, Karasu and Bandirma (note that the port in Karasu is to be newly built upon the decision of building the new plant there). The remaining plants and consumption centers are mapped to the nearest port. Hence, Karabuk is mapped to Eregli, Bursa to Bandirma, and Diyarbakir to Iskenderun. The geographical locations of the plants, mines, ports and consumption centers are shown on the map in Figure 4. The resulting model consists of 2280 variables and 1494 constraints.
163
Table 3 Capacities a of existing plants for scenario 1 (million tons/year) Plant
Base year
Liquid metal
Molten steel
Final product
Eregli
1984 1987 1990 1984 1987 1990 1984 1987 1990 1984 1987 1990
1.750 1.820 2.230 0.560 0.750 0.900 1.300 2.150 2.450 -
1.700 2.000 2.400 0.510 0.665 0.700 0.900 2.065 2.300 1.400 2.000 2.000
1.500 1.900 2.860 0.440 0.555 0.595 0.750 1.800 2.020 1.750 1.750 1.750
Karabuk Iskenderun Nonintegrated a
Capacities remain the same after the year 1990.
demand level and the last two are for the maximum demand level. The scenarios are the following: Scenario 1. No new plants are to be built. The capacities of the existing plants are to be expanded to the values given in Table 3. Scenario 2. While the capacities of the existing plants are to be kept as in Scenario 1, two new non-integrated plants are to be built: A sponge iron plant and an electric arc furnace, of 0.325 million t o n s / y e a r and 0.350 million t o n s / y e a r final products capacity, respectively.
5. The solution
Scenario 3. N o new plants are to be built. The capacities of the existing plants are to be expanded as follows: Iskenderun plant is to be expanded in 1993 to produce 3.2 million t o n s / y e a r liquid metal, 4.25 million t o n s / y e a r molten steel and 3.6 million t o n s / y e a r final products. The added capacities will be utilized 50% in the year 1993, 90% in the year 1996 and at full capacity in the year 1999. The existing non-integrated plants are to be expanded to 2.4 million t o n s / y e a r molten steel and 2.15 million t o n s / y e a r final products capacity in the year 1987 and 2.7 million t o n s / y e a r molten steel and 2.4 million t o n s / y e a r final products capacity in the year 1990.
The model is solved for seven scenarios that are determined by the decision maker. Each scenario is a plan that is possible to adopt. The first five scenarios are designed to study the case of normal
Scenario 4. A new integrated plant is to be built to start production in the year 1996 or 1999. Capacity utifization ratio of the new plant will be 50% in its start of production period and 100% in the
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S. Kuru / Long term planning of a major national sector using LP
following periods. As for the existing plants, the capacity of the Iskenderun plant is to be expanded as in Scenario 3 while the capacities of the other plants are to be as in Scenario 1. Scenario 5. A new integrated plant is to be built as in Scenario 4 while the capacities of the Iskenderun plant, Eregli plant and non-integrated plants are to be as in Scenario 3. Scenario 6. The best of the first five scenarios with maximum demand level. Scenario 7. Scenario 6 plus building a new integrated plant at the most appropriate time and location as determined from the solutions of the first five scenarios. Scenario 1 is designed to study the case of no major investment. Capacity increase through the years is due to minor investment decisions such as modernization, bottleneck elimination and small expansions that have been decided on in previous plans. Scenarios 2 and 3 are designed to study the case of meeting the demand by investing for nonintegrated plants and by expanding the capacities of existing plants, respectively. Scenarios 4 and 5 are designed to study the case of investing for a new integrated plant. Capacity expansions of existing plants are also considered both in Scenario 4 and Scenario 5 (the Eregli plant is not expanded in Scenario 4). Note that the location, the capacity and the start of production periods of new non-integrated plants are determined a priori. For the new integrated plant, while the capacities of the plant units are determined by the model, the start of production year is fixed by the scenario. The location, on the other hand, is determined by solving the scenario for the five alternative locations. Table 4 gives the objective function values for the best solution of each of the scenarios. The values represent the sum of the cost terms for the base year of each period, expressed in terms of the prices reduced to the year 1983. Note that the first five scenarios are solved using the normal level of demand, while the last two are solved using a demand level 10% higher than the normal level. Also note that Scenario 4 is solved for 5 times for period 5, once for building the new integrated plant at each of the five alternative locations, and
Table 4 Objective function values for the scenarios (billion TL) Scenario
N u m b e r of solutions obtained
Objective function value for the best solution
1 2 3 4
1 1 1 5 for 1 for 3 for 1 for 1 1
1399.3851 1402.3940 1321.9048 1394.9054
5 6 7
period period period period
5 6 5 6
1364.9855 1791.7844 1689.2509
that it is solved once for period 6, which is for building the new integrated plant at the location that is the best for period 5. Scenario 5, on the other hand, is solved 3 times for period 5, once for building the new integrated plant at each of the best three locations determined by the solutions of Scenario 4 at period 5. The solution of Scenario 5 for period 6 is for the location that is the best for period 5. The solutions of Scenario 4 for period 5 indicated that Sivas is the best location for building the new integrated plant, followed by Samsun and Izmir. Thus Scenario 4 is solved for Sivas for period 6 and Scenario 5 is solved for these three locations. Being the best solution of Scenario 5 at period 5, Scenario 5 is solved for Sivas for period 6. As seen from Table 4, the best of the first five scenarios is Scenario 3 followed by Scenarios 5, 4, 1 and 2 in this order. This shows that no major investment is necessary to meet the demand. The difference between investing for a new plant and
Table 5 Breakdown of major cost components for Scenario 3 (billion TL) Period Cost of Operating Imported Transport- Total national costs material ation cost raw cost cost materials 1 2 3 4 5 6
66.78 108.19 127.09 148.54 162.64 177.08
53.60 69.69 92.37 127.18 122.14 129.65
108.26 109.91 139.18 195.02 313.46 498.08
14.44 21.13 27.64 29.86 33.96 37.71
253.08 308.83 386.28 500.60 632.20 842.52
S. Kuru / Long term planning of a major national sector using LP
increasing the capacities of the existing plants is not of primary importance. While the location does not make that much difference, the period of start of production is important for the case of investing for a new plant. If a new integrated plant is to be built it should start production at period 5 and it should produce both flat and long products. The solutions of the last two scenarios showed that, in case of a high level demand, building a new integrated plant is necessary. As for the numerical results of the scenario solutions we give only the breakdown of major cost components for Scenario 3, which is the best scenario. The values are given in 1983 prices without a net present value reduction. As seen, the imported material cost is the major cost of the sector, which is followed by the operating costs, the cost of raw materials of national origin and the transportation cost. Within the imported material cost, the cost of raw materials is the major constituent at the early periods but the cost of final products becomes dominant towards the final periods. The cost of coal dominates the others in the cost of raw materials of national origin. Similarly, the cost of raw material transportation dominates the other transportation costs. Some of the other results obtained from the solutions of the model may be summarized as follows: National resources are preferred as the supply of raw materials. As for importing intermediate products, the amount of ingot imported is negligible but the Eregli plant imports an imported amount of slab. Final products are both imported and exported. The demands for hot rolled sheets are totally met by national production. These products are also exported. Plates, cold rolled sheets, and rods and bars, on the other hand, are imported at some periods. For the details of the scenario solutions see [11].
6. Conclusion The model developed for the Turkish iron and steel sector has proved to be very useful for studying the long term investment planning of the sector and gave valuable information for an analysis of it. The model, which is a linear programming model of considerable size (1494 constraints and 2280 variables), is solved using the FMPS mathematical programming package available on
165
the U N I V A C 1100 system at Bogazici University Computer Center. The model showed that investing for a new integrated plant is not necessary if the demand follows the projected trajectory upto the year 2000. Increasing the production capacities of the existing plants is sufficient for this case. In case of an unexpected increase in demand, on the other hand, a new integrated plant that would produce both flat and long products should be built to start production by the year 1996. The location of the new integrated plant is not of prime importance and can be decided on by considering other factors that are not represented in the model. The transportation cost is the determinant of the location of the new integrated plant. As such, Sivas and Samsun have advantages over the others because both are close to the consumption centers at the Central and Eastern Anatolia regions, and to the major ore mines. Sivas has the advantage of being right on the largest ore mines region, whereas Samsun has the advantage of being a port city.
Acknowledgements This work was supported by The State Planning Organization (DPT) of Turkey. The views and conclusions in this article are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of DPT.
References [1] Organization for Economic Cooperation and Development, The Iron and Steel lndustry in 1983, Paris, 1985. [2] Manners, G., The Changing World Market for Iron Ore 1950-1980, Johns Hopkins Press, Baltimore, MD, 1971. [3] Kavrakoglu, I., "A dynamic optimization model for energy policy analysis", in: I. Kavrakoglu (ed.), Mathematical Modelling of Energy Systems, Sijthoff and Noordhoff, Alphen aan de Rijn, 1981, 131-135. [4] Russel, C.S., and Vaughan, W.J., "An analysis of the historical choice among technologies in the U.S. steel industry: Contributions from a linear programming model", The Engineering Economist 22/1 (1976) 1-26. [5] Lawrence, K.D., Lawrence, S.M., and Reeves, G.R., "Aggregate industrial expansion: A multiple-objective linear programming approach", The Engineering Economist 25/3 (1979) 197-207.
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[6] Abe, M.A., "Dynamic micro-economic models of production, investment and technological change of the U.S. and Japanese iron and steel industries", in: G.C. Judge and T. Takayama (eds.), Studies in Economic Planning Over Space and Time, North-Holland, Amsterdam, 1973, 344-367. [7] Nelson, J.P., "An interregional recursive linear programming model of the U.S. iron and steel industry", in: G.G. Judge and T. Takayama (eds.), Studies in Economic Planning Over Space and Time, North-Holland, Amsterdam, 1973, 368-393. [8] Sawey, R.M., and Zinn, C.D., "Mathematical model for long range expansion planning of generation and
transmission in electric utility systems", IEEE TransaCtions on Power Apparatus and Systems 96 (1977) 657-666. [9] Peterson, E.R., "A dynamic programming model for the expansion of electric power systems", Management Science 20 (1973) 656-664. [10] Schmitz, K., and Schwefel, H.P., "Finding reasonable energy policies by means of a dynamic simulation model", in: Proceedings of the International Symposium Simulation '77, ACTA Press, 1977. [11] Kuru, S., "Expanded master plan model of the Turkish iron and steel sector" (research report in Turkish), Bogazici University, 1985.
153
Case Study
Long term planning of a major national sector using a linear programming model Selahattin KURU
Department of Computer Engineering~ Bogazici University, Bebek, Istanbul, Turkey Abstract: This article discusses a linear programming model for the long term planning of the Turkish iron and steel industry. Iron and steel industry is a complicated industry. It is rich in terms of products and production technologies, and has a large transportation problem associated with it. The model is formulated as a cost minimization problem and is solved in terms of several scenarios, each representing a different strategy for capacity expansion, using a mathematical programming package. Keywords: Sectoral planning, investment, linear programming, iron and steel
1. Introduction
The volume of iron and steel consumption is considered as an indicator of the level of national development. The present level of iron and steel consumption in Turkey is about 100 kg per capita, which is a typical value for a developing country [1]. This value is about 500 for developed countries, and Turkey plans to increase it to 300 in the year 2000. This corresponds to the equivalent of 17 million tons of molten steel consumption a year. The demand for iron and steel products may be met by imports or by increasing the production level, which requires investing for capacity increase of existing plants or building new plants. The subject of this case study is to evaluate different investment alternatives in detail for the planning of the Turkish iron and steel sector up to the year 2000. The work is done for The State Planning Organization of Turkey, which is responsible for developing sectoral and national investment plans.
Received December 1986; revised October 1987
The study of investment involves the determination of the location, the capacity, and the start of the production year for plants to be newly installed, and the capacity increase, and the year of capacity increase for existing plants. Also required is the determination of the production levels of plant units, the flow of raw materials and intermediate and final products within the sector, and the export and import levels.
2. The structure of the sector
The iron and steel sector is a complicated sector. It produces many different products using many different production technologies. Ore mines and consumption centers are distributed over the whole country. Furthermore, the private sector in Turkey has a large number of small plants whose operations are very hard to follow. The sector can be represented as a network as shown in Figure 1, where nodes denote plants, mines, e x p o r t / i m p o r t ports and consumption centers, and arcs denote the flow of material between nodes. As can be seen in the figure, there is
0377-2217/88/$3.50 © 1988, ElsevierSciencePubfishers B.V. (North-Holland)
154
S. Kuru / Long term planning of a major national sector using LP
PORTS K rfw materials
MINES
"I" "I"- \ | |fi I
finol
interm i product s
rawmateriats finalproducts I PLANTS I
ICONSUMPTION I CENTERS
I intermadlate products Figure 1. Network representation of the sector
the flow of raw materials from mines and ports to plants, the flow of final products from plants and ports to consumption centers and from plants to ports, and the flow of intermediate products between plants and ports (both directions) and between plants. The arc denoting the flow of intermediate products between plants also represent the flow of intermadiate products within plants. Note that mines and consumption centers are source and sink nodes, respectively, and plant units are intermediate nodes. A plant unit is actually a conversion point where incoming materials change form. One characteristic of a plant unit is that it may involve incoming flows of more than one type of material, but outgoing flows are always of one single type of material. Plants can be divided into two categories according to the types of final products they produce: long products plants and flat products plants. Long products include sections, rods and bars, and flat products include sheets and plates. For both product groups plants are either integrated plants, consisting of all the major production units, or small plants, consisting of some of the production units. Integrated plants produce essentially all final products of a group starting from iron ore whereas small plants either use an intermadiate product as the input material or produce only a few product types of the product group. There are many variations for small plants such as electric arc furnaces, sponge iron plants, and rolling mills. We will call all these variations non-integrated plants.
The flow diagram of a typical integrated plant producing flat products is given in a simplified form in Figure 2a. The plant may be divided into three main sections: Blast furnaces, steel converters and rolling mills. Blast furnaces convert iron ore into liquid metal. Iron ore may be in the form of lumps, dust, pellet or sinter. Pellet and sinter are pretreated ore. Pellet is produced at an ore mine site. Sinterizing, on the other hand, is performed at the sinter unit of a plant. The other major input to blast furnaces is coke, which is used to reduce iron ore (iron oxide) into metalic iron by a chemical reaction called reduction. Coke is produced from coking coal in the coke works unit of the plant. Most of the liquid metal produced in a plant goes directly into steel converters while the rest is converted into pig iron in the pig iron unit. Steel converters convert liquid metal into molten steel by reducing its carbon content. Lime, which is produced in the lime unit from limestone is used in this process. Another input to steel converters is scrap. Metallurgically, scrap is steel and needs only reshaping for converting it into final products. The price of scrap is low compared to the cost of producing liquid metal from ore, hence making it a desirable raw material. Scrap comes from two sources: Recycling of rolling mills, where it is obtained as a by-product of the shaping process, and from outside. Blast furnaces and steel converters are basically the same for both types of integrated plants. The rolling mill section, which is the section equipped
S. Kuru / Long term planning of a major national sector using LP
c°al~
scrap IliquidI "4 STEEL molten )J BLAST steel FURNACESI I "I CONVERTERS
c°k~ e
jumpore polle
15 5
~
PIG IRON I > "1 UN,T I 'g
dustI ore .... Islnter
Iron
~
plates
_ ~ CASTING SLAB~ (a)
)
L hot slab
rolled
sheets
sheets
c°al~coke lliquid~rap I PLANTI--~I STEEL lumpore )I BLAST metal steel ] ]CONVERTERS ~ FURNACES
Imo,ten
dus~l . . . . ore
Is~nter
--I
PIG UNIT IRON I > mg iroiq ingot
I INGOT CASTING
sectlons
>
rodsandbars
>
(b) Figure 2. Simplified flow diagrams of an integrated plant. (a) Flat products plant; (b) Long products plant
with units for producing specific products, is necessarily different for the two product groups. This section varies considerably even for plants producing products of the same product group. The rolling mills section of a flat products plant typically consists of slab casters, plate mills, and hot and cold rolls. For a long products plant the rolling mills section consists of ingot casting units, section mills, and rods and bars mills. Figure 2b shows the flow diagram of a long products plant in a simplified form. As can be seen in Figure 2 an intermediate product is exported from or im-
ported to the plant, i.e. slab in the case of flat products plants and steel ingot in the case of long products plants. The major unit in a non-integrated plant is the electric arc furnace, which converts scrap into steel. Steel may be in the form of ingot or it may undergo continuous casting for slab production. While ingot is reshaped into sections and bars in rolling mills, slab is converted into plates in hot rolls. The sponge iron technology, which is a relatively new technology, is based on converting pellet into sponge iron, which corresponds to the
156
S. Kuru / Long term planning of a major national sector using LP
mgot I scrap 1
SECTIONS MILLS
] sections )
ELECTRIC FURNACE ARC RODS AND BARS MILLS
(a)
I rods& bars).
ot
scrap
ELECTRIC ARC FURNACE
Imolten | I steel ICASTING ~ UNIT
HOT ROLL
hotrolled ) sheets
(b)
ingot scrap
t
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sp°nge ~ODS& iron \II ELECTRICIinn°t/I ARC ~ e~b ! r°dsand bars
UNIT
(c)
IFURNACE ii IMILLS ,,~ ingot
Figure 3. Simplified flow diagrams of a non-integrated plant. (a) Long products plant; (b) Flat products plant; (c) Sponge iron plant
process performed in blast furnaces in integrated plants. Sponge iron then undergoes steel conversion process in an electric furnace. These variations are shown in Figure 3. Iron and steel products are consumed over the whole country. This is especially true for long products since these are mainly consumed by building and construction industries. Flat products, on the other hand, are the main input of the manufacturing industry, thus being consumed mostly at industrial regions. As stated earlier, iron ore and coal are the principal raw materials of the sector. Coking coal is available in one region in Turkey, namely the Zonguldak region. Iron ore, on the other hand, is available in many regions. Ore mines differ greately in terms of production capacity and ore quality. Some ore mines have facilities to convert dust ore into pellet. Ore quality is mainly measured in terms of its tenor, which is the iron content. Tenor is typically around 50%. Size is of secondary importance. Ore lumps are prefered because of ease of transporta-
tion and handling. Dust ore is utilized by converting it into pellet or sinter. The sinterizing process also reduces the sulphur content of the ore. The iron and steel sector introduces a heavy demand on the transportation sector because of the huge amount of material transported within the sector. This includes the transportation of raw materials between mines and plants and between ports and plants, the transportation of intermediate products between plants and between ports and plants, and the transportation of final products between plants and consumption centers, between plants and ports and between ports and consumption centers. The transportation of raw materials within the country is always by rail because the mines are located inland. This means that railroad construction should be completed before a new mine starts production. Railroad reaches a mine region up to a location called a loading station. Mine is transported to the loading station from individual mines of the region by trucks. The transportation of intermediate and final
S. Kuru / Long term planning of a major national sector using LP
products within the country is by rail or by sea, depending on the location of the plant and the consumption center. Export and import of raw materials and intermadiate and final products is by sea. Because of the volume of international trade in the sector, plants are usually built on the coast. Ports utilized by the sector are usually among the largest in a country. For a detailed discussion of both the technological and economical aspects of the iron and steel industry see Manners [2].
3. The approach Long term sectoral planning has been approached in numerous ways. Methodologies include linear [3,4,5], recursive linear [6,7], mixed integer [8], and dynamic programming [9] as well as simulation models [10]. We have adopted linear programming and mixed integer programming methodologies at different stages of the study. We will first briefly discuss how the model evolved into its final form by noting the key issues addressed at those stages and then give the final form of the model in the next section. The study was carried out in cooperation with the experts from The State Planning Organization, from a private engineering and consultancy firm and from the university. The first version of the model was developed in about four months. Many meetings were held during this period trying to determine the details of the model. The experts from the State Planning Organization and the consultancy firm were eager to represent all the fine details of the sector in the model while those from the university tried to keep it small and manageable. Finally the following strategy was adopted: Start with a small linear programming model and expand it into a full size mixed integer programming model gradually. This strategy has two benefits: ease of debugging the model at all stages, and better understanding of the significance of including different phenomena in the model. In the first stage, a linear programming model was constructed. This model ignored the transportation problem completely. In other words, the model assumed that the whole country was a single consumption center, that all of the existing plants formed a single large plant, that all of the ore mines formed a single large ore mine etc. The
157
solution of this model showed that no major inv e s t m e n t - - t o build a new p l a n t - - w a s necessary before the year 1993. The required capacity increase up to the year 1993 could be met by small modernization investments and by eliminating bottlenecks in the existing plants arising from the incompatibilities in the capacities of different units. This is a very important result for two reasons: First, it eliminates the need for 0-1 variables up to the year 1993 for the mixed integer programming model to be constructed at the next stage, and secondly it allows one to decouple the plan periods up to the year 1993. The next stage involved solving mixed integer programming models. The first version of the mixed integer programming model ignored the transportation model completely as the linear programming model. The purpose of this model was to study the types of plants to be built and the times to build them. The model was solved for three levels of demand. It was shown that a new integrated plant was necessary to build in case the demand follows a trajectory at a higher-than-expected level. For the case the demand follows its expected trajectory, on the other hand, a nonintegrated plant could well do. The second version of the mixed integer model included the transportation problem. The purpose of this model was to see whether the location of the new plant was significant. The solution of the model showed that the model was much more sensitive to the time of building the new plant than the location where to build it. Although the second version of the mixed integer programming model represented the final form of the model, it was decided to convert it into a set of linear programming problems for the following reason. The experts from the State Planning Organization did not feel comfortable enough to accept the solution of a mixed integer programming model, but instead, they wanted to see and study in detail several of the linear programming solutions that are considered by the mixed integer programming algorithm (the branch and bound algorithm) while trying to reach the optimum. The linear programming problems actually contained 0-1 variables to represent investment decisions, but their values were fixed externally using special constraints. A given set of values for the integer variables represented an alternative plan for the sector. These alternative plans, which are called
158
S. Kuru / Long term planning of a major national sector using LP
scenarios, are discussed in detail in Section 5. The next section discusses the linear programming formulation of the model.
4. The model
The linear programming model of the Turkish iron and steel industry uses a cost function representing the net present value of all costs associated with the operation of the sector. This includes operating costs, investment cost, transportation cost, imported materials costs and income of exported materials. The investment cost is expressed in terms of the capacity of individual plant units. Although the cost of a plant unit is not a linear function of its capacity, it is approximated by a linear function of its capacity observing that its slope does not vary considerably. The relations governing the behaviour of the sector are basically material balance relations. Speaking in terms of the network representation of the sector in Figure 1, all nodes but those representing plant units are either source or sink nodes. One relation states that the flow from a source node or to a sink node cannot exceed its capacity. A plant unit, which is an intermediate node, actually converts one or more types of material into another type of material. Thus we need two material balance relations for each plant unit, one stating that the flow of material out from a unit cannot exceed its production level, and the other stating that the required amount of each type of input material for a unit is related to the production level of the unit by a conversion factor. Aside from material balance relations there are some technological and policy related constraints imposed on the behaviour of the sector. The fact that the scrap ratio in the feed to a steel converter cannot exceed 20% is such a technological constraint. An example of a policy related constraint is the requirement that there is an upper limit on the amount of final products imported. One point that needs elaboration is the representation of investment alternatives for capacity increase of plant units for new or existing plants. For some units such as rolling mills, the capacity increase takes discrete values while for others such as blast furnaces it is a continuous quantity. By making the following observation discrete values
are approximated as continuous values, thus making it possible to avoid the use of integer variables. Rolling mills of many different capacities are available through vendors. Thus, given the capacity of a rolling mill section, it is always possible to find a combination of mill capacities, which are themselves discrete, whose sum is very close to the continuous value determined by the model. Furthermore, the number of mills required is usually few. We should also note that the error introduced by this approximation is well within the precision limits of such a model. Another point that should be noted is that the model represents only one new plant alternative. This may seem a rather serious restriction but actually it is not. This is because the mixed integer programming model discussed in the previous section showed that at most one new plant was necessary to meet the increasing demand up to the year 2000. The model is represented in terms of several index sets that are used to represent coefficients, parameters, and variables. The following notation is used for indices: period t, T; plant i, u; plant unit type j, e; consumption center n; mine m; port l; raw material type h; final product type k; intermediate product type r. Note that a plant unit is specified by giving the index values of the plant and the unit type. This corresponds to introducing a new set which is the cartesian product of the sets representing plants and unit types. The objective function, which is formulated as a cost function for the sector, is given below. Note that this formulation is used for better readability rather than computational efficiency. Coefficients are factored out of summation signs as much as possible in the formulation used in actual computations.
MinU=~td,(~ij(aijtAijt) +EEE(bmht-q-Cmijht)Fmijht h m ij
(1) (2)
S. Kuru / Long term planning of a major national sector using LP + EEE(bl,~t+Clijht)Flijht h
l
+ EE~_.(btrt+c,ijrt)Ftijr, r
1
(4)
/j
+E E Z k
(3)
ij
1
+
(5)
n
-E~_.E(b,n-c,j,rt)F~jm
(6t
r ij I
--Y'E~_.(b,k,--Ci,,k,)F~,tkt
(7)
k ij l Jr- E E ECijertFijer, r
ij
"1- E E ECijnktFijnkt k
/j
(9)
n
+ ~_, E W,Ttf~TZ,T i
(8)
ue
(10)
T=I t
+ E
E WiTtgijTEiT
(11)
/j T = I
Term (1) of the objective function represents the operating cost of all plant units in terms of their production levels Aij,. The unit production cost a includes all costs (i.e. labor, energy, etc.) associated with producing a unit amount of output material but the cost of input materials. It is assumed that a,j, is independent of A~j,, thus eliminating nonlinearities. This assumption is justified by observing a negligible linearization error. Term (2) represents the cost of raw materials of national origin. The cost is expressed as a linear combination of flow of raw materials Fm,jm from national mines to plants. The coefficients b,~m and c,~om denote the unit price of a particular type of raw material at the mine site and the cost of transporting one unit of raw material from the mine site to the plant, respectively, bmm reflects all the costs associated with producing one unit of raw material at the mine site. The next three terms, terms (3), (4) and (5), represent the cost of imported materials, which include the cost of raw materials and intermediate products imported to plants and final products imported to consumption centers. Each of these terms is expressed in terms of the flow of the raw material under consideration, i.e. Fl~jm, Ftijr , or Ft, k,. Again the coefficients have a unit price
159
Clijht, Clijrt o r Clnkt. The unit prices, which are fixed by the world market, are C I F prices at the nearest national port. The unit transportation cost is the cost of transporting one unit of material from the port to the destination. Terms (6) and (7) represent the negative cost of exported materials, which include intermediate and final products (export of raw materials is not allowed as required by the State Planning Organization). Again, each term is expressed in terms of the flow of the material exported, i.e. Ftjlr t OF F, jlk ,. Coefficients b and c are the same as those for imported materials. Note that the cost of transportation from a plant to a port is positive. Terms (8) and (9) represent the cost of transporting nationally produced intermediate products and final products within the country, expressed in terms of the flow of the respective material Fijuert a n d F,j,~. The last two terms represent the investment costs. Investment costs have a fixed cost component and a variable cost component, which are represented by terms (10) and (11), respectively. The fixed cost component is expressed in terms of the integer variable Ziv which represents the investment decision for plant i at period T and £ r which is the cost of the investment. Integer variables Z,r are treated as continuous variables by the model but they are each forced to take the value 0 or 1 using a constraint. This point will be discussed further later. The cost is distributed over the life cycle of the plant and the portion of f,r charged to period t is expressed in terms of WiT,. w,r t is computed for a plant as the ratio of its nominal production capacity at period t and the sum of its nominal production capacities over its whole life. The alternative to this strategy is to charge the investment cost to a particular period such as the period when production starts or to several periods depending on the cash flow of the investment and to subtract the present worth of the remaining value at the final period. The variable cost component of the investment cost is expressed in terms of the production capacities introduced by the investment. Here gilT is the incremental cost of investment for one unit of increase in the production capacity of a particular plant unit and Eo r is the increase in production capacity of that plant unit. Eij r and Z~T are related to each other through a constraint which will be discussed later. component
S. Kuru / Long term planning of a major national sector using LP
160
Cost terms are reduced to net present cost values through coefficients d. d t is computed using a discount factor, which is taken as the difference between the interest rate and the inflation rate. As stated earlier, material balance relations constitute the major constraints of the linear programming problem. The constraints are given below.
EFiijnkt+EFlnkt~Dnkt ij
forallk, n , t ,
(1)
for all m, h, t,
(2)
l
~Fmijh, <~Gmh, /j
E E Umijhtgmijht "~- E E UlijhtFlijht m h
1 h
-'}- EUieijrtFieijrt "1- EVueijrtFueijrt ie ue
"~- E OlijrtFlijrt -- Aijt >/0
for all 0", t,
(3)
1
Aijt - ( ~n Fijnkt "~- ~l Fijlkt q- ~1Fijlrt
+ ~ F,jiert + ~ F~j~t) <~0 for all /j', t, ie ue (4)
E~j, -yjZ~t >~0
for all/j, t,
(5)
Eij t - YjZit <~0
for all 0, t,
(6)
Z a =
for all i, t,
(7)
for all
(8)
t
Aijt -- E XiTtEijT < Bijt
0", t.
T=I
Constraint (1) is the material balance relation for consumption centers. It states that the demand D, kt of a consumption center for a particular product type should be met in the form of shipments F,j~kt from all plants and imports Fl~kt. Constraint (2) is the material balance relation for mines. It states that the amount of a particular raw material type Fm~/ht shipped from a mine site to all plants cannot exceed its production capacity Gmm for that raw material type. Constraints (3) and (4) are the material balance relations for plant units. Constraint (3) states that a sufficient amount of input material should be supplied to a particular plant unit for it to produce output material of amount equal to its production level Aij,. Depending on the plant unit type input materials may be of more than one
type, and each type may be supplied from more than one source (i.e. it could be ore supplied from mines Fmijht or by import Ftijh t, or different intermediate products supplied from another unit of the same plant F~eijrt, from another plant Fueijn or by import Fti/~t). For each of the input material source there is a conversion factor v which determines the amount of output material to be obtained from one unit amount of the particular input material. Although the constraint is written for a plant unit type producing intermediate product r it is also valid for plant unit types producing final products. Note that this constraint is repeated for every different type of input material for a plant unit type. The second material balance relation for plant units, constraint (4), states that the output of a plant unit, whether it is a final product or an intermediate product, cannot exceed the production level Aiyt of the plant unit. The final products are shipped to consumption centers (F~ynkt) or exported (F, jlkt), while the intermediate products are either used by other units of the same plant (F//ien), or shipped to other plants (Fiy,,ert) or exported (F/fin) . Constraints (5), (6) and (7) are capacity increase constraints and represent investment decisions. Constraint (7) sets the investment decision variable Z~t to 0 or 1. If it is set to 1, then constraints (5) and (6) force E~jt, the capacity introduced by the investment, to take a value between yj and Yj, the minimum and the maximum feasible capacity increases for plant unit type j, respectively. An alternative formulation would be not to introduce constraint (7) and declare Z~t as 0-1 variables. This would convert the linear programming model into a mixed integer programming model. As explained in section 3, the linear programming formulation is prefered for the following reasons: First, it allows the decision maker to see some of the intermediate solutions of the mixed integer programming problem. The decision maker wants to see these solutions to increase his confidence on the choice made by the model. He may even choose to adopt a near optimal solution judging that the near optimal solution is better than the optimal solution because of the things that are not reflected in the model. Secondly, the decision maker has a good understanding of the problem so that he usually performs better to
S. Kuru / Long term planning of a major national sector using LP
reduce the number of linear programming problems to be solved than a typical mixed integer programming strategy such as the branch and bound method. The last constraint, constraint (8), states that the production level A~jt of a plant unit cannot exceed its production capacity. The production capacity of a plant unit at period t is expressed as the sum of two capacity values: Its existing capacity Bij , at this period and the sum of the capacity increases E0~- up to period t. The coefficient x i r t is introduced to reflect the fact that only a portion of the capacity increase E~jT introduced by an investment at period T can be utilized at period t. Although a function of t, the existing capacity B i j t of a plant unit at period t does not include capacity increases introduced by investments. B i j t is usually equal to the capacity available at the first period for all periods, but rarely it changes with t for reasons not reflected in the model. Note that the only constraint causing coupling between constraints is constraint (8). If this constraint did not exist it would be possible to solve the problem separately for each period. This property of the model suggests a natural decomposition scheme for the problem. The cardinalities of the index sets are typically of O(10) and for some sets the cardinality is of O(100). This causes the model to become very large in terms of the number of constraints and the number of variables. The variables F alone quickly exceed thousands in number. It should be noted that the model is for the most general case--i.e, every plant has all types of units and produces all types of final products using all types of raw materials, final products may be shipped to any consumption center or exported via any port, raw materials may be supplied from any mine or imported via any port, etc. Actually many of the variables do vanish when the model is formulated for a specific country, thus reducing the size of the problem considerably. But the model is still too large to debug and to afford the excessive amount of computation time. As stated earlier, a much smaller version of the model was solved at the first stage and the model evolved to its full size gradually. Grouping of ore mines, consumption centers and non-integrated plants are based on the results obtained from the solutions of the models that are used at the early stages. The final form of the model is based on the
161
following index sets: period (base years) intermediate products
final products
plants
plant units consumption centers
raw materials mines
ports
: {1984,1987,1990,1993, 1996 1999}; : {coke, sinter, liquid metal, molten steel, recycle scrap, ingot, hot rolled sheet, sponge iron} ; : {pig iron, plates, hot rolled sheets, cold rolled sheets, sections, rods and bars}; : {Eregli, Karabuk, Iskenderun, Istanbul-1, Izmir-1, a new integrated plant, Istanbul-2, Izmir-2} ; : as in Figures 2 and 3; : (Istanbul, Bursa, Izmir, Ankara, Adana, Kayseri, Samsun, Erzurum, Diyarbakir } ; : {coal, lump ore, dust ore, pellet, scrap}; : {Zonguldak, Divrigi, Hasancelebi, Kangal, Avnik, Yesilhisar, Yahsihan, Akcay } ; : {Eregli, Iskenderun, Istanbul, B a n d i r m a , Izmir, Samsun, Karasu}.
As it is seen each period covers three consecutive years and is represented by a base year, which is the middle year of the period. The first period 1984 is included in the model in order to satisfy the continuity of the model with the near past. Intermediate products, final products and plant units correspond to the flow diagrams given in Figures 2 and 3. Of the three existing integrated plants, Karabuk and Iskenderun produce long products whereas Eregli produces flat products. The existing electric arc furnaces, all of which produce long products in accordance with the flow diagram given in Figure 3a, are groupped as two plants, one in Istanbul and one in Izmir. Investment alternatives considered are as follows. All of the existing plants may undergo capacity expansion at any period. As new plants, one integrated plant and two non-integrated plants
S. Kuru / Long term planning of a major national sector using LP
162
Table 1 Distribution of demand among consumption centers (%)
Table 2 Final product demand (million tons/year)
Consumption center
Pig iron
Plates
Hot rolled sheets
Cold rolled sheets
Sections
Rods & bars
Base year
Pig iron
Plates
Hot rolled sheets
Cold rolled sheets
Sections
Rods & bars
Istanbul Bursa hmir Ankara Adana Kayseri Samsun Erzurum Diyarbakir
40 8 10 30 2 10 . . .
29 2 9 50 4 6 . . .
63 2 8 20 1 6 . . .
54 5 17 15 5 4
25 5 18 20 12 5 4 5 6
23 6 19 21 10 5 7 4 5
1984 1987 1990 1993 1996 1999
0.466 0.542 0.654 0.820 1.036 1.328
0.325 0.418 0.536 0.694 0.904 1.181
0.648 0.834 1.071 1.387 1.812 2.372
0.699 0.886 1.141 1.453 1.893 2.439
0.290 0.423 0.587 0.739 0.943 1.210
1.506 2.084 2.801 3.538 4.506 5.754
. . .
period when it starts production are to be determined by the model. The locations of new non-integrated plants are fixed as Izmir and Istanbul. One of them is to produce long products from scrap as given in Figure 3b and the other is a sponge iron plant as given in Figure 3c~ None of these two plants may start production before the year 1993. The demand for iron and steel products is
are considered. The new integrated plant may produce both flat and long products and may start production at any period not before the year 1996. Five alternative locations are determined for this plant: Samsun, Sivas, Bandirma, Izmir and Karasu. Its location, the capacities of its units, and the
\ 5 Gtm.~un I tO,
Ere-L"
•
Er2uru-% El
?
~nk~ra
\ ,
\
\
~H~sancetcbi Yoh.
/
LEGEND
:
I~
existing integrated plant a l t e r n a t i v e l o c a t i o n for i n t e g r a t e d
•
existing
non-integrated
o
consumption
• •
ore m i n e coal m i n e port railway
plant
center
Figure 4. Geographical distribution of plants, mines, consumption centers and ports
plant
s. Kuru / Long term planning of a major national sector using LP
assumed to be distributed between consumption centers as in Table 1. As seen from the table, long products are consumed in all 9 consumption centers whereas flat products are consumed in only 6 of them. The demand levels for each final product type are determined by projecting the consumption of the previous years. It is decided to run the model for three different demand levels, normal, maximum and minimum, in order to safeguard the investment decisions for unforeseen changes in the world iron and steel trade. Normal level of demand is given in Table 2. The maximum and the minimum levels are taken as 10% above and below the normal level. The mapping between mine regions and raw material types is as follows: coal : Zonguldak; lump ore: Divrigi, Hasancelebi, Kangal, Yesilhisar, Yahsihan, Akcay; dust ore : Divrigi, Hasancelebi, Avnik, Yesilhisar, Akcay; pellet : Divrigi, Hasancelebi, Avnik; scrap : available at plant site. As it is seen, the number of coking coal mines is reduced to one, lump ore mines to 6, dust ore mines to 5 and pellet plants to 3. This reduces the size of transportation matrix considerably. The transportation matrix is further reduced by mapping plants to ports and consumption centers to ports. A plant or a consumption center at a coastal city uses its own port. This is the case with Eregli, Iskenderun, Istanbul, Izmir, Samsun, Karasu and Bandirma (note that the port in Karasu is to be newly built upon the decision of building the new plant there). The remaining plants and consumption centers are mapped to the nearest port. Hence, Karabuk is mapped to Eregli, Bursa to Bandirma, and Diyarbakir to Iskenderun. The geographical locations of the plants, mines, ports and consumption centers are shown on the map in Figure 4. The resulting model consists of 2280 variables and 1494 constraints.
163
Table 3 Capacities a of existing plants for scenario 1 (million tons/year) Plant
Base year
Liquid metal
Molten steel
Final product
Eregli
1984 1987 1990 1984 1987 1990 1984 1987 1990 1984 1987 1990
1.750 1.820 2.230 0.560 0.750 0.900 1.300 2.150 2.450 -
1.700 2.000 2.400 0.510 0.665 0.700 0.900 2.065 2.300 1.400 2.000 2.000
1.500 1.900 2.860 0.440 0.555 0.595 0.750 1.800 2.020 1.750 1.750 1.750
Karabuk Iskenderun Nonintegrated a
Capacities remain the same after the year 1990.
demand level and the last two are for the maximum demand level. The scenarios are the following: Scenario 1. No new plants are to be built. The capacities of the existing plants are to be expanded to the values given in Table 3. Scenario 2. While the capacities of the existing plants are to be kept as in Scenario 1, two new non-integrated plants are to be built: A sponge iron plant and an electric arc furnace, of 0.325 million t o n s / y e a r and 0.350 million t o n s / y e a r final products capacity, respectively.
5. The solution
Scenario 3. N o new plants are to be built. The capacities of the existing plants are to be expanded as follows: Iskenderun plant is to be expanded in 1993 to produce 3.2 million t o n s / y e a r liquid metal, 4.25 million t o n s / y e a r molten steel and 3.6 million t o n s / y e a r final products. The added capacities will be utilized 50% in the year 1993, 90% in the year 1996 and at full capacity in the year 1999. The existing non-integrated plants are to be expanded to 2.4 million t o n s / y e a r molten steel and 2.15 million t o n s / y e a r final products capacity in the year 1987 and 2.7 million t o n s / y e a r molten steel and 2.4 million t o n s / y e a r final products capacity in the year 1990.
The model is solved for seven scenarios that are determined by the decision maker. Each scenario is a plan that is possible to adopt. The first five scenarios are designed to study the case of normal
Scenario 4. A new integrated plant is to be built to start production in the year 1996 or 1999. Capacity utifization ratio of the new plant will be 50% in its start of production period and 100% in the
164
S. Kuru / Long term planning of a major national sector using LP
following periods. As for the existing plants, the capacity of the Iskenderun plant is to be expanded as in Scenario 3 while the capacities of the other plants are to be as in Scenario 1. Scenario 5. A new integrated plant is to be built as in Scenario 4 while the capacities of the Iskenderun plant, Eregli plant and non-integrated plants are to be as in Scenario 3. Scenario 6. The best of the first five scenarios with maximum demand level. Scenario 7. Scenario 6 plus building a new integrated plant at the most appropriate time and location as determined from the solutions of the first five scenarios. Scenario 1 is designed to study the case of no major investment. Capacity increase through the years is due to minor investment decisions such as modernization, bottleneck elimination and small expansions that have been decided on in previous plans. Scenarios 2 and 3 are designed to study the case of meeting the demand by investing for nonintegrated plants and by expanding the capacities of existing plants, respectively. Scenarios 4 and 5 are designed to study the case of investing for a new integrated plant. Capacity expansions of existing plants are also considered both in Scenario 4 and Scenario 5 (the Eregli plant is not expanded in Scenario 4). Note that the location, the capacity and the start of production periods of new non-integrated plants are determined a priori. For the new integrated plant, while the capacities of the plant units are determined by the model, the start of production year is fixed by the scenario. The location, on the other hand, is determined by solving the scenario for the five alternative locations. Table 4 gives the objective function values for the best solution of each of the scenarios. The values represent the sum of the cost terms for the base year of each period, expressed in terms of the prices reduced to the year 1983. Note that the first five scenarios are solved using the normal level of demand, while the last two are solved using a demand level 10% higher than the normal level. Also note that Scenario 4 is solved for 5 times for period 5, once for building the new integrated plant at each of the five alternative locations, and
Table 4 Objective function values for the scenarios (billion TL) Scenario
N u m b e r of solutions obtained
Objective function value for the best solution
1 2 3 4
1 1 1 5 for 1 for 3 for 1 for 1 1
1399.3851 1402.3940 1321.9048 1394.9054
5 6 7
period period period period
5 6 5 6
1364.9855 1791.7844 1689.2509
that it is solved once for period 6, which is for building the new integrated plant at the location that is the best for period 5. Scenario 5, on the other hand, is solved 3 times for period 5, once for building the new integrated plant at each of the best three locations determined by the solutions of Scenario 4 at period 5. The solution of Scenario 5 for period 6 is for the location that is the best for period 5. The solutions of Scenario 4 for period 5 indicated that Sivas is the best location for building the new integrated plant, followed by Samsun and Izmir. Thus Scenario 4 is solved for Sivas for period 6 and Scenario 5 is solved for these three locations. Being the best solution of Scenario 5 at period 5, Scenario 5 is solved for Sivas for period 6. As seen from Table 4, the best of the first five scenarios is Scenario 3 followed by Scenarios 5, 4, 1 and 2 in this order. This shows that no major investment is necessary to meet the demand. The difference between investing for a new plant and
Table 5 Breakdown of major cost components for Scenario 3 (billion TL) Period Cost of Operating Imported Transport- Total national costs material ation cost raw cost cost materials 1 2 3 4 5 6
66.78 108.19 127.09 148.54 162.64 177.08
53.60 69.69 92.37 127.18 122.14 129.65
108.26 109.91 139.18 195.02 313.46 498.08
14.44 21.13 27.64 29.86 33.96 37.71
253.08 308.83 386.28 500.60 632.20 842.52
S. Kuru / Long term planning of a major national sector using LP
increasing the capacities of the existing plants is not of primary importance. While the location does not make that much difference, the period of start of production is important for the case of investing for a new plant. If a new integrated plant is to be built it should start production at period 5 and it should produce both flat and long products. The solutions of the last two scenarios showed that, in case of a high level demand, building a new integrated plant is necessary. As for the numerical results of the scenario solutions we give only the breakdown of major cost components for Scenario 3, which is the best scenario. The values are given in 1983 prices without a net present value reduction. As seen, the imported material cost is the major cost of the sector, which is followed by the operating costs, the cost of raw materials of national origin and the transportation cost. Within the imported material cost, the cost of raw materials is the major constituent at the early periods but the cost of final products becomes dominant towards the final periods. The cost of coal dominates the others in the cost of raw materials of national origin. Similarly, the cost of raw material transportation dominates the other transportation costs. Some of the other results obtained from the solutions of the model may be summarized as follows: National resources are preferred as the supply of raw materials. As for importing intermediate products, the amount of ingot imported is negligible but the Eregli plant imports an imported amount of slab. Final products are both imported and exported. The demands for hot rolled sheets are totally met by national production. These products are also exported. Plates, cold rolled sheets, and rods and bars, on the other hand, are imported at some periods. For the details of the scenario solutions see [11].
6. Conclusion The model developed for the Turkish iron and steel sector has proved to be very useful for studying the long term investment planning of the sector and gave valuable information for an analysis of it. The model, which is a linear programming model of considerable size (1494 constraints and 2280 variables), is solved using the FMPS mathematical programming package available on
165
the U N I V A C 1100 system at Bogazici University Computer Center. The model showed that investing for a new integrated plant is not necessary if the demand follows the projected trajectory upto the year 2000. Increasing the production capacities of the existing plants is sufficient for this case. In case of an unexpected increase in demand, on the other hand, a new integrated plant that would produce both flat and long products should be built to start production by the year 1996. The location of the new integrated plant is not of prime importance and can be decided on by considering other factors that are not represented in the model. The transportation cost is the determinant of the location of the new integrated plant. As such, Sivas and Samsun have advantages over the others because both are close to the consumption centers at the Central and Eastern Anatolia regions, and to the major ore mines. Sivas has the advantage of being right on the largest ore mines region, whereas Samsun has the advantage of being a port city.
Acknowledgements This work was supported by The State Planning Organization (DPT) of Turkey. The views and conclusions in this article are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of DPT.
References [1] Organization for Economic Cooperation and Development, The Iron and Steel lndustry in 1983, Paris, 1985. [2] Manners, G., The Changing World Market for Iron Ore 1950-1980, Johns Hopkins Press, Baltimore, MD, 1971. [3] Kavrakoglu, I., "A dynamic optimization model for energy policy analysis", in: I. Kavrakoglu (ed.), Mathematical Modelling of Energy Systems, Sijthoff and Noordhoff, Alphen aan de Rijn, 1981, 131-135. [4] Russel, C.S., and Vaughan, W.J., "An analysis of the historical choice among technologies in the U.S. steel industry: Contributions from a linear programming model", The Engineering Economist 22/1 (1976) 1-26. [5] Lawrence, K.D., Lawrence, S.M., and Reeves, G.R., "Aggregate industrial expansion: A multiple-objective linear programming approach", The Engineering Economist 25/3 (1979) 197-207.
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[6] Abe, M.A., "Dynamic micro-economic models of production, investment and technological change of the U.S. and Japanese iron and steel industries", in: G.C. Judge and T. Takayama (eds.), Studies in Economic Planning Over Space and Time, North-Holland, Amsterdam, 1973, 344-367. [7] Nelson, J.P., "An interregional recursive linear programming model of the U.S. iron and steel industry", in: G.G. Judge and T. Takayama (eds.), Studies in Economic Planning Over Space and Time, North-Holland, Amsterdam, 1973, 368-393. [8] Sawey, R.M., and Zinn, C.D., "Mathematical model for long range expansion planning of generation and
transmission in electric utility systems", IEEE TransaCtions on Power Apparatus and Systems 96 (1977) 657-666. [9] Peterson, E.R., "A dynamic programming model for the expansion of electric power systems", Management Science 20 (1973) 656-664. [10] Schmitz, K., and Schwefel, H.P., "Finding reasonable energy policies by means of a dynamic simulation model", in: Proceedings of the International Symposium Simulation '77, ACTA Press, 1977. [11] Kuru, S., "Expanded master plan model of the Turkish iron and steel sector" (research report in Turkish), Bogazici University, 1985.