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Research Article ISSN 2229 – 3795

ASIAN JOURNAL OF MANAGEMENT RESEARCH 297 Volume 2 Issue 1, 2011

Optimization of material requirement planning by Goal programming model Davood Gharakhani

Faculty of Management and Accounting, Islamic Azad University (IAU), Qazvin Branch, Qazvin, Iran

[email protected]

ABSTRACT

Material requirement planning (MRP) is a plan for the production and purchase of the components used in making items in the master production schedule. It shows the quantities needed and when manufacturing intends to make or use them. In this paper, as an extension of work reported by Yenisey (2006), where optimization of material flow in MRP had been presented, a goal programming model is used to minimize production cost, minimize holding cost and minimize costs of the extra time used by resources, and the costs of the lazy time of resources. The aim of this paper is to demonstrate the usefulness and significance of the Goal programming model for Optimization of Material Requirement Planning. A set of real data from an automobile Gearbox manufacture is used to test the effectiveness and the efficiency of the proposed model. The model of the example case is solved and the computational results are given. The corresponding results show that the proposed models can help manufacturers make better decisions when they have multiple objectives.

Keywords: Material requirement planning, Goal programming, Optimization, capacity of resources

1. Introduction

Traditionally, manufacturing companies have controlled their parts through the reorder point (ROP) technique. Gradually, they recognized that some of these components had dependent demand, and material requirements planning (MRP) evolved to control the dependent items more effectively. MRP has been a very popular and widely used multilevel inventory control method since 1970s. The application of this popular tool in materials management has greatly reduced inventory levels and improved productivity (Wee and Shum, 1999).The introduced MRP was the first version of MRP system, named as Materials Requirements Planning (MRP I). Later, several MRP systems were extended into other versions including Manufacturing Resources Planning (MRP II) and Enterprise Resources Planning (ERP) (Browne et al., 1996).

MRP is a commonly accepted approach for replenishment planning in major companies. The MRPbased software tools are accepted readily. Most industrial decision makers are familiar with their use. The practical aspect of MRP lies in the fact that this is based on comprehensible rules, and provides cognitive support, as well as a powerful information system for decision



making. Some instructive presentations of this approach can be found in Baker (1993), Zipkin (2000), Tempelmeier (2006), Dolgui and Proth (2010) and Graves (2011).

Material requirements planning which is a method based on planning the requirements according to the master production schedule (MPS) which is prepared depending on customers’ demands. Two basic data are necessary for MRP: (1) the MPS, and (2) Bill of materials (BOM). MPS is a plan showing the product which will be produced when and in what quantity based on forecasting or received customer orders. BOM shows which subcomponent or raw material is used for which product and in what quantity. Required material quantities are calculated by hierarchically multiplying the production quantities in MPS by unit usage coefficients in BOM.MRP determines the quantity and timing of the acquisition of dependent demand items needed to satisfy master schedule requirements. One of its main objectives is to keep the due date equal to the need date, eliminating material shortages and excess stocks. MRP breaks a component into parts and subassemblies, and plans for those parts to come into stock when needed. Material requirement planning systems help manufactures determine precisely when and how much material to purchase and process based upon a timephased analysis of sales orders, production orders, current inventory and forecasts. They ensure that firms will always have sufficient inventory to meet production demands, but not more than necessary at any given time.MRP will even schedule purchase orders and/or production orders for Justintime receipt. The aim of this paper is to demonstrate the usefulness and significance of the Goal programming model for Optimization of Material Requirement Planning. In production planning, attention the capacity of available resources is very important. Therefore, in this paper, capacities of resources were added to the model proposed by Yenisey. In fact, for production in different periods of planning horizon should be observed capacity of available resources. This research considers capacity of available of two types of sources, these sources are respectively :( 1) Manpower, (2) Machinery. The goal programming technique appears to be an appropriate, powerful, and flexible one for decision analysis to help the troubled modern decision maker who is burdened with achieving multiple conflicting objectives under complex environmental constraints (Schniederjans, 1995; Tamiz et al., 1998; Aouni and Kettani, 2001).

However, application of GP to the reallife problems may be faced with two important difficulties. One of which is expressing the decision making's vague goals and/or constraints mathematically and the second is the need to optimize all goals simultaneously and as such the solution would be the best for the decision making. This paper is organized as follows. Firstly, in Section 2, presents the Research background. In Section 3, presents the Goal programming approach. Then, with use a mathematical programming model as proposed by Yenisey, a Goal programming model for optimization of MRP is formulated in Section 4. Section 5 shows the architecture used to implement and solve the proposed model with use real data from an automobile Gearbox manufacturer. In Section 6, this paper offers some conclusions and directions for further research.

2. Research background

Yenisey (2006) applied a flownetwork model and solved a linear programming method for MRP problems that minimized the total cost of the MRP system. Mula et al. (2006) provided a new linear programming model for mediumterm production planning in a capacityconstrained MRP with a multiproduct, multilevel, and multi period production system. Their proposed model



comprised three fuzzy sub models with flexibility in the objective function, market demand, and capacity of resources.

Wilhelm and Som (1998) present an inventory control approach for an assembly system with several types of components. Their model focuses on a single finished product inventory, so the interdependence between inventory levels of different components is once again neglected. Axsäter (2005) considers a multi level assembly system where operation times are independent random variables. The objective is to choose starting times (release dates) for different operations in order to minimize the sum of the expected holding and backlogging costs. The paper (Louly and Dolgui, 2002) considers the case of the objective function minimizing the sum of average holding and backlogging costs. While Louly et al. (2008) study the case when backlogging cost is replaced with a service level constraint. The obtained models for optimal planned lead times represent generalizations of the discrete Newsboy model.

Kanet and Sridharan (1998) examined late delivery of raw materials, variations in process lead times, interoperation move times and queue waiting times in MRP controlled manufacturing environment. To model such environment, they represented demand by interarrival time rather than defined from the master production schedule.

Kumar (1989) studies a single period model (one assembly batch) for a multicomponent assembly system with stochastic component lead times and a fixed assembly due date and quantity. The problem is to determine the timing of each component order so that the total cost composed of the component holding and product tardiness costs is minimized. Chauhan et al. (2009), presents an interesting singleperiod model. Their approach considers a punctual fixed demand for one finished product. Multiple types of components are needed to assembly this product. The objective is to determine the ordering time for each component such as to minimize the sum of expected holding and backlogging costs.

Van Donselaar and Gubbels (2002) compare MRP and line requirements planning (LRP) for planning orders. Their research basically focuses on minimizing the system inventory and system nervousness. They also discuss and propose LRP technique to achieve their goals. Minifie and Davies (1990) developed a dynamic MRP controlled manufacturing system simulation model to study the interaction effects of demand and supply uncertainties. These uncertainties were modelled in terms of changes in lotsize, timing, planned orders and policy fence on several system performance measures, namely late deliveries, number of setups, ending inventory levels, component shortages and number of exception reports. Billington et al. (1983) suggest a mathematical programming approach for scheduling capacity constrained MRP systems. They propose a discretetime, mixedinteger linear programming formulation. In order to reduce the number of variables, and thus the problem size, they introduce the idea of product structure compression.

3. Goal programming approach

Goal programming (GP) is a multiobjective optimization tool. Goal programming models are very similar to linear programming models. However, there are some differences. While linear programs have only one objective to be maximized or minimized, goal programs consider multiple goals that are often in conflict with each other. As goal programs have several goals



with tradeoffs, in most instances, all of these goals cannot be realized at the same time. Thus, goal programs aim to minimize the deviations among the target goals and the actual results considering the priorities assigned. The objective function of a goal programming model is expressed in terms of the deviations from the desired goals and deviations from the desired goals are penalized (Cohon, 2004).

The general aim of GP is the optimization of several conflicting goals precisely defined by the decision maker(s) by minimizing the deviations from the target values. The original objectives are expressed as a linear equation with target values and two auxiliary variables. These two auxiliary variables represent under achievement of the target value by negative deviation (d − ) and over achievement of the target value by positive deviation (d + ). If the desire is not to underachievement the goal, d − should be driven to zero. To the contrary, if d + is driven to zero, the overachievement of the goal will not be realized. The unwanted deviations between target values of objectives are minimized hierarchically. Hence, the goals of primary importance are satisfied first, and it is only then the goals of second importance are considered, and so forth.

It may be noted that the structure of a GP model involves two types of constraint: system and goal constraints. The system constraints are those which are more restrictive in nature and have to be satisfied before the goal constraints, as they represent the existing capabilities, rather than what we would like to achieve. From the above discussion, it can be deduced that deviational variables are mutually exclusive.

This relationship is mathematically expressed as:

The steps needed to structure a GP model are threefold:

(1) Goals are identified and expressed as constraints. (2) Goals are analysed to determine the correct deviational variables needed for them, di , di + , or both. (3) A hierarchy of importance among goals is established by assigning to each of them a pre emptive priority factor, Pj.

These preemptive priority factors reflect the hierarchical relationships in such a way that P1 represents the highest priority, P2 the second highest, and so on. In other words, the Ps indicates a simple ordinal ordering of the goals.

Once the above steps are completed, the problem can be quantified as a GP model as follows:



4. Model formulation

In this paper, a goal programming model is used to solve the multiobjective Material requirement planning problem. The company has indicated three goals to be achieved:

(1) Minimization of production cost (production goal); (2) Minimization of holding cost (holding goal); and (3) Minimization of costs of the extra time used by sources, and the costs of the lazy time of sources (sources goal).

In the following, the parameters and the variables for the model are defined. Mathematical formulation of the proposed model, including various goal constraints related to the respective goals, system constraints, and the achievement function are also described. Model of this paper can be interpreted at three hierarchical levels. The first level is for products, second is for sub components, and finally, the third one is for raw materials.

A threelevel hierarchical product tree made of products, subcomponents and raw materials is used for the research. Product supplies in every period of planning horizon trigger the system as pulling required products in order to satisfy the demands. Subcomponents according to these supplies and raw materials depending on subcomponents flow through the network. The plant may have some initial inventory on hand and a policy of transferring some inventory from a period to others following. An upper bound made of zero can be put on these flows in order not to allow onhand inventory in stock policy. The initial onhand inventories’ flows have upper bounds representing the amounts being held in stock. If the decisionmaker wants to hold inventory at the end of planning horizon, he/she can put lower bounds on onhand inventory flows going beyond the last period.

The parameter kr must be defined as follows for each raw material before the formulation is given. kr =max CLc+RLr|(c, r) BOMC.

4.1 Notations In this section, the notation used by Yenisey (2006) is given, which is applied to minimize total costs in the MRP system.

Indexes:

p index for products i,j index for periods c index for subcomponents r index for raw materials

Costs:

pcc unit cost of product p including all production costs other than raw materials’ and subcomponents’ purchasing costs



ccc unit cost of subcomponent c rcc unit cost of raw material r php holding cost of product p chc holding cost of subcomponent c rhr holding cost of raw material r pihp holding cost of product p for initial period cihc holding cost of subcomponent c for initial period rihr holding cost of raw material r for initial period

Parameters:

PDp,i demand for product p in period i TPISp total initial onhand inventory level of product p TCISc total initial onhand inventory level of subcomponent c TRISr total initial onhand inventory level of raw material r TPFSp total final onhand inventory level of product p TCFSc total final onhand inventory level of subcomponent c TRFSr total final onhand inventory level of raw material r cup,c unit usage of subcomponent c for product p ruc,r unit usage of raw material r for subcomponent c CLc lead time of subcomponent c RLr lead time of raw material r

Variables: Basic decision variables:

PPp,j quantity of product p that must be procured in period j CPc,j quantity of subcomponent c that must be procured in period j RPr,j quantity of raw material r that must be procured in period j

Secondary decision variables:

PIp,i,j inventory of product p carried from period i to period j CIc,i,j inventory of subcomponent c carried from period i to period j RIr,i,j inventory of raw material r carried from period i to period j PISp,i quantity of product p to be used in period i from initial onhand inventory CISc,i quantity of component c to be used in period i from initial onhand

inventory RISr,i quantity of raw material r to be used in period i from initial onhand

inventory PFSp,i quantity of product p to be saved in period i for onhand inventory beyond

planning horizon CFSc,i quantity of component c to be saved in period i for onhand inventory

beyond planning horizon RFSr,i quantity of raw material r to be saved in period i for onhand inventory



beyond planning horizon

Side decision variables:

PSp,i supply of product p in period i CDp,c,i demand for subcomponent c to be used in product p in period i RDc,r,i demand for raw material r to be used in subcomponent c in period i Sets: P set of products C set of subcomponents R set of raw materials I set of periods in planning horizon

BOMP =(p,c)|p P,c C, c is a subcomponent used in product PSet of Billof materials for relations between subcomponents and products BOMC = (c, r) |c C, r R, r is a raw material used in subcomponent c Set of Billof materials for relations between raw materials and subcomponents.

Additional notation:

In addition to the abovementioned notation some additional notation is applied in this section in order to incorporate the limits of capacity of sources into model and also, change the flownetwork model to goal programming model.

Indexes:

s index for sources j index for periods

Costs:

Ctups,j Under time hour cost of the source s on period j Ctops,j Overtime hour cost of the source s on period j

Parameters:

APp,s Required time of the source s for unit of production of the product p CAPs,j Available capacity of the source s in the period j

Variable:

TUPs,j Under time hours of the source s on period j TOPs,j Overtime hours of the source s on period j

Auxiliary variables

dp − deviation variable of underachievement of Goalp (the production goal)



dp + deviation variable of overachievement of Goalp (the production goal) dh − deviation variable of underachievement of Goalh (the holding goal) dh + deviation variable of overachievement of Goalh (the holding goal) ds − deviation variable of underachievement of Goals (the sources goal) ds + deviation variable of overachievement of Goals (the sources goal)

4.2 Goal constraints and objective functions

4.2.1 Goal 1: The production goal

The company evaluates production cost for all periods and there is an expected production cost target to be achieved at the end of the planning horizon. The production goal constraint is illustrated below;

This first component in constraint (1) is the total production cost of products including all production costs other than raw materials’ and subcomponents’ purchasing costs. The second component is the total subcomponent cost including purchasing cost for items procured from outside while production cost for those produced within the plant. The third component is the total material cost including purchasing cost for items procured from outside while production cost for those produced within the plant. Parameter GOALp denotes the acceptable production cost set by management. A positive deviational variable, dp + , represents the overachievement of the aspiration level of production goal and a negative deviational variable, dp − , represents the underachievement of the aspiration level of production goal. This gives dp + . dp − = 0.

4.2.2 Goal 2: The holding goal

The company evaluates holding cost for all periods and there is an expected holding cost target to be achieved at the end of the planning horizon. The goal constraint is formulated below.



Constraint (2) pursues to achieve the holding costs goal. The total holding costs include, holding costs of products, subcomponents and raw materials. Holding costs for periods in the planning horizon, php, chc and rhr, include only inventory carrying cost. They do not include any production or purchasing cost. These holding costs are also valid for the inventories being saved for the periods beyond the last period of the planning horizon. However, the holding costs defined for the inventories being carried from the periods before the first period of the planning horizon, pihp, cihc and rihr include all costs made of both production or purchasing cost and inventory carrying costs after they were procured. Parameter GOALh denotes the acceptable holding cost set by management. A positive deviational variable, dh + , represents the over achievement of the aspiration level of holding goal and a negative deviational variable, dh − , represents the underachievement of the aspiration level of holding goal. This gives dh + . dh − = 0.

4.2.3 Goal 3: The sources goal

Management is pursuing to achieve an acceptable level for costs of the extra time used by sources, and the costs of the lazy time of sources. Therefore, the sources goal constraint is formulated below;

The first component in constraint (3) represents the total costs of the extra time used by sources, and the costs of the lazy time of sources. Parameter GOALs denotes the acceptable sources cost set by management. A positive deviational variable, ds + , represents the overachievement of the aspiration level of sources goal and a negative deviational variable, ds − , represents the under achievement of the aspiration level of sources goal. This gives ds + . ds − = 0.

The achievement function of the multiobjective material requirement planning problem is formulated as follows:

Minimize dp + , dh + , ds +

4.3 Constraints



Equations (4) to (15) were used as mentioned by Yenisey. However, since goal programming model requires all the equations that were used previously by Yenisey, all the constraints used in the singleobjective optimization case are also retained and additional equations (i.e. equations (1) to (3), which are required for minimizing the total cost and achieve to goals. also, since The production in every period is limited by the availability of a group of shared resources. equation (16) considers the limits of capacity of these resources.

This research considers capacity of available of two types of sources, these sources are respectively :( 1) Manpower, (2) Machinery. The decision variables TUPs, j and TOPs, j are not limited by any established parameter but are penalized with the corresponding costs in the equation (3). The limitation of these variables for specific applications could be easily considered taking into account that if those limits are exceeded the solution of the model could be no feasible. It is necessary to underline that an adjustment is made for indices used in model in order to make them consistent, i.e. min I=max CLc+RLr|(c, r) BOMC +1. This equation is used as



a basis for the adjustments made in Eqs. (13) and (14). Hence, it is guaranteed that only positive indices are used and the first period of planning horizon’s index is one.

5. Example

A threelevel hierarchical model made is as an example for this research. The BOM structure for the example case is given in Fig. 1. The example consists of one product, six subcomponents and two raw materials. In Fig. 1, numbers in brackets by nodes represent the lead times for the items, and the numbers in parentheses near arcs represent the unit usage of the subitems. Additionally, the full formulation of the example is given in Appendix A.

The structure of the example has very high flexibility in order to include more periods, products and subitems. The model of the example case was solved by Lindo/PC running on a notebook. The optimum achieved at 104 rd iteration. The model made of 142 constrains and 278variables. The number of non zeroes was 732. The solution time was less than 1 s.

Figure1: The BOM of the example case

Company has set target values, to be achieved under each of the three goals, they are: Production cost (4,500,000,000), holding cost (20,000,000) and sources cost (100,000,000). The optimum values of the basic decision variables and target values are summarized in the Table 1. The signs of ‘‘N/A’’ in the cells of the Table 1 mean that the supply for the intersection of corresponding period and item cannot exist due to the structure of the sample problem. The complete solution is provided in Appendix B.

In this run, the goals are to minimize the overachievement of Production cost, holding cost and sources cost. The positive deviation variable dp + is 28, 400,000. This shows that production cost



is more overachieved. The positive deviation variable dh + is 0. The goal for holding cost has been achieved. Finally, the positive deviation variable ds + is 8, 000,000. This shows that sources cost is more overachieved.

Table 1: Production/procurement volumes of the materials and Target value

6. Conclusion and scope for future research

The planning and use of the production resources and the planning for the provisioning/production of materials is a considerably complex process. Many manufacturing environments, such as the automobile industry, require this kind of planning for dozens of thousands of components and subassemblies. In this paper, as an extension of work reported by Yenisey, a goal programming model is presented to minimize production cost, minimize holding cost and minimize costs of the extra time used by resources, and the costs of the lazy time of resources. Since, the production in every period is limited by the availability of a group of shared resources. This paper considers the limits of capacity of these resources. This model can effectively find Production/procurement volumes of the materials. The model was implemented in an automobile Gearbox manufacturer cases successfully.

This study has some propositions for further research. (a)Types of uncertainty could be incorporated, such as, uncertainty in demand, uncertainty in lead times, uncertainty in resources capacity or uncertainty due to quality variations. (b) The proposed model in this paper can be the construction blocks for a decision making support system for production planning with imprecise data. (c) Further research can focus on a nonlinear goal programming model. (d) Further research can use from fuzzy coefficients in model.

periods1 2 3 4 5 6 7 8

Products 1 N/A N/A N/A N/A 1100 0

8500 9500 8000

Components 1 N/A N/A N/A 36500 0 0 0 N/A 2 N/A N/A N/A 28000 0 0 8000 N/A 3 N/A N/A N/A 35200 0 1200 0 N/A 4 N/A N/A N/A 36500 0 0 0 N/A 5 N/A N/A 36200 0 0 0 N/A N/A 6 N/A N/A 18900 0 0 0 N/A N/A

Raw materials 1 191500 0 0 8000 N/A N/A N/A N/A 2 N/A 191300 0 0 8000 N/A N/A N/A

Production cost (4,528,400,000), holding cost (20,000,000) and sources cost (108,000,000)

N/A: The value cannot exist due to the sample problem’s structure.



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Appendix A: The complete model of the sample problem

P = 1 c = 1, 2, 3, 4, 5, 6 r = 1, 2

BOMP= (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)

BOMC= (1, 1), (1, 2), (2, 1), (2, 2), (3, 1) (3, 2) (4, 1), (4, 2), (5, 1), (5, 2), (6, 1), (6, 2)

CL1=1, CL2=1, CL3=1, CL4=1, CL5=2, CL6=2

RL1=2, RL2=1

Min I=max CLc+RLr|(c, r) BOMC +1

= max CL1+RL1, CL1+RL2 , CL2+RL1 , CL2+RL2 , CL3+RL1 , CL3+RL2 , CL4+RL1 , CL4+RL2 ,

CL5+RL1 , CL5+RL2 , CL6+RL1 , CL6+RL2+1

=max3,2,3 ,2 ,3 ,2 ,3 ,2 ,4 ,3 ,4 ,3 +1=5

I= 5, 6, 7, 8 (Since the planning horizon consists of four periods)

k1=2, k2 = 2

Min dp + + dh + + ds +

Subject to

PP1,5+PP1,6+PP1,7+PP1,8+CP1,4+CP1,5+CP1,6+CP1,7+CP2,4+CP2,5+CP2,6+CP2,7+CP3,4+CP3,5+CP3,6+ CP3,7+CP4,4 + CP4,5 +CP4,6+CP4,7+CP5,3+CP5,4+CP5,5+CP5,6+CP6,3+CP6,4 + CP6,5+CP6,6+RP1,1+RP1,2 +RP1,3+RP1,4 +RP2,2 +RP2,3 +RP2,4 +RP2,5 dp + + dp − =GOALp

PI1,5,6+PI1,5,7+PI1,5,8+PI1,6,7+PI1,6,8+PI1,7,8+CI1,5,6+CI1,5,7+CI1,5,8+CI1,6,7+CI1,6,8+CI1,7,8+CI2,5,6+CI2,5, 7+CI2,5,8+CI2,6,7+CI2,6,8+CI2,7,8+CI3,5,6+CI3,5,7 +CI3,5,8+CI3,6,7+CI3,6,8+CI3,7,8+CI4,5,6+CI4,5,7+CI4,5,8+CI4,6,7+CI4,6,8+CI4,7,8+CI5,5,6 +CI5,5,7+CI5,5,8+CI5,6,7+CI5,6,8+CI5,7,8+CI6,5,6+CI6,5,7+CI6,5,8+CI6,6,7+CI6,6,8+CI6,7,8+RI1,3,4+RI1,3,5+R I1,3,6+RI1,4,5 +RI1,4,6+RI1,5,6+RI2,3,4+RI2,3,5+RI2,3,6 +RI2,4,5+RI2,4,6+RI2,5,6+PFS1,5+PFS1,6+PFS1,7+PFS1,8+CFS1,5+CFS1,6+CFS1,7 +CFS1,8+CFS2,5+CFS2,6+CFS2,7+CFS2,8+CFS3,5+CFS3,6+CFS3,7+CFS3,8+CFS4,5+CFS4,6+CFS4,7+ CFS4,8+CFS5,5 +CFS5,6+CFS5,7+CFS5,8+CFS6,5+CFS6,6+CFS6,7+CFS6,8+RFS1,3+RFS1,4+RFS1,5+RFS1,6+RFS2,3+ RFS2,4+RFS2,5+RFS2,6+PIS1,6 +PIS1,7+PIS1,8+CIS1,6+CIS1,7+CIS1,8+CIS2,6+CIS2,7+CIS2,8+CIS3,6+CIS3,7+CIS3,8+CIS4,6+CIS4,7 +CIS4,8+CIS5,6+CIS5,7+CIS5,8+CIS6,6+CIS6,7+CIS6,8+RIS1,4+RIS1,5+RIS1,6+RIS2,4+RIS2,5+RIS2,6+ TPIS1+TCIS1+TCIS2+TCIS3 +TCIS4+TCIS5+TCIS6+TRIS1+TRIS2 dh + + dh − = GOALh



TUP1,5+TUP1,6+ TUP1,7+TUP1,8+ TUP2,5+TUP2,6+ TUP2,7+ TUP2,8+ TOP1,5+TOP1,6+ TOP1,7+ TOP1,8+TOP2,5 +TOP2,6+ TOP2,7+ TOP2,8 ds + + ds − = GOALs

PS1,5+PI1,5,6+PI1,5,7+PI1,5,8+PFS1,5PP1,5PIS1,5=0 PS1,6+PI1,6,7+PI1,6,8+PFS1,6PI1,5,6PP1,6PIS1,6=0 PS1,7+PI1,7,8+PFS1,7PI1,5,7PI1,6,7PP1,7PIS1,7=0 PS1,8+PFS1,8PI1,5,8PI1,6,8PI1,7,8PP1,8PIS1,8=0

PS1, 5>=PD1, 5 PS1, 6>=PD1, 6 PS1, 7>=PD1, 7 PS1, 8>=PD1, 8

PIS1, 5+PIS1, 6+PIS1, 7+PIS1, 8TPIS1=0 PFS1, 5PFS1, 6PFS1, 7PFS1, 8+TPFS1=0

CD1, 1, 5PP1, 5=0 CD1, 2, 5PP1, 5=0 CD1, 3, 5PP1, 5=0 CD1, 4, 5PP1, 5=0 CD1, 5, 5PP1, 5=0 CD1, 6, 5PP1, 5=0 CD1, 1, 6PP1, 6=0 CD1, 2, 6PP1, 6=0 CD1, 3, 6PP1, 6=0 CD1, 4, 6PP1, 6=0 CD1, 5, 6PP1, 6=0 CD1, 6, 6PP1, 6=0 CD1, 1, 7PP1, 7=0 CD1, 2, 7PP1, 7=0 CD1, 3, 7PP1, 7=0 CD1, 4, 7PP1, 7=0 CD1, 5, 7PP1, 7=0 CD1, 6, 7PP1, 7=0 CD1, 1, 8PP1, 8=0 CD1, 2, 8PP1, 8=0 CD1, 3, 8PP1, 8=0 CD1, 4, 8PP1, 8=0 CD1, 5, 8PP1, 8=0 CD1, 6, 8PP1, 8=0

CD1,1,5+CI1,5,6+CI1,5,7+CI1,5,8+CFS1,5CP1,4CIS1,5=0 CD1,2,5+CI2,5,6+CI2,5,7+CI2,5,8+CFS2,5CP2,4CIS2,5=0 CD1,3,5+CI3,5,6+CI3,5,7+CI3,5,8+CFS3,5CP3,4CIS3,5=0 CD1,4,5+CI4,5,6+CI4,5,7+CI4,5,8+CFS4,5CP4,4CIS4,5=0 CD1,5,5+CI5,5,6+CI5,5,7+CI5,5,8+CFS5,5CP5,3CIS5,5=0



CD1,6,5+CI6,5,6+CI6,5,7+CI6,5,8+CFS6,5CP6,3CIS6,5=0 CD1,1,6+CI1,6,7+CI1,6,8+CFS1,6CI1,5,6CP1,5CIS1,6=0 CD1,2,6+CI2,6,7+CI2,6,8+CFS2,6CI2,5,6CP2,5CIS2,6=0 CD1,3,6+CI3,6,7+CI3,6,8+CFS3,6CI3,5,6CP3,5CIS3,6=0 CD1,4,6+CI4,6,7+CI4,6,8+CFS4,6CI4,5,6CP4,5CIS4,6=0 CD1,5,6+CI5,6,7+CI5,6,8+CFS5,6CI5,5,6CP5,4CIS5,6=0 CD1,6,6+CI6,6,7+CI6,6,8+CFS6,6CI6,5,6CP6,4CIS6,6=0 CD1,1,7+CI1,7,8+CFS1,7CI1,5,7CI1,6,7CP1,6CIS1,7=0 CD1,2,7+CI2,7,8+CFS2,7CI2,5,7CI2,6,7CP2,6CIS2,7=0 CD1,3,7+CI3,7,8+CFS3,7CI3,5,7CI3,6,7CP3,6CIS3,7=0 CD1,4,7+CI4,7,8+CFS4,7CI4,5,7CI4,6,7CP4,6CIS4,7=0 CD1,5,7+CI5,7,8+CFS5,7CI5,5,7CI5,6,7CP5,5CIS5,7=0 CD1,6,7+CI6,7,8+CFS6,7CI65,7CI1,6,7CP6,5CIS6,7=0 CD1,1,8+CFS1,8CI1,5,8CI1,6,8CI1,7,8CP1,7CIS1,8=0 CD1,2,8+CFS2,8CI2,5,8CI2,6,8CI2,7,8CP2,7CIS2,8=0 CD1,3,8+CFS3,8CI3,5,8CI3,6,8CI3,7,8CP3,7CIS3,8=0 CD1,4,8+CFS4,8CI4,5,8CI4,6,8CI4,7,8CP4,7CIS4,8=0 CD1,5,8+CFS5,8CI5,5,8CI5,6,8CI5,7,8CP5,6CIS5,8=0 CD1,6,8+CFS6,8CI6,5,8CI6,6,8CI6,7,8CP6,6CIS6,8=0

CIS1, 5+CIS1, 6+CIS1, 7+CIS1, 8TCIS1=0 CIS2, 5+CIS2, 6+CIS2, 7+CIS2, 8TCIS2=0 CIS3, 5+CIS3, 6+CIS3, 7+CIS3, 8TCIS3=0 CIS4, 5+CIS4, 6+CIS4, 7+CIS4, 8TCIS4=0 CIS5, 5+CIS5, 6+CIS5, 7+CIS5, 8TCIS5=0

CIS6, 5+CIS6, 6+CIS6, 7+CIS6, 8TCIS6=0

CFS1, 5CFS1, 6CFS1, 7CFS1, 8+TCFS1=0 CFS2, 5CFS2, 6CFS2, 7CFS2, 8+TCFS2=0 CFS3, 5CFS3, 6CFS3, 7CFS3, 8+TCFS3=0 CFS4, 5CFS4, 6CFS4, 7CFS4, 8+TCFS4=0 CFS5, 5CFS5, 6CFS5, 7CFS5, 8+TCFS5=0 CFS6, 5CFS6, 6CFS6, 7CFS6, 8+TCFS6=0

RD1, 1, 4CP1, 4=0 RD1, 1, 5CP1, 5=0 RD1, 1, 6CP1, 6=0 RD1, 1, 7CP1, 7=0 RD2, 1, 4CP2, 4=0 RD2, 1, 5CP2, 5=0 RD2, 1, 6CP2, 6=0 RD2, 1, 7CP2, 7=0 RD3, 1, 4CP3, 4=0 RD3, 1, 5CP3, 5=0 RD3, 1, 6CP3, 6=0 RD3, 1, 7CP3, 7=0



RD4, 1, 4CP4, 4=0 RD4, 1, 5CP4, 5=0 RD4, 1, 6CP4, 6=0 RD4, 1, 7CP4, 7=0 RD5, 1, 3CP5, 3=0 RD5, 1, 4CP5, 4=0 RD5, 1, 5CP5, 5=0 RD5, 1, 6CP5, 6=0 RD6, 1, 3CP6, 3=0 RD6, 1, 4CP6, 4=0 RD6, 1, 5CP6, 5=0 RD6, 1, 6CP6, 6=0 RD1, 2, 4CP1, 4=0 RD1, 2, 5CP1, 5=0 RD1, 2, 6CP1, 6=0 RD1, 2, 7CP1, 7=0 RD2, 2, 4CP2, 4=0 RD2, 2, 5CP2, 5=0 RD2, 2, 6CP2, 6=0 RD2, 2, 7CP2, 7=0 RD3, 2, 4CP3, 4=0 RD3, 2, 5CP3, 5=0 RD3, 2, 6CP3, 6=0 RD3, 2, 7CP3, 7=0 RD4, 2, 4CP4, 4=0 RD4, 2, 5CP4, 5=0 RD4, 2, 6CP4, 6=0 RD4, 2, 7CP4, 7=0 RD5, 2, 3CP5, 3=0 RD5, 2, 4CP5, 4=0 RD5, 2, 5CP5, 5=0 RD5, 2, 6CP5, 6=0 RD6, 2, 3CP6, 3=0 RD6, 2, 4CP6, 4=0 RD6, 2, 5CP6, 5=0 RD6, 2, 6CP6, 6=0

RD1,1,4+RD2,1,4+RD3,1,4+RD4,1,4+RD5,1,3+RD6,1,3+RI1,3,4+RI1,3,5+RI1,3,6+RFS1,3RP1,1RIS1,3=0 RD1,1,5+RD2,1,5+RD3,1,5+RD4,1,5+RD5,1,4+RD6,1,4+RI1,4,5+RI1,4,6+RFS1,4RI1,3,4RP1,2RIS1,4=0 RD1,1,6+RD2,1,6+RD3,1,6+RD4,1,6+RD5,1,5+RD6,1,5+RI1,5,6+RFS1,5RI1,3,5RI1,4,5RP1,3RIS1,5=0 RD1,1,7+RD2,1,7+RD3,1,7+RD4,1,7+RD5,1,6+RD6,1,6+RFS1,6RI1,3,6RI1,4,6RI1,5,6RP1,4RIS1,6=0 RD1,2,4+RD2,2,4+RD3,2,4+RD4,2,4+RD5,2,3+RD6,2,3+RI2,3,4+RI2,3,5+RI2,3,6+RFS2,3RP2,2RIS2,3=0 RD1,2,5+RD2,2,5+RD3,2,5+RD4,2,5+RD5,2,4+RD6,2,4+RI2,4,5+RI2,4,6+ RFS2,4RI2,3,4RP2,3RIS2,4=0 RD1,2,6+RD2,2,6+RD3,2,6+RD4,2,6+RD5,2,5+RD6,2,5+RI2,5,6+RFS2,5RI2,3,5RI2,4,5RP2,4RIS2,5=0 RD1,2,7+RD2,2,7+RD3,2,7+RD4,2,7+RD5,2,6+RD6,2,6+RFS2,6RI2,3,6RI2,4,6RI2,5,6RP2,5RIS2,6=0



RIS1, 3+RIS1, 4+RIS1, 5+RIS1, 6TRIS1=0 RIS2, 3+RIS2, 4+RIS2, 5+RIS2, 6TRIS2=0

RFS1, 3RFS1, 4RFS1, 5RFS1, 6+TRFS1=0 RFS2, 3RFS2, 4RFS2, 5RFS2, 6+TRFS2=0

PP1, 5+TUP1, 5TOP1, 5= CAP1, 5 PP1, 6+TUP1, 6TOP1, 6= CAP1, 6 PP1, 7+TUP1, 7TOP1, 7= CAP1, 7 PP1, 8+TUP1, 8TOP1, 8= CAP1, 8 PP1, 5+TUP2, 5TOP2, 5= CAP2, 5 PP1, 6+TUP2, 6TOP2, 6= CAP2, 6 PP1, 7+TUP2, 7TOP2, 7= CAP2, 7 PP1, 8+TUP2, 8TOP2, 8= CAP2, 8

Appendix B: The complete solution of the sample problem

Variable Value Variable Value Variable Value Variable Value

dp + 28400000 CI5,5,6 7700 CIS3,7 600 CD1,3,8 8000 dh + 0 CI5,5,7 9500 CIS3,8 0 CD1,4,8 8000 ds + 8000000 CI5,5,8 8000 CIS4,6 0 CD1,5,8 8000 PP1,5 11000 CI5,6,7 0 CIS4,7 500 CD1,6,8 8000 PP1,6 8500 CI5,6,8 0 CIS4,8 0 CIS1,5 500 PP1,7 9500 CI5,7,8 0 CIS5,6 800 CIS2,5 0 PP1,8 8000 CI6,5,6 7900 CIS5,7 0 CIS3,5 0 CP1,4 36500 CI6,5,7 0 CIS5,8 0 CIS4,5 0 CP1,5 0 CI6,5,8 0 CIS6,6 600 CIS5,5 0 CP1,6 0 CI6,6,7 0 CIS6,7 0 CIS6,5 0 CP1,7 0 CI6,6,8 0 CIS6,8 0 TCFS1 0 CP2,4 28000 CI6,7,8 8000 RIS1,4 0 TCFS2 0 CP2,5 0 RI1,3,4 0 RIS1,5 1000 TCFS3 0 CP2,6 0 RI1,3,5 200 RIS1,6 0 TCFS4 0 CP2,7 8000 RI1,3,6 0 RIS2,4 0 TCFS5 0 CP3,4 35200 RI1,4,5 0 RIS2,5 1200 TCFS6 0 CP3,5 0 RI1,4,6 0 RIS2,6 0 RD1,1,4 36500 CP3,6 1200 RI1,5,6 0 TPIS1 2000 RD1,1,5 0 CP3,7 0 RI2,3,4 0 TCIS1 500 RD1,1,6 0 CP4,4 36500 RI2,3,5 0 TCIS2 1000 RD1,1,7 0 CP4,5 0 RI2,3,6 0 TCIS3 600 RD2,1,4 28000 CP4,6 0 RI2,4,5 0 TCIS4 500 RD2,1,5 0 CP4,7 0 RI2,4,6 0 TCIS5 800 RD2,1,6 0 CP5,3 36200 RI2,5,6 0 TCIS6 600 RD2,1,7 8000 CP5,4 0 PFS1,5 0 TRIS1 1000 RD3,1,4 35200 CP5,5 0 PFS1,6 0 TRIS2 1200 RD3,1,5 0 CP5,6 0 PFS1,7 0 dh − 0 RD3,1,6 1200



CP6,3 18900 PFS1,8 0 TUP1,5 0 RD3,1,7 0 CP6,4 0 CFS1,5 0 TUP1,6 0 RD4,1,4 36500 CP6,5 0 CFS1,6 0 TUP1,7 0 RD4,1,5 0 CP6,6 0 CFS1,7 0 TUP1,8 0 RD4,1,6 0 RP1,1 191500 CFS1,8 0 TUP2,5 0 RD4,1,7 0 RP1,2 0 CFS2,5 0 TUP2,6 2000 RD5,1,3 36200 RP1,3 0 CFS2,6 0 TUP2,7 0 RD5,1,4 0 RP1,4 8000 CFS2,7 0 TUP2,8 4000 RD5,1,5 0 RP2,2 191300 CFS2,8 0 TOP1,5 13000 RD5,1,6 0 RP2,3 0 CFS3,,5 0 TOP1,6 7500 RD6,1,3 18900 RP2,4 0 CFS3,6 0 TOP1,7 12500 RD6,1,4 0 RP2,5 8000 CFS3,7 0 TOP1,8 7000 RD6,1,5 0 dp − 0 CFS3,8 0 TOP2,5 0 RD6,1,6 0 PI1,5,6 0 CFS4,5 0 TOP2,6 0 RD1,2,4 36500 PI1,5,7 0 CFS4,6 0 TOP2,7 1000 RD1,2,5 0 PI1,5,8 0 CFS4,7 0 TOP2,8 0 RD1,2,6 0 PI1,6,7 0 CFS4,8 0 ds − 0 RD1,2,7 0 PI1,6,8 0 CFS5,5 0 PS1,5 11000 RD2,2,4 28000 PI1,7,8 0 CFS5,6 0 PIS1,5 0 RD2,2,5 0 CI1,5,6 26000 CFS5,7 0 PS1,6 10500 RD2,2,6 0 CI1,5,7 0 CFS5,8 0 PS1,7 9500 RD2,2,7 8000 CI1,5,8 0 CFS6,5 0 PS1,8 8000 RD3,2,4 35200 CI1,6,7 17500 CFS6,6 0 TPFS1 0 RD3,2,5 0 CI1,6,8 0 CFS6,7 0 CD1,1,5 11000 RD3,2,6 1200 CI1,7,8 8000 CFS6,8 0 CD1,2,5 11000 RD3,2,7 0 CI2,5,6 7500 RFS1,3 0 CD1,3,5 11000 RD4,2,4 36500 CI2,5,7 9500 RFS1,4 0 CD1,4,5 11000 RD4,2,5 0 CI2,5,8 0 RFS1,5 0 CD1,5,5 11000 RD4,2,6 0 CI2,6,7 0 RFS1,6 0 CD1,6,5 11000 RD4,2,7 0 CI2,6,8 0 RFS2,3 0 CD1,1,6 8500 RD5,2,3 36200 CI2,7,8 0 RFS2,4 0 CD1,2,6 8500 RD5,2,4 0 CI3,5,6 8500 RFS2,5 0 CD1,3,6 8500 RD5,2,5 0 CI3,5,7 7700 RFS2,6 0 CD1,4,6 8500 RD5,2,6 0 CI3,5,8 8000 PIS1,6 2000 CD1,5,6 8500 RD6,2,3 18900 CI3,6,7 0 PIS1,7 0 CD1,6,6 8500 RD6,2,4 0 CI3,6,8 0 PIS1,8 0 CD1,1,7 9500 RD6,2,5 0 CI3,7,8 0 CIS1,6 0 CD1,2,7 9500 RD6,2,6 0 CI4,5,6 17500 CIS1,7 0 CD1,3,7 9500 RIS1,3 0 CI4,5,7 0 CIS1,8 0 CD1,4,7 9500 RIS2,3 0 CI4,5,8 8000 CIS2,6 1000 CD1,5,7 9500 TRFS1 0 CI4,6,7 9000 CIS2,7 0 CD1,6,7 9500 TRFS2 0 CI4,6,8 0 CIS2,8 0 CD1,1,8 8000 CI4,7,8 0 CIS3,6 0 CD1,2,8 8000

No of Iterations = 104



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