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International Journal of Production Research

ISSN: 0020-7543 (Print) 1366-588X (Online) Journal homepage: http://www.tandfonline.com/loi/tprs20

A bilinear programming model and a modifiedbranch-and-bound algorithm for productionplanning in steel rolling mills with substitutabledemand

Rami As’ad & Kudret Demirli

To cite this article: Rami As’ad & Kudret Demirli (2011) A bilinear programming model anda modified branch-and-bound algorithm for production planning in steel rolling mills withsubstitutable demand, International Journal of Production Research, 49:12, 3731-3749, DOI:10.1080/00207541003690116

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Published online: 30 Jun 2010.

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International Journal of Production ResearchVol. 49, No. 12, 15 June 2011, 3731–3749

A bilinear programming model and a modified branch-and-bound

algorithm for production planning in steel rolling

mills with substitutable demand

Rami As’ad and Kudret Demirli*

Fuzzy Systems Research Laboratory, Department of Mechanical and Industrial Engineering,Concordia University, 1515 St. Catherine W., Montreal, PQ, H3G 1M8, Canada

(Received 6 July 2009; final version received 29 January 2010)

In this paper, we address an instance of the dynamic capacitated multi-itemlot-sizing problem (CMILSP) typically encountered in steel rolling mills.Production planning is carried out at the master production schedule level,where the various end items lot sizes are determined such that the total cost isminimised. Through incorporating the various technological constraints associ-ated with the manufacturing process, the integrated production–inventoryproblem is formulated as a mixed integer bilinear program (MIBLP). Typically,such class of mathematical models is solved via linearisation techniques whichtransform the model to an equivalent MILP (mixed integer linear program) at theexpense of increased model dimensionality. This paper presents an alternativebranch-and-bound based algorithm that exploits the special structure of themathematical model to minimise the number of branches and obtain the boundat each node. The performance of our algorithm is benchmarked against that of aclassical linearisation technique for several problem instances and the obtainedresults are reported.

Keywords: steel mills; demand substitution; mixed integer bilinear program;modified branch and bound algorithm

1. Introduction

Due to the ever-increasing level of competitiveness and the newly emerging productioncontrol techniques, industrial enterprises are forced to revise their strategies continuouslyand further optimise their operational processes accordingly. The iron and steel industry,for instance, is characterised by being both capital and energy intensive and, as such,the importance of effective production planning in such industry is by no means less thanthat in any other industry. For a rolling mill producing between 300,000 and one milliontons of steel annually, the capital investment is measured in tens of millions of dollars(Denton et al. 2003). Furthermore, the steel industry represents one of the backboneindustries greatly affecting a nation’s economic growth and pace of development. Needlessto say, a substantial portion of today’s indispensable products that are used on a dailybasis and serve multiple purposes have steel ingredient in them in one form or another.In North America only, more than 100 million tons of steel are produced annually withan estimated value of over 50 billion dollars (Denton et al. 2003).

*Corresponding author. Email: [email protected]

ISSN 0020–7543 print/ISSN 1366–588X online

� 2011 Taylor & Francis

DOI: 10.1080/00207541003690116

http://www.informaworld.com

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Upon realising the major investments associated with the construction and operationsof steel plants, the main concern of steel manufacturers has been the adoption of the latesttechnology advances in the production process as well as finding better ways to managethe rapid increase in product variety. In particular, the steel mill under considerationproduces round steel bars (rebars), where product differentiation increases as the steelbillets (i.e., the raw material) proceed on their journey through the rolling stands towardsthe finished product. As such, the raw material-finished product combination not onlydetermines to a great extent the number of rolling passes that the product has to gothrough, and hence the energy consumption, but also the quantity of both types ofmaterials to be kept in stock, hence the inventory holding and opportunity costs.

This paper presents a realistic case study taken from a steel mill that produces a widevariety of steel bars from different sized steel billets. The production planning problem istackled at the master production schedule (MPS) level, where the daily/weekly productionlot sizes for the various end items (rebars) are determined such that end customer demandsover the planning horizon are fulfilled at a minimal total cost. In particular, there are foursets of decisions that the model seeks to optimise: (1) which products to produce in eachperiod; (2) how much of each shall be produced; (3) the allocation of the productsto satisfy the customer’s demand (since demand substitution is allowed); and (4) theraw-material finished-product combination (i.e., which raw material shall be used toproduce which product). The contributions of this paper are twofold. First, it presentsa mixed integer bilinear program (MIBLP) that captures the combined effect of severalinterrelated factors encountered on a daily basis in the steel mill. Second, it adaptsthe classical branch and bound technique to efficiently handle the special structure of thedeveloped mathematical model.

The remainder of this paper is organised as follows. In Section 2, we provide a briefreview of the literature addressing the capacitated multi-item lot-sizing problem(CMILSP) in the general context as well as the more related work focusing on productionplanning models in the steel industry. Section 3 explains the manufacturing process anddefines the problem at hand along with the stipulated assumptions. The mathematicalmodel is derived in Section 4 and the proposed branch-and-bound (B&B) based solutionalgorithm is detailed in Section 5. A numerical comparison between the classicallinearisation approach and the proposed B&B algorithm is carried out in Section 6.Finally, Section 7 provides a brief conclusion and highlights future research directions.

2. Literature review

Since the production planning problem addressed in this paper is an instance of thedynamic lot sizing problem (DLSP), this review will be divided into two subsections.The first aims at addressing the issues encountered at the steel mill but were dealt within the general context of the DLSP. The second subsection, however, focuses on steelrelated production planning problems.

2.1 Dynamic lot sizing related literature

The well known dynamic lot sizing problem (DLSP) roots back to the seminal papersof Wagner-Whitin (1958), and Manne (1958) in which this line of research was initiated.In its most basic form, the DLSP seeks the optimal timing and level of production such

3732 R. As’ad and K. Demirli

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that the total setup, production and holding costs are minimised. In particular, lot sizingdecisions are carried out for discrete time periods extending over a finite time horizonwhere the demand for each time period might assume a different value, hence the termdynamic. This distinguishes the DLSP from the continuous time scale, constant demandand infinite planning horizon lot sizing problems such as the economic order quantity(EOQ) and the economic lot scheduling (ELS) models.

Since it was first introduced, the DLSP continues to pose as one of the long standingresearch fields that have received a great deal of interest from industrial practitionersas well as researchers. The literature reviewed in this subsection, however, focuses oncapacitated multi-item lot-sizing models addressing specific issues such as setup times,overtime production, backlogging, and demand substitution. Although it has been widelyexplored, there has been little literature regarding problems such as CLSP (capacitated lotsizing problem) with backlogging or with setup times and setup carry-over (Karimi et al.2003). Figure 1, which combines and extends the work of Karimi et al. (2003), andHaase (1994), provides a classification of the various lot sizing problems based on thecharacteristics of each. For a recent and comprehensive review of lot sizing models fromboth modelling and solution algorithms perspectives, interested readers are referred toJans and Degraeve (2007, 2008). The interaction between the model formulation and thesolution algorithm for the DLSP has been studied by Alfieri et al. (2002) through the useof LP-based rounding heuristics.

In most practical situations, a setup is incurred whenever the manufacturing processswitches between two different products. This setup consumes partial capacity and henceit needs to be explicitly accounted for in the mathematical model. Ozdamar and Bozyel(2000) considered the case of CMILSP with overtime and setup time decisions, andpresented several meta-heuristics to solve the problem. Starting from an initial lot-for-lotapproach, Trigeiro et al. (1989) developed a heuristic algorithm for the CMILSPwith setup times based on the Silver-Meal lot sizing heuristic. Hindi et al. (2003)addressed the same problem and obtained a lower bound on the value of the objectivefunction by Lagrangian relaxation with sub-gradient optimisation. Recently, Absi andKedad-Sidhoum (2008) tackled a more generalised version of the CMILSP with setuptimes in which the demand can be totally or partially lost.

The limited capacity of today’s manufacturing facilities poses as a challenge towardson time fulfilment of various customers’ demands. In general, the unmet portion of the

Lot-sizingProblems

Planning horizonBig bucket vs small

bucket

Number of levels Single-level vs

multi-level

Number of products Single-item vs multi-

item

Capacity restrictionUncapacitated vs

capacitated

Nature of the product

Demand pattern - Static vs dynamic

- Deterministic, probabilistic or fuzzy

Setup structure Simple vs complex

Service policy Backlogging vs lost

sales

Figure 1. Characteristics of the dynamic lot-sizing problem.

International Journal of Production Research 3733

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demand is either lost to another competitor in what is called ‘lost sales’, or is satisfiedat a later period of time in what is known as ‘backlogging’ or ‘backordering’. The firstoption entails an associated lost profit and is usually adopted in the retailing business.However, in the steel industry, most customers are long term customers and, as such,manufacturers might make use of their powers alongside customers’ loyalty to backloga portion of the demand at a certain additional cost, called ‘backlogging cost’. An easyand robust alternative to the Wagner-Whitin backorder algorithm, proposed by Webster(1989), was introduced by Gupta and Brennan (1992). Also, Lagrangian relaxationwas used by Miller and Yang (1994) to exploit the underlying network structure of theCMILSP with backlogging.

Some products, such as integrated circuits and steel bars, are produced in differentgrades with varying performance characteristics. In such a situation, the manufacturermay occasionally choose to downgrade a product instead of backordering the demand fora similar product with the lower grade. The term ‘downgrading’ has been previouslyestablished in the literature, and it refers to instances where class j product is used to satisfythe excess demand for that of class i, where i � j. For example, Bitran and Dasu (1992)presented the case of the semiconductor industry and called the demand substitutionstructure where a higher quality chip satisfies the demand for the lower one ‘downgrading’.For the same industry, Bassok et al. (1999) addressed this type of substitution structureand called it ‘downward substitution’. In general, the issue of demand substitutionhas been considered in a variety of contexts for traditional production planning and theavailable literature could be broadly classified into three streams of work, as pointed outby Rajam and Tang (2001). However, most papers concentrate on the problem in thecontext of single period models (Li et al. 2007). Balakrishnan and Geunes (2000)considered the requirement planning problem with substitutions in a multi-period horizonand derived a dynamic-programming (DP) algorithm that obtains the production andsubstitution quantities in each period. Li et al. (2006, 2007) dealt with the DLSP in thecontext of a hybrid manufacturing/remanufacturing system with product substitution.In their analysis, a new product is offered in place of a remanufactured one when thereis a remanufactured product shortage. They developed a DP algorithm for theuncapaciatetd case, and a genetic algorithm for the capacitated one.

2.2 Production planning related literature in steel plants

In spite of the significance of steel industry, planning and scheduling problems in ironand steel production have not drawn as wide attention of the production and operationsmanagement researchers as many other industries such as metal cutting and electronicsindustry (Tang et al. 2001). As pointed out by Dutta and Fourer (2001), very little workhas been done in the area of inventory control, manufacturing control and multi-periodlinear programming modelling in the steel industry. The first attempt towards formulatingthe production process at a steel plant as a linear program was made by Fabian (1958),where he developed an integrated LP model for iron making, steel making and rollingoperations. A review of the most recent planning and scheduling tools as applied tointegrated steel plants was conducted by Tang et al. (2001).

With the increasing steel products variety, it becomes increasingly important for steelmanufacturers to adopt new strategies towards improving the service level and reducingthe service time. Sharma and Sinha (1991) discussed the various issues affecting the choice


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of an optimum product mix in a steel plant, and described an optimisation modelfor determining such mix. The network approach was used by Sasidhar and Achary (1991)to deal with the problem of production planning at a steel mill with the objective ofmaximising capacity utilisation. In their work, production is planned according tocustomers’ priorities, where different customers are assigned different priorities. Chen andWang (1997) established a strategic linear programming model for a steel plant from asupply chain perspective, where the model seeks the optimal production plan, raw materialsupply and finished product distribution policy. However, the optimal solution of theproblem is found with a reduced number of variables and heuristics are not presented for amore realistic solution (Zanoni and Zavanella 2005). Based on the input-output concept,Li and Shang (2001) developed a production planning model for a large steel corporationin China. Denton et al. (2003) built a decision support system that allows inventoryplanners to analyse various scenarios and identify which slabs shall be produced accordingto a make-to-stock (MTS) strategy. Zanoni and Zavanella (2005) established a linearprogramming model that gives the optimal production sequence of the billets, ordered bythe customers, while taking into account the limitations in warehouse space availability.Recently, Kerkkanen (2007) analysed the case of a small make-to-order (MTO) steel milland came up with a new inventory policy in order to enable a comparison between a puremake-to-order (MTO) system and a hybrid MTO/MTS system.

This paper investigates one of the potential areas requiring future research explicitlystated in Dutta and Fourer (2001): ‘Simultaneous optimization of product-mix, inventory,and transportation problems over multiple periods’. We formulate the short termmulti-input multi-output production planning problem encountered in a medium-sizedsteel mill as a multi-period mixed integer bilinear programming (MIBLP) model that takesinto account the technological constraints associated with the manufacturing process.

3. Problem definition and the production process

The steel mill under consideration produces reinforced round steel bars (rebars) from anexternally supplied raw material, which is steel billets having a square cross-sectional area.The steel billets (rebars) are purchased (produced) in two different steel grades (grades 40and 60) and have several dimensions that are the same for both grades (Tables 1 and 2).Due to technical considerations regarding yield and scrap rate, there exist somerestrictions on the possible billet-rebar combinations. For instance, a billet of dimensions100mm� 100mm� 6m (index i¼ 6 in Table 1) is not to be used as an input materialin the manufacturing of a 32mm diameter steel bar. The two steel grades differ mainly

Table 1. Raw material dimensions.

Index (i) Width (mm) Height (mm) Length (m)

1 130 130 122 130 130 83 130 130 64 125 125 85 125 125 66 100 100 6


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in the chemical composition, metallurgical structure and carbon content, which lead to

varying mechanical properties and performance. In particular, grade 60 has higher values

for yield strength and ultimate tensile strength (YS¼ 421N/mm2, UTS¼ 621N/mm2) ascompared to those for grade 40 (YS¼ 300N/mm2, UTS¼ 500N/mm2). As such, grade

60 steel is considered to be of better quality and is thus sold at a higher price.Clearly, a steel bar of a certain grade can only be produced from a billet of that

particular grade. The production process starts by placing several billets, with uniform

dimensions, into the furnace where they are heated up to 1200�C. The heated billets are

then taken out of the furnace to the rolling mill where the hot rolling operation takesplace followed by the bar rolling operation. After rolling the billets into long bars of the

desired cross-sectional area and a standard length, the bars are pushed on the cooling bedto allow the product to cool down. The next step is to bind the bundle of products and

label the bundle. At last, the bars are either stored in the warehouse or shipped directly to

the customer.The processing times depend on which rebar is being produced from which billet

(i.e., on the dimensions of both). This is because a billet may be required to pass through

several rolling stands depending on the desired diameter of the rebar to be produced.

In addition, the higher the number of rolling passes that a billet has to go through, themore return scrap is expected. However, billets of bigger dimensions are cheaper to buy,

and bars of smaller diameters are sold to the customer at a more expensive price. This mayjustify the production of a small diameter rebar from a bigger dimension billet in spite

of excess processing time and higher chances of scrap produced. The operations at the steel

mill are characterised by the following features:

(1) Setup: one of the most distinguishing and complicating features of steel mills

operation is the setup time structure. The setup activities include those associatedwith the furnace, such as placing the new batch of billets inside and adjusting the

settings, as well as those activities carried out on the rest of the production line,

such as rolls and stands changing, guides and grooves changing, runner way andtime billet changing, and speed reference adjustments. Hence, when the setup is

carried out between batches, the setup time depends on the raw material

(as the number of billets placed in the furnace depends on the size) and on thefinished product (as the rollers and speed have to be adjusted according to the

Table 2. Finished product dimensions.

Index ( j ) Diameter (mm)

1 322 283 254 225 206 187 168 149 1210 10


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rebar to be produced). The between batches setup is referred to as ‘minor setup’in what follows.On the other hand, when the setup is carried out at the beginning of the day, it istime dependent rather than product dependent. At the beginning of each businessday, the furnace temperature has to be elevated gradually to 1200�C before startingproduction. This temperature elevation time depends mainly on the idle time sincethe production of the last batch in the previous day. The equation for the bestfitted line between the idle time and the setup time can be obtained using simpleregression as shown in Figure 2. A setup conducted at the beginning of the dayis referred to as ‘major setup’. It should be noted that such setup time structure isfrequently encountered in other industries such as metal rolling (other than steel)and plastics manufacturing in which the product undergoes a heat-treatment phaseduring the production process.

(2) Product substitution: the company has the option to:

(a) Fulfil the unmet demand for an out-of-stock lower grade rebar (i.e., grade 40)with a same sized rebar of the higher grade (i.e., grade 60) in the same timeperiod as to meet the promised delivery schedule, which results in:

. Increased customer expectation for future shipments;

. Lost profit (due to selling a higher quality product at the price of the lowerquality one).

(b) Backlog and match the order with the delivery at a later period in time. In thiscase,

. A backlogging cost is incurred.

. Might eventually lead to loss of customer goodwill.

(3) Overtime: as the setup time depends on the working hours, it would make sensefor the company to consider the option of working overtime especially in periodsof excess demand. Although it costs more to produce on overtime basis, it might beeconomical to do so as this avoids the backlogging cost and reduces the furnacesetup time at the beginning of the next business day.

Figure 2. The linear relationship as obtained from regression analysis.


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(4) Yield: the number of steel bars produced from the same billet is a function of thedimensions of both the billet and the rebar to be produced. Moreover, themanufacturing process in the steel industry, like all other industries, is not a perfectone in the sense that it produces a portion of defective items. However, theresulting nonconforming items can be sold as scrap steel to other manufacturers.The percentage of scrap produced depends on the billet-rebar combination.

(5) Time dependent raw material purchasing cost and finished product selling prices:

in reality, the market prices of the reinforced steel bars have been subjectedto drastic variations in the last decade or so. A similar argument applies to thepurchasing prices of the steel billets. Hence, such variation in prices has to be takeninto consideration to better reflect the reality. In fact, to serve a broader rangeof planning purposes, the proposed mathematical formulation assumes that all thecost parameters are time dependent.

Having established the features of the problem, we next state the assumptions underwhich the mathematical model in the next section is developed:

(1) Batch production: in accordance with the economies of scale and to utilise theavailable capacity to its fullest, the production takes place in batches of size 60 tonseach (i.e., the furnace capacity).

(2) Batch uniformity: this applies to both the billets placed at once in the furnace and tothe steel bars produced from the same batch. Neither billets nor bars of differentdimensions are produced from within the same batch.

(3) Deterministic demand: the steel industry is characterised by having customersthat are in most cases long term loyal customers (Chen and Wang 1997,Kerkkanen 2007). This justifies the assumption of a deterministic demand ratefor short term planning purposes such as the operational model presented in thispaper.

(4) One way substitutability: a grade 60 steel bar of a certain diameter could be usedto fulfil a portion of the unmet demand for a similar steel bar with grade 40 giventhat such substitution is suitable for the engineering application, but not the otherway around. This ‘downgrading’ is motivated by a variety of reasons, for example,to prevent customer dissatisfaction, to reduce setup costs, or to reduce inventorycosts (Bitran and Dasu 1992).

4. Model development

The problem at hand is a typical instance of the well known dynamic lot-sizing problem.Our objective is to formulate the problem as a mathematical model, where the rawmaterial purchasing quantities, regular time and overtime based production quantities,inventory levels, backorder and substitution quantities for each product in each timeperiod are determined such that the total cost over the planning horizon is minimised.

The following notation is used in developing the mathematical model. It is importantto note that quantities of materials, whether billets or rebars, are measured in tons andthe production rate is measured in tons per hour. An index, whether i for raw material(RM) or j for finished product (FP), refers to the same dimension in both steelgrades. This notation will greatly assist in formulating the problem as a mathematicalmodel.


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Indices:

i index of available RM (billets) dimensions (cross sectional area and length),

i ¼ 1, . . . , I;j index of FP (rebars) dimensions (cross-sectional area), j ¼ 1, . . . , J;k index of steel grade, k ¼ 1 for grade 60 and k ¼ 2 for grade 40, with grade

60 being better;t index of time periods (days), t ¼ 1, . . . ,T, where T is the planning horizon.

Parameters:

CRkit cost of purchasing one unit of RM i of steel grade k in time period t;

ORkit fixed cost of ordering RM i of steel grade k in time period t (independent of the

order quantity);IRk

it cost of holding one unit of raw material i of steel grade k in stock for one time

period (from t to tþ 1);PCijt cost of producing one unit of FP j from RM i in time period t;POt overtime production cost per hour in period t;SCt production line setup cost per hour in period t;IFk

jt cost of holding one unit of FP j of steel grade k in stock for one time period

(from t to tþ 1);BCk

jt cost of backlogging one unit of FP j of steel grade k for one time period (from t

to tþ 1);SPk

jt selling price of one unit of FP j of steel grade k in period t;

Mkit upper limit on the supply capacity of RM i of steel grade k in period t;�ij yield resulting from producing FP j using RM i (independent of the steel grade),

05 �ij 5 1 (where �ij ¼ 0 if RM i is not to be used in the production of FP j);�ij rate of producing FP j from RM i;

Dkjt anticipated demand for FP j of steel grade k in time period t;At available regular production time in period t (in hours);Aot maximum allowable overtime production hours in period t;bt fixed production batch size (in tons);

STij minor setup time for a batch of FP j produced from RM i.

Decision variables:

Qkit quantity of RM i of steel grade k purchased in period t;

Gkit 1, if RM i of steel grade k is purchased in period t; 0, otherwise;

Sbkijt 1, if a setup for FP j made from RM i both having steel grade k is carried out

at the beginning of period t (major setup); 0, otherwise;Sdk

ijt number of minor setups for FP j produced from RM i both having steel grade k

conducted during period t (between batches);Skijt total setup time in period t for FP j produced from RM i both having steel

grade k;

Xkijt quantity of FP j of steel grade k produced from RM i during period t;

Ot overtime production capacity used in period t (in hours);Wk

jt quantity of FP j of steel grade k used to satisfy the demand for the corresponding

product (i.e., same dimensions) with steel grade 40 in time period t;I kit inventory level for RM i of steel grade k at the end of period t;


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I kjt inventory level for FP j of steel grade k at the end of period t;

Bkjt backlogging level for FP j of steel level k at the end of period t.

The total cost is composed of RM ordering, RM purchasing, RM inventory holding,setup, regular time production, overtime production, FP holding and backorder costs.Another term added to the objective function is the lost profit due to selling a portionof high quality steel at the price of the lower grade one.

min Z ¼XIi¼1

XJj¼1

X2k¼1

XTt¼1

SCtSkijt þ PCijtX

kijt

� �

þXJj¼1

X2k¼1

XTt¼1

IFkjt I

kjt þ BCk

jtBkjt

� �

þXTt¼1

POtOt

þXIi¼1

X2k¼1

XTt¼1

ORkitG

kit þ CRk

itQkit þ IRk

it Ikit

� �

þXJj¼1

XTt¼1

W1jt SP1

jt � SP2jt

� �: ð1Þ

Subject to:

Qkit �Mk

itGkit , 8i, t, k ð2Þ

XIi¼1

XJj¼1

X2k¼1

Sbkijt ¼ 1, 8t ð3Þ

Skijt ¼ 0:4� Sbk

ijt 24�XIi 0¼1

XJj 0¼1

X2k 0¼1

Xk 0

i 0j 0, t�1

�i 0j 0þ Sk 0

i 0j 0, t�1

! ! !

þ STij � Sdkijt, 8i, j, t, k

ð4Þ

XIi¼1

XJj¼1

X2k¼1

Xkijt

�ijþ Sk

ijt

!� At þOt, 8t ð5Þ

Ot � Aot, 8t ð6Þ

Xkijt ¼ bt � Sbk

ijt þ Sdkijt

� �� ij, 8i, j, t, k ð7Þ

I kit ¼ I ki,t�1 þQkit � bt �

XJj¼1

Sbkijt þ Sdk

ijt

h i, 8i, t, k ð8Þ

I1j,t � B1j,t ¼ I1j,t�1 � B1

j,t�1 þXIi¼1

X1ijt �W1

jt �D1jt, 8j, t ð9Þ


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I2j,t � B2j,t ¼ I2j,t�1 � B2

j,t�1 þXIi¼1

X2ijt �W2

jt, 8j, t ð10Þ

W1jt þW2

jt ¼ D2jt, 8j, t ð11Þ

I kj0 ¼ Bkj0 ¼ Bk

jT ¼ 0, 8j, k ð12Þ

Sdkijt,Q

kit ,X

kijt,Ot,W

kjt , I

kit , I

kjt ,B

kjt � 0, 8i, j, t, k ð13Þ

Sdkijt is integer, 8i, j, t, k ð14Þ

Gkit ,Sb

kijt 2 0, 1f g, 8i, j, t, k: ð15Þ

Constraint set (2) ensures that the quantity of each RM i purchased in time period t, ifany, is limited by the supplier capacity in that period. Only one major setup for a certainRM, FP and steel grade combination can take place during a time period, which is statedin constraint (3). Constraint set (4) specifies the total setup time for a certain (i, j, k)combination in any period, which is simply the sum of both major and minor setup times.As per constraint (5), the total production and setup times for all products should notexceed the available uptime, whether regular production time or overtime, in any timeperiod. The maximum number of overtime hours worked per day is bounded by a certainallowable value as specified by constraint (6). Since steel bars industry is a batch process,the production quantity of a certain FP from a particular RM, Xk

ijt, is a function of thenumber of batches produced, batch size and the yield, which is stated in constraint (7).The raw material balance equation is given by constraint (8), and the finished productsbalance equations for both steel grades are determined by constraints (9)–(11). Withoutloss of generality, the initial inventory and backorder levels are set to zero as shownin constraint (12) which also ensures that the demand for each FP throughout the planninghorizon is met. Constraints (13)–(15) represent the non-negativity, binary and integralityrestrictions on the respective decision variables. It should also be noted that, in any timeperiod, the usage rate of a specific raw material to produce the various finished productsshould not exceed the purchased quantity of that raw material in that same period inaddition to the previously available stock. That is:

bt �XJj¼1

Sbkijt þ Sdk

ijt

h i� Qk

it þ I ki,t�1, 8i, t, k: ð16Þ

However, constraint set (16) is implied by constraint set (8) along with thenon-negativity restriction on the variables I kit � 0.

The model developed above is, in a sense, a typical dynamic CMILSP formulation withvarious practical extensions incorporated into the model. A much simpler version, which isthe capacitated single item lot-sizing problem, is NP-hard in general. It is even NP-hard forvery special cases (Bitran and Yanasse 1982).

5. Solution algorithm

The developed mathematical model is a mixed integer bilinear program (MIBLP) in whichthe binary variable Sbk

ijt is multiplied by the continuous variables Xk 0

i 0j 0, t�1 and Sk 0

i 0j 0, t�1,


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one at a time, as can be seen in Equation (4). The bilinearity property these models possessis due to the fact that, for fixed Sbk

ijt values, the original model reduces to a mixed integerlinear program (MILP), and for fixed Xk 0

i 0j 0, t�1 and Sk 0

i 0j 0, t�1 values, it again reduces to aMILP in the space of the other decision variables. Such bilinearity causes the model to benon-convex, which can be seen in Equation (4) and in the objective function. This explainsthe incapability of commercial software (e.g., AMPL/CPLEX and LINGO) to obtain thesolution to such class of models even for small problem instances. Also, since it is notknown in advance how much capacity should be allocated to each steel grade, the problemis non-separable into two smaller problems each dealing with one grade at a time eventhough both grades share the same combinations of dimensions for both RM and FP.Having established the complexity of the model, we next start with the classical approachused to solve such models (i.e., linearisation technique) followed by our modified branchand bound based solution algorithm.

5.1 Linearisation approach

Typically, the bilinearity existing in MIBLPs is resolved using the classical linearisationapproach established in the literature (e.g., Peterson 1971, Glover 1975, Adams andSherali 1990, Adams and Forrester 2007). Following these approaches, the originalproblem is transformed to an equivalent MILP through the introduction of auxiliaryvariables and constraints. Employing the technique of Glover (1975), Equation (4) isreplaced by the following sets of constraints in the linearised version of the model:

Skijt ¼ 9:6� Sbk

ijt � 0:4� ykijt þ STij � Sdk

ijt, 8i, j, t, k ð4:1Þ

LkijtSb

kijt � yk

ijt � UkijtSb

kijt, 8i, j, t, k ð4:2Þ

Xi 0j 0k 0

Xk 0

i 0j 0, t�1

�i 0j 0þ Sk 0

i 0j 0, t�1

!�Uk

ijt 1� Sbkijt

� �� yk

ijt �Xi 0j 0k 0

Xk 0

i 0j 0, t�1

�i 0j 0þ Sk 0

i 0j 0, t�1

!

� Lkijt 1� Sbk

ijt

� �, 8i, j, t, k ð4:3Þ

where Lkijt and Uk

ijt represent the lower and upper bounds on the term:

Xi 0j 0k 0

Xk 0

i 0j 0, t�1

�i 0j 0þ Sk 0

i 0j 0, t�1

!,

respectively. While constraint (4.2) is the binding constraint once the binary variable Sbkijt

equals zero, constraint (4.3) is the binding one if the value of the binary variable Sbkijt is

equal to one. Since the newly added variable, ykijt, appears in the objective function

with a negative sign, the model will seek the maximum possible value for this variable.This indicates that the left hand side inequalities of constraints (4.2) and (4.3) areredundant and can be dropped out of the model.

It should be noted that the linearisation approach, in general, suffers from two severeshortcomings. First, it involves a radical increase in the number of problem variables andconstraints and, as such, the gains to be derived from dealing with linear functionsare quite likely to be nullified by the increased problem size (Glover 1975). In particular,


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the linearised model for our problem involves the addition of I� J� T� K variables and

2� I� J� T� K constraints. Secondly, the success obtained via such an approach is

highly dependent on the specific problem (Adams and Sherali 1990). The performance of

the linearisation approach for several problem instances is reported in Table 5.

5.2 Modified branch-and-bound algorithm

The shortcomings associated with the linearisation approach necessitate the development

of an alternative and a more efficient solution procedure. This section highlights the

proposed solution algorithm which is a modified version of the long established

branch-and-bound (B&B) technique. The motive behind such an algorithm is that these

B&B methods can often be tailored to exploit special problem structures, thereby allowing

these structures to be handled with greater efficiency and reduced computer memory

(Avriel and Golany 1996). The first implementation of a B&B algorithm dates back to

Land and Doig (1960) for the linear case. Since then, the algorithm has seen various

improvements and was used to efficiently solve problems of different natures including

mixed integer non-linear programs (e.g., Leyffer 2001). However, to the best of our

knowledge, B&B based techniques have not yet been implemented for the solution of

MIBLPs.The basic idea here is to get rid of bilinearity through proper substitution of the

complicating binary variables, Sbkijt, while simultaneously obtaining the bound at each

node via such substitution. Once the values of these variables are set to either zero or one,

the resulting reduced size MILP becomes a lot easier to solve. Fortunately, constraint (3)

states that among all possible RM, FP and steel grade combinations, there exists only one

possible combination for which a major setup could take place in day t. Hence, setting

Sbk0

i 0j 0t ¼ 1 for a certain value of i 0, j 0 and k 0 entails that Sbkijt ¼ 0 for either one of i 6¼ i 0,

j 6¼ j 0, k 6¼ k 0 and for the same t. This reduces the number of possibilities (i.e., branches to

be explored) to I� J� K for each t. Moreover, the possible number of Sbkijt ¼ 1

combinations at optimality is now ðIJK ÞT as a direct result of constraint (3). However,

fully exploring this many possibilities may turn out to be a tedious and a time consuming

task especially for large scale models. Therefore, a clever enumeration algorithm for such

possibilities is in need. The proposed B&B based solution algorithm is stated formally as

follows.

Step 1: Set Sbkijt ¼ 0, 8i, j, k, t, throughout the model, ignore Equation (3) and then solve.

The resulting MILP problem is a relaxed version of the original problem and its

solution provides a lower bound on the optimal value for the original problem since the

major setup cost is set to zero. However, its solution is not feasible to the original problem

as it violates Equation (3).

Step 2: Set t ¼ 1 and substitute Sbkij1 ¼ 1 for each branch emanating from the original

node, with I� J� K possible branches, while keeping the substitution Sbkijt ¼ 0 8i, j, k

and t4 1.The resulting solution to each of these sub-problems gives a lower bound on the

optimal objective function value of the original problem. Among the resulting I� J� K

sub-problems, branch from the one with the lowest value of the objective function, as this

is the most promising node (ties are broken arbitrarily).


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Step 3: Repeat Step 2 for t ¼ 2, 3, . . . ,T with each value of t corresponding to one levelin the tree. Again, there would be I� J� K possible branches from each node. Clearly,the lower bound on the optimal objective function value increases as we go down the treesince larger portions of the major setup cost are being accounted for.

Step 4: The solution with the minimum objective function value obtained at the lowestlevel of the tree (corresponding to t ¼ T) is called the incumbent, which is used as an upperbound during the search of the unexplored branches of the tree. The search continues withother branches and the value of this incumbent is compared with the bounds obtained ateach node in order to make the fathoming decisions. If the value of the objective functioncalculated at a particular node is larger than the incumbent, that branch is fathomed.Otherwise, the value of the incumbent is updated whenever a lower value incumbent isattained, and then the new incumbent is used to make the fathoming decisions.

It should be noted that at each node of the tree, the MILP model is solved withoutEquation (3) but it is this constraint that drove the branching scheme in the first place.The tree exploring strategy is depth-first since this allows a faster recovery of anincumbent, which can be used to make the fathoming decisions. The width of the treedepends on the values of I, J and K, and the number of different levels is equal to Tþ 1.Note that the obtained MILP models are directly solved using AMPL/CPLEX 11.0 solver.

For instance, Figure 3 depicts a partial tree resulting from the application of theproposed B&B algorithm for a small problem instance (I¼K¼ 2, J¼ 3 and T¼ 4).Implementing the algorithm yields a tree with five levels (since Tþ 1¼ 5) and 12 nodes(since I� J� K ¼ 12) at each level starting from level 2. The nodes shown with a solid lineare the ones yielding the minimum objective function value (i.e., most promising) amongall other nodes in the same level. The branches in the solid line represent the pathconnecting these nodes which leads to the incumbent at the bottom of the tree. Hence,the values of the major setup variables corresponding to the incumbent in Figure 3 areSb2221 ¼ Sb1232 ¼ Sb2113 ¼ Sb2224 ¼ 1. The incumbent value can now be used to make the

2111 1Sb = 1

121 1Sb = 2221 1Sb = 1

231 1Sb =1111 1Sb = 2

231 1Sb =

2112 1Sb = 1

122 1Sb = 2222 1Sb = 1

232 1Sb =1112 1Sb = 2

232 1Sb =

2113 1Sb = 1

123 1Sb = 2223 1Sb = 1

233 1Sb =1113 1Sb = 2

233 1Sb =

2114 1Sb = 1

124 1Sb = 2224 1Sb = 1

234 1Sb =1114 1Sb = 2

234 1Sb =

0kijtSb =, , ,i j t k∀

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

Figure 3. Partial tree resulting from applying the proposed B&B algorithm for a small probleminstance.


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fathoming decisions for the unexplored branches of the tree, which could eventually resultin an optimal solution different from the current incumbent (indicating that a lower valueincumbent has been found).

At this point, a clear distinction has to be made between the classical B&B algorithmand the one proposed in this paper. The classical B&B algorithm utilises LP-relaxation ofthe binary variables Sbk

ijt in order to obtain the bound at the initial node (i.e., top-mostlevel of the tree). Such a relaxation does not resolve either bilinearity or non-convexityembedded in the model, and hence the resulting model is still not directly solvable usingoptimisation software. In fact, the relaxed version of the model is now ‘pure bilinear’ (twocontinuous variables are multiplied by one another) instead of being ‘mixed-integerbilinear’, and there exists a reformulation-linearisation technique established in theliterature for this class of bilinear models (Sherali and Alameddine 1992). Table 3 providesmore insights into the differences between the classical B&B algorithm and the modifiedone.

6. Experimental analysis

At this juncture, a comparison between the performance of the proposed B&B algorithmand the generic linearisation approach discussed in Section 5.1 is due. For a realisticproblem size (I¼T¼ 5, J¼ 7 and K¼ 2), the linearisation approach adds 350 variablesand 700 constraints to the original model. On the other hand, the B&B based algorithminvolves the solution of a reduced size MILP at each node (350 less binary variables and355 less constraints as compared to the original problem).

Table 3. A comparison between the classical B&B algorithm and the modified B&B algorithmproposed in this paper.

Classical B&B Modified B&B

Branching � Two branches emanate from eachnode corresponding to one of thebinary variables being assigned avalue of either zero or one (exceptat an incumbent where no furtherbranching takes place).

� Hence, the number of nodes at anylevel of the tree is twice that of thehigher level.

� Utilises constraint (3) to obtainI�J�K branches from each node,where each branch corresponds toone of the binary variables beingequal to one.

� Each level in the tree corresponds toa single t value.

� This branching scheme results in atree with I�J�K nodes at each levelof the tree.

Bounding Relaxes the binarity restriction on allbinary variables (except the branchingvariable) and allows those variablesto assume any value between zero andone (i.e., LP relaxation).

Assumes the values of all binary vari-ables (other than the branching vari-able) are set to zero (those in the samelevel are set to zero due to constraint(3) and others to obtain the boundand reduce the original problem toa MILP).

Incumbent Corresponds to an integer solutionobtained at any node (this mightoccur at any level of the tree).

Obtained only at the lowest level of thetree, since that is when constraint (3)is satisfied.


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Both algorithms were coded using AMPL programming language (Fourer et al. 2003)

and solved using CPLEX 11.0 solver, where the solver option is set to solve integer

problems using the built-in branch and cut algorithm. For the sake of comparison,

10 different problem instances are tested and each problem instance is solved to optimality

under the same set of input parameters for both algorithms. The numerical experiments

are implemented on a single CPU with 4-2.2 GHz AMD Opteron 64-bit processors and

16 GB RAM. The values for the different input parameters are generated within certain

range of intervals, as shown in Table 4.

Table 5. Numerical comparison between the linearisation approach and the proposed B&Balgorithm.

Probleminstance

Problem size(I� J�T�K)

Linearisation Modified branch & bound

% solutiontime

savingsNo. ofvariables

No. ofconstraints

Solutiontime(sec.)

No. ofvariables1

No. ofconstraints2

Solutiontime(sec.)

1 (1� 2� 2� 2) 74 140 0.42 58 106 8.59 –2 (1� 3� 3� 2) 159 297 0.71 123 222 106.32 –3 (2� 3� 3� 2) 267 507 1836 195 360 1501 184 (2� 3� 4� 2) 358 678 17,961 262 482 9305 485 (3� 4� 4� 2) 644 1232 94,027 452 844 68,113 286 (4� 4� 4� 2) 828 1592 125,691 572 1076 88,476 307 (4� 5� 4� 2) 1010 1946 169,829 690 1302 106,177 378 (5� 5� 4� 2) 1234 2386 * 834 1582 121,830 4329 (5� 6� 4� 2) 1456 2820 * 976 1856 134,252 42510 (5� 7� 5� 2) 2101 4071 * 1401 2666 174,514 43

Notes:1Number of variables in the MILP solved at each node; 2Number of constraints in the MILPsolved at each node; *Code execution was interrupted after 50 hours of run time with no resultsobtained.

Table 4. Selected range of values for input parametersin the test problems.

Input parameter Range of values

CRkit (550, 850)

ORkit (2000, 2800)

IRkit (15, 25)

PCijt (10, 40)POt (150, 450)SCt (400, 1000)IFk

jt (30, 45)BCk

jt (25, 35)

SPkjt (1500, 2000)

Mkit (150, 300)

�ij (0.82, 0.98)�ij (49, 58)Dk

jt (0, 120)STij (0.25, 1.0)


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The obtained results for the 10 problem instances are reported in Table 5. It shouldbe noted that the generation of the input parameters within different ranges of intervalswould influence the time it takes both algorithms to render an optimal solution. As can beseen in Table 5, the linearisation approach attains the optimal solution in less time ascompared to the B&B algorithm for small problem instances (problems 1 and 2). However,as the problem size increases, the B&B algorithm tends to outperform the correspondinglinearisation approach, although a larger problem size yields a bigger tree with, mostlikely, more nodes to be explored. This time reduction may be attributed to two reasons.First, the binary variables Sbk

ijt are considered as parameters at each node of theB&B algorithm while they remain as decisions variables in the linearisation approach.Second, the constraints added through the linearisation approach (constraints (4.1)–(4.3))are more involved as they all contain either the binary variable Sbk

ijt or the integer variableSdk

ijt or even both, which requires longer solution time from the IP solver. The savingsin the solution time could amount to 48% as shown in the last column of the table.

7. Conclusion and future research directions

This paper presented an operational MIBLP model for production planning purposes atthe master production schedule level in a steel rolling mill. The model studies the combinedeffect of practical constraints encountered on a daily basis, such as scrap rate, complexsetup time structure, overtime, backlogging and product substitution, on the planningdecisions. As this class of models is not directly solvable using off-the-shelf optimisationpackages, a modified branch-and-bound solution methodology was developed. Thealgorithm utilises the special problem structure to minimise the number of branches andobtain the bound at each node. It also has the ability to solve realistic problem sizesinvolving five different billets, seven different rebars, two steel grades with a planninghorizon of one week (five working days). Upon conducting numerical comparisons, thealgorithm outperforms one of the classical linearisation approaches typically used to solvethese models, with a reduction in the solution time of up to 48%. Hence, our algorithmprovides an efficient alternative for solving bilinear models in which the number ofpossible combinations for the values of the complicating binary variables is limited.

There exist several areas to which the proposed B&B based algorithm is applicable.Examples include, but not limited to: (1) small-bucket dynamic lot-sizing problems wherea single product may be produced in each time period; (2) portfolio management problemswhere a new investment opportunity is decided upon at the beginning of each year; and(3) vendor selection problems where a product may be solely supplied by a single sourcefor a specific time period.

The future effort will be directed towards tackling the above problem in a rollinghorizon context which allows for the uncertainties associated with the demand and theproduction capacity to be accounted for. This can be better captured through the useof fuzzy tools by which experts opinion and subjective judgment can be incorporated intothe decision making process.

Acknowledgment

The authors would like to acknowledge the support of the Natural Sciences and EngineeringResearch Council of Canada under grant number RGPIN 184143. We are also thankful to twoanonymous referees for their constructive comments.


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References

Absi, N. and Kedad-Sidhoum, S., 2008. The multi-item capacitated lot sizing problem with

setup times and shortage costs. European Journal of Operational Research, 185 (3), 1351–1374.Adams, W.P. and Sherali, H.D., 1990. Linearization strategies for a class of zero-one mixed integer

programming problems. Operations Research, 38 (2), 217–226.

Adams, W.P. and Forrester, R.J., 2007. Linear forms of nonlinear expressions: new insights on old

ideas. Operations Research Letters, 35 (4), 510–518.

Alfieri, A., Brandimarte, P., and D’Orazio, S., 2002. LP-based heuristics for the capacitated

lot-sizing problem: the interaction of model formulation and solution algorithm. International

Journal of Production Research, 40 (2), 441–458.Avriel, M. and Golany, B., eds., 1996. Mathematical programming for industrial engineers. 1st ed.

New York: Marcel Dekker.

Balakrishnan, A. and Geunes, J., 2000. Requirements planning with substitutions: exploiting

bill-of-materials flexibility in production planning. Manufacturing and Service Operations

Management, 2 (2), 166–185.Bassok, Y., Anupindi, R., and Akella, R., 1999. Single-period multi-product inventory models with

substitution. Operations Research, 47 (4), 632–642.Bitran, G.R. and Yanasse, H.H., 1982. Computational complexity of the capacitated lot size

problem. Management Science, 28 (10), 1174–1186.

Bitran, G.R. and Dasu, S., 1992. Ordering policies in an environment of stochastic yields and

substitutable demands. Operations Research, 40 (5), 999–1017.

Chen, M. and Wang, W., 1997. A linear programming model for integrated steel production and

distribution planning. International Journal of Production and Operations Management, 17 (6),

592–610.Denton, B., Gupta, D., and Jawahir, K., 2003. Managing increasing product variety at integrated

steel mills. Interfaces, 33 (2), 41–53.

Dutta, G. and Fourer, R., 2001. A survey of mathematical programming applications in integrated

steel plants. Manufacturing and Service Operations Management, 3 (4), 387–400.

Fabian, T., 1958. A linear programming model of integrated iron and steel production. Management

Sciences, 4 (4), 415–449.Fourer, R., Gay, D.M., and Kernighan, B.W., 2003. AMPL: a modeling language for mathematical

programming. 2nd ed. Pacific Grove, CA: Brooks/Cole–Thomson Learning.Glover, F., 1975. Improved linear integer programming formulations of nonlinear integer problems.

Management Science, 22 (4), 455–460.Gupta, S.M. and Brennan, L., 1992. Heuristic and optimal approaches to lot-sizing incorporating

backorders: an empirical evaluation. International Journal of Production Research, 30 (12),

2813–2824.Haase, K., 1994. Lot sizing and scheduling for production planning. Berlin: Springer.

Hindi, K.S., Fleszar, K., and Charalambous, C., 2003. An effective heuristic for the CLSP with setup

times. Journal of the Operational Research Society, 54 (5), 490–498.

Jans, R. and Degraeve, Z., 2007. Meta-heuristics for dynamic lot sizing: a review and

comparison of solution approaches. European Journal of Operational Research, 177 (3),

1855–1875.Jans, R. and Degraeve, Z., 2008. Modeling industrial lot sizing problems: a review. International

Journal of Production Research, 46 (6), 1619–1643.

Karimi, B., Fatemi Ghomi, S.M.T., and Wilson, J.M., 2003. The capacitated lot sizing problem:

a review of models and algorithms. The International Journal of Management Science, 31 (5),

365–378.Kerkkanen, A., 2007. Determining semi-finished products to be stocked when changing the

MTS-MTO policy: case of a steel mill. International Journal of Production Economics,

108 (1–2), 111–118.


Dow

nloa

ded

by [

IIT

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ian

Inst

itute

of

Tec

hnol

ogy

- M

umba

i] a

t 04:

20 1

7 Se

ptem

ber

2015

Land, A.H. and Doig, A.G., 1960. An automatic method for solving discrete programmingproblems. Econometrica, 28 (3), 497–520.

Leyffer, S., 2001. Integrating SQP and branch-and-bound for mixed integer nonlinear programming.Computational Optimization and Applications, 18 (3), 295–309.

Li, H. and Shang, J., 2001. Integrated model for production planning in a large iron and steelmanufacturing environment. International Journal of Production Research, 39 (9), 2037–2062.

Li, Y., Chen, J., and Cai, X., 2006. Uncapacitated production planning with multiple product types,

returned product remanufacturing and demand substitution. OR Spectrum, 28 (1), 98–122.Li, Y., Chen, J., and Cai, X., 2007. Heuristic genetic algorithm for capacitated production planning

problems with batch processing and remanufacturing. International Journal of Production

Economics, 105 (2), 301–317.Manne, A.S., 1958. Programming of economic lot sizes. Management Science, 4 (2), 115–135.Miller, H.M. and Yang, M., 1994. Lagrangian heuristics for the capacitated multi-item lot sizing

problem with backordering. International Journal of Production Economics, 34 (1), 1–15.Ozdamar, L. and Bozyel, M., 2000. The capacitated lot sizing problem with overtime decisions and

setup times. IIE Transactions, 32 (11), 1043–1057.Peterson, C.C., 1971. A note on transforming the product of variables to linear forms in linear

programs. Working paper. Purdue University, West Lafayette, IN.Rajam, K. and Tang, C.S., 2001. The impact of product substitution on retail merchandising.

European Journal of Operational Research, 135 (3), 582–601.

Sasidhar, B. and Achary, K.K., 1991. A multiple arc network model of production planningin a steel mill. International Journal of Production Economics, 22 (3), 195–202.

Sharma, A. and Sinha, K., 1991. Product mix optimization: a case study of integrated steel plants

of SAIL. OPSEARCH, 28, 188–201.Sherali, H. and Alameddine, A., 1992. A new reformulation-linearization technique for bilinear

programming problems. Journal of Global Optimization, 2 (4), 379–410.Tang, L., et al., 2001. A review of planning and scheduling systems and methods for integrated steel

production. European Journal of Operational Research, 133 (1), 1–20.Trigeiro, W.W., Thomas, L.J., and McClain, J.O., 1989. Capacitated lot sizing with setup times.

Management Science, 35 (3), 353–366.

Wagner, H.M. and Whitin, T.M., 1958. Dynamic version of the economic lot size model.Management Science, 5 (1), 89–96.

Webster, F.M., 1989. A backorder version of the Wagner-Whitin discrete demand EOQ algorithm

by dynamic programming. Production and Inventory Management Journal, 30 (4), 1–5.Zanoni, S. and Zavanella, L., 2005. Model and analysis of integrated production-inventory system:

the case of steel production. International Journal of Production Economics, 93–94, 197–205.


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