R.K. Gupta* and Pravin Chandra**


With the fall of East European Socialist-Bloc and opening up of the Asian markets, the trade barriers began falling during the 1980’s and continued throughout the 1990’s. This development lead to organizations having a supply chain, that criss-crossed the whole globe. The proliferation of trade agreements has thus changed the global business scenarios. The Integrated Supply Chain Management (ISCM) is now not only a problem of integrated logistics (as a process) but also demands that the supply chain management (SCM) must look into the ramifications of these arrangements on the cost of transportation (including tariffs or duties) of products within a trade zone and outside it, besides, developing logistics strategies. The field has thus developed in the last few years for bridging the gap between demand and supply vis-à-vis efficiency and cost trade-offs. The SCM now not only involves the  “management of logistic function”, as was done in the past (to achieve internal efficiency of operations) but, includes the management and co-ordination of activities, upstream and downstream linkage(s) in the supply chain. The integrated supply chain management, in particular include :

Planning and Managing supply and demand; Warehouse Management; Optimal Inventory control; Transportation and Distribution, Delivery and customer’s delight following the basic principles of supply chain management viz. working together;  Enhancing revenue; Cost control; Assets utilization besides, customer’s satisfaction.

The last two decade has seen the rise of a plethora of acronyms always used in conjunction with production, operational management and control. To name a few JIT (Just-In-Time); TQM (Total-Quality-Management); ZI (Zero-Inventory); ECR (Efficient Consumer Response); VMI (Vendor Managed Inventory). All these have now been integrated within the domain of Supply Chain Management  Process.

With the growth in the Information Technology and easy accessibility of computing power, The development and implementation of objective based modelling system(s) have been changed to a new environment, for integrating quantitative and simulation models, as a backend system for both horizontally diversified and vertically integrated Supply Chain Management System(s).

Though, the SCM have found the versatility of applications, more so in the private sector enterprises (business environment) for cost cutting and for having a competitive advantage. In the government set-up though the basic objective, is not maximization of profit, but the social-economic development of people. Even, if the objectives of these two mutually exclusive categories of enterprises, are different, they share some features:

·          Satisfying the end-consumer(s) by providing the right product, in right condition at the right time to fulfil the social obligation towards society.

·          The optimum allocation of limited resources.

Thus, the SCM has many applications in the government environment too. The paper highlights some of the typical applications in the government sector of the SCM paradigm. What is essential in the SCM is to establish operationally feasible link(s) between various key component for achieving overall efficiency and cost trade-off. The use of quantitative methods in SCM is evaluated, embedding of these models in Decision Support System (DSS) have been discussed. The major component of SCM is multi-objective transportation and distribution function for time and cost trade-off. The Multiple Criterion Decision Making (MCDM) model for the component of SCM viz. Transportation and Distribution, system as a DSS have been described in detail - a major backend system of Integrated Supply Chain Management process (ISCMP).



*      Senior Technical Director and Head, Analytics and Modelling Division, National Informatics Centre, ‘A’ Block, CGO, Lodhi Road, N. Delhi - 11003, Tel. : 4362530 (O) 4672885 (R) Email : Or


**     M.Tech. Trainee at NIC during June-December’ 1997, and at present Assistant Professor (Information Technology), Lal Bahadur Shastri Institute of Management, Shastri Sadan, Sector - III, R.K.Puram, N.Delhi - 110022, Tel. 6172407 (O) 91-532971 (R) Email :


Supply Chain Management (SCM) can be best described as the natural extension of the downsizing (right-sizing) and re-engineering performed by the organization(s) in the past. Downsizing and re-engineering transformed the enterprises into “lean and mean competitive units”, by cost cutting and process simplifications. These operations (of downsizing and re-engineering) involved the “optimization” (in terms of the number of persons involved, the time taken, the complexity of the work etc.) of business “units” (functional and/or administrative domains) over which the organizations had full control. These strategies did lead to increased productivity and profitability of the organizations but as the benefits of these levelled off, it was realized that the approach to the way organizations work needed to be changed. The above changes were a by-product of the “isolationist” (closed system) world picture of the enterprises involved in the full value chain; with organizations (the system) trying to survive in an hostile environment; assuming that all other participants in the value chain were adversaries with whom the organization must compete, even though the operations performed by the separate organizations may be supplementary in nature rather than complementary. The realization that this world picture was an impediment to the growth of organizations prompted the enterprises to start seeking “strategic alliances” with other organizations. The formation of these alliances required a basis (a common ground) which would be acceptable to each and every partner in the alliance. This common basis is/was supplied by the participation of the organizations in the value chain (the demand-supply chain). The participants in the chain, suppliers, sub-contract suppliers, inhouse product processes, transportation, distribution, warehouses, and the end customer, generally, perform mutually exclusive tasks and thus do not compete directly with each other.

The present paper explores the following issues:

-           The need for supply chain management.

-           Type of supply chain management model(s)

-           Framework of the supply chain management model(s).

-           Issues in the design of supply chain management framework.

-           Quantitative methods and supply chain management (SCM).

-           Information technology as a supply chain management enabler.

-           Design of a Multiple Criteria DSS for transportation and distribution.

-           Relevance of the supply chain management paradigm to the government sector / public-sector enterprises.

Issues in SCM

A supply chain encompasses all the activities, functions and facilities involved in producing and delivering a product and/or service, from suppliers (and their suppliers) to the customers. The supply chain management (SCM) paradigm is geared towards optimizing each component of what used to be called (Production and) Operations management (production, warehousing, inventory, transportation and distribution etc.) and the inter-links between these components synergistically[21]. In the 70’s and the 80’s, various models for production and operations control and management were developed : Just-In-Time (JIT) Inventory management model, Vendor Managed Inventory (VMI) model, Zero Inventory (ZI) model, Total Quality Management (TQM) etc[1]. These models focussed on the various components of the supply chain in isolation, this implies that these models were oriented towards the optimization of a sub-part of the system whereas the SCM paradigm aims at the optimization of the full chain. This leads to trade-offs among the different components of the supply chain. For example, JIT would require a factory to keep inventories low and produce and distribute products in a timely manner, however JIT ignores many other aspects which cannot be seen independently, e.g. if the availability of the input materials is uncertain and irregular, the factory may need to insure smooth and continuous production. Similarly, regional stocking may permit reductions in transportation costs through increased shipment consolidation, as well as expanded sales through better delivery performance. These improvements may be accomplished with only moderate increases in inventory and warehousing cost(s). However, in an environment where different functional units manage the various logistics activities independently, an organization is less likely to properly analyze such important trade-offs.

Fig. -1 : Interdependence of supply chain with other functional domains in an enterprise.

Moreover, these models also ignore the interdependency of  production and operations functions with other domains within an organization, such as marketing and finance. Marketing decisions have serious impact on logistics function and vice-versa. For example, a marketing promotion campaign should be coordinated with production planning, since a higher demand may be expected. On the other hand, when raw materials are cheap, or when the factory temporarily has an over-capacity, the marketing department may decide to cut prices and/or start other promotion campaigns during these periods to increase demands. Also, financial decisions are driven by production and logistics decisions. Production of new products require the investment in raw materials and consume other change-over costs. Financial managers have to be aware of the increased demand for capital to finance the production plan. Likewise, the delivery of finished products generate financial income, so the forecast demand can be used to calculate/forecast the accounts payable and receivable in the future. The above description means that production, finance and marketing decisions cannot be made independently (fig.1). All these decisions are driven by the activities in the supply chain of a manufacturing company[1]. Fig.-1 shows a simple representation of the interdependence of the supply chain and the other functional domains in the organization. The links between the (other) functional domains - marketing, sales, human resources etc. - are not shown. The linkage between the supply chain components and the other functional domains relies heavily on information sharing to have an effective impact.

One other major factor in the current scenario is the globalization of the supply chain. With the fall of the East-European socialist bloc and the opening of the Asian market, the trade barriers began falling in the 1980’s and the 90’s. This lead to organizations having a supply chain that criss-crossed the globe. The proliferation of trade agreements - EC, ASEAN, NAFTA, APEC, etc. - has changed the global market. SCM now has become not only a problem of logistics but also demands that supply chain management must look into the ramifications of these agreements on the cost of transportation (including tariffs or duties) of products within a trade zone and outside it[1].

Furthermore, organizations now acknowledge that efficient consumer response (ECR) can lead to competitive edge. SCM is tantamount to coordinating all the operations of an organization with the operations of the suppliers and customers. Effective SCM strategies are essential for successful implementation of ECR programmes[22]. Thus, a production planning and control model that focuses on all the aspects of the operations and distribution activities and links with other functional domains such as finance and marketing is needed. The supply chain management model should also perform the task of managing and coordinating activities upstream and downstream in the supply chain. Of course, such a model in its entirety becomes very complex and can not be used without a sufficient computational infrastructure.

Supply-Demand Nexus

To have an effective supply chain management framework; organizations must have a clear understanding of the supply - demand nexus and its implications for strategy and implementation. There is an interdependent relationship between supply and demand; organizations need to understand customer demand so that they can manage it, create future demand and, of course, meet the level of desired customer satisfaction. Demand defines the supply chain target, while supply side capabilities support, shape and sustain demand[1].

When one considers how tangentially marketing and operations area of an organization typically interact (in practice), it becomes obvious that putting together the supply-demand can only occur in the context of overall perspective. The wide gap between the supply and demand sides of an organization can only be bridged by a comprehensive umbrella strategy. This can be done by developing a holistic strategic framework that leverages the generation and understanding of demand effectiveness with supply efficiency. Such a framework provides a strategic anchor to prevent the supply and demand components of a business from drifting apart.

The basis of such a holistic strategy framework is the integrated supply and demand model (Fig.-2). The model is designed around two key principles. First, in the present scenario where vertically integrated supply chains (VISC) are a rarity, if not non-existent; organizations must bring a multi-enterprise view to their supply chains. They must be capable of working co-operatively with other organizations in the chain rather than seeking to outdo them. Secondly, they must recognize the distinct supply and demand processes that must be integrated in order to gain the greatest value.

Fig. -2 : The Integrated Demand-Supply Model

Source: This model is based on the work done by Bill Copacino.[5]

Thus involving three key elements :

·         the core process of the supply and demand chains viewed from a broad cross-enterprise vantage point rather than as discrete function. To gain the maximum benefits, organizations need to identify the core processes across the demand and supply chain, as well as exploring the impact of each of these processes on the different functions.

Fig. -3 : Integrating processes in the supply and demand chains

Source : This model is based on the work done by Jeff Beech[1]

·         the integrating processes that create the links between the supply and demand chains (fig - 3). This implies that the planning processes (which involves development of channel strategies, planning of manufacturing, inventory, distribution and transportation, demand planning and forecasting; and marketing and promotional planning) and service processes (which includes functions such as credit, order management, load planning, billing and collection, etc.) must be integrated. This integration must be done across the boundaries of the enterprises. If each participating organization in the chain formulates its own plans on the basis of its own private information, then there is no way to integrate the supply and demand chain processes that they share.

·         the supporting information technology (IT) infrastructure that makes such integration possible. While information technology is needed to handle routine transactions in an efficient manner, it can also play the a critical role in facilitating the timely sharing of planning, production and purchasing information; capturing and analyzing production, distribution and sales data at new levels of detail and complexity. Information technology provides an integrating tools that makes it possible to convert data into meaningful pictures of business processes, markets and consumers that are needed to feed company strategies in order to develop competitive advantage.

On the administrative side, such elements as flow path economics, which help organizations understand the real drivers of costs, and new performance and measurement standards that align functions in accordance with total process goals that are critical to achieving integration.

SCM Framework

A framework to understand the various issues involved in SCM is provided by the pyramid structure for the SCM paradigm (fig. 4) The pyramid allows issues to be analysed on four levels:

·         Strategic : On the strategic, level it is important to know how SCM can contribute to the enterprises’ basic “value proposition” to the customers? Important questions  that are addressed at this level include : What are the basic and distinctive service needs of the customers? What can SCM do to meet these needs? Can the SCM capabilities be used to provide unique services to the customers? etc.

·         Structural : After the strategic issues are dealt with, the next level question(s) that should be asked are : Should the organization market directly or should it use distributors or other intermediaries to reach the customers? What should the SCM network look like? What products should be sourced from which manufacturing locations? How many warehouses should the company have and where should the be located? What is the mission of each facility (full stocking, fast moving items only, cross-docking etc.)? etc.

·         Functional : This is the level where operational details are decided upon. Functional excellence requires that the optimal operating practices for transportation management, warehouse operations, and materials management (which includes forecasting, inventory management, production scheduling, and purchasing) are designed. These strategies should keep in view the trade-offs that may need to be made for the overall efficiency of the system. Achieving functional excellence also entails development of a process-oriented perspective on replenishment and order fulfillment so that all activities involved in these functions can be well integrated.

Fig. -4 : SCM Framework Pyramid

Source : Based on work done by William C. Copacino[5]

·         Implementation : Without successful implementation, the development of SCM strategies and plans is meaningless. Of particular importance are the organizational and information systems issues. Organizational issues centers on the overall structure, individual roles and responsibilities, and measurement systems needed to build an integrated operation. Information systems are “enablers” for supply chain management operations and therefore must be carefully designed to support the SCM strategy. Supply chain managers must consider their information needs relative to decision support tools, application software’s, data capture, and the system’s overall structure.

It is important to note that the decisions made within the SCM strategy pyramid are interdependent. That is, it must be understood what capabilities and limitations affect the functional and implementation decisions and consider those factors while developing a supply chain management strategy and structure.

The SCM models used in practice lie in a continuum between two extreme models : on one end of the spectrum lies the vertically integrated supply chain model in which the organization has direct control over each and every component of the supply chain, while on the other end of the spectrum lies the horizontally diversified supply chain model (ideally) in which the number of participant is as large as the number of distinct parts of the supply chain. In an vertically integrated supply chain system, the organization can control every component of the chain and can make various changes to the system to optimize the chain very easily. But in a horizontally diversified supply chain the tendency will be to optimize only the functions that the organization is involved in, thus conscious efforts must be made by the various participants in the supply chain for the integration of their respective components in the supply chain. If an organization can be identified as the major/dominant partner in the supply chain, then this organization has to take an initiative in seeking the co-operation of the other participants in the supply chain.

The type and structure of the supply chain that is established depends on many factors, some of the major factors are :

·         Geographical : If the supply chain is stretched across the globe then it may not be possible to incorporate some of the principles of lean production like JIT delivery, flexible manufacturing, and co-ordination among suppliers and customers. It can lead to uncertain transportation schedules, unpredictable lead time and may need larger inventory carriage.

·         Cultural : The difference in the “culture” of the participants in the chain (the difference can be due to geographical factors or corporate practices) can lead to friction and distrust. This may hamper the development of close ties.

·         Government Legislation : The laws of the country may prohibit the sharing of information about some facet of the supply chain and thus, may lead to a restrictive participation by one or more participant in the supply chain.

Fig. -5 : Spectrum of alliances in the supply chain.

·         Time : Just as among individuals, organizations require time before trust can be built up. The first phase in any relationship is manifest as confrontation, that essentially means that participants in the chain try to win at the cost of other participants. And, the last phase is exemplified by total trust and working together of organizations. The information sharing behaviour in the first phase is almost zero, while in the integrated relationship the information sharing is mutual and free about the common concerns. In between the two phases lie a continuum of phases (see fig. 5).

Quantitative Methods and SCM

‘SCM’ requires extensive decision support tools for the effective monitoring, control and management of the supply chain, that is tools for channel design, transportation and distribution planning, inventory control etc. Various analytical and quantitative methods form the core of these decision support system(s). The quantitative models used in SCM are in general large linear programming models viz. model(s) for job scheduling, transportation and distribution, warehouse/facility location etc. All these models have one intrinsic limitation : they are, more often than not, single objective/criteria optimization methods. But, it is very rarely, in real life, that one encounters single criterion problems, by default all real life problems are multiple criteria decision making (MCDM) problems.

The MCDM solution methodologies address the multiple objective programming problem, viz.

            max { fi((x) = zi } ,  1 £ i £ k

            such that x Î S

where k (> 1) the number of criterion to be optimized, z’s are the criterion functions and S is the constraint set. Without the knowledge of the decision makers utility function, the methods search the “space of trade-offs” among the criterion to arrive at a pareto optimal solution to the problem using only the implicit information present[3,9,12, 24,25]. In practice, interactive procedures have proven to be the most effective in searching the trade-off space for the final solution. MCDM has two distinct halves. One half is multiple-attribute decision analysis and the other half is multiple-objective mathematical programming. Multiple-attribute decision analysis is most often applicable to problems with a small number of alternatives in an environment of uncertainty. Multiple-objective mathematical programming is most often applied to deterministic problems in which the number of feasible alternatives is large. MCDM techniques have not yet become widespread in managerial decision making (except maybe, the use of goal programming techniques). Below we review some of the areas (related to supply chain) where the use of MCDM methods have been reported:

The use of multiple objective have been reported for production planning in a multiple product, multiple period aggregate production planning by Jasskelainen[11], Lee and Jasskelainen[18]; by Wallenius[26] to solve a single product aggregate production planning. The classical Holt quadratic model of the problem of scheduling aggregate production and work force has been approximated by a linear goal programming model by Goodman[8].

Lawrence and Burbridge[15] use the multiple objective linear programming (MOLP) method for co-ordinated production and logistics planning. The decision making utilises several key objectives : a) maximising total sales revenue for specific location and customer; b) minimising total cost of cost of production and distribution; and c) maximising production of a particular item at a particular location.

The “blending of materials problem” is solved by using MOLP and modified goal programming by Lawrence and Burbridge[16]. Stainton[23] uses a heuristic approach to solve the multiple objective production scheduling problem for a large food manufacturer. Lee and Moore[19] use linear goal programming for optimisation of transportation problem while Charnes, Cooper[4] present an assignment problem which is a variant of the transportation model.

Other techniques that can be used are : a) Neural network[2,6,10] based techniques for the evaluation of alternatives in conjunction with MCDM solution generators (using neural networks to model the decision makers utility function); b) using neural networks for demand forecasting (it has been experimentally demonstrated that neural network based forecasting techniques are better and more robust than forecasting methods based on econometric modelling and/or statistical time series forecasting techniques and  c) the use of fuzzy-neural network or genetic algorithms based[7,13] methods to incorporate uncertainty in the decision making process.

These models can be incorporated in the (standard two-layered) architecture[24] for the development of interactive decision support system(s). Where the database refers to a repository of relevant data for the solution of the problem and the modelbase refers to the database of relevant analytical, fuzzy, neural network or genetic algorithm based models parameters that the user can choose from to solve the problem. Its the opinion of many that interactive methodologies are the best for the solution of multiple criteria decision problems.


Fig. -6 : Architecture for a DSS

Information Technology and SCM

Information technology (IT) includes a set of powerful tools that can lead to the failure or success of a supply chain process. With the development of information systems (IS) and information technologies the use of information sharing and decision making is growing at a very fast pace. IT solutions are no longer likely to provide strategic advantage, but imply the business basics. The competitive advantage for organization(s) originates from development of creative information technology strategies and implementing them. IS’s enable existing strategies to be realized, Information flows  provide the linkage that allows the supply chain to operate efficiently.

Technologies like internet, intranet, extranets and groupwares[20] facilitate the sharing of information using (distributed) common databases (with access control to the database for checking unauthorized access). These allows sharing the information not just within the functional divisions of an enterprise but upstream and downstream the supply chain. Electronic Data Interchange (EDI) can be used to place orders, inventory database can be shared between the manufacturer and the supplies for efficient implementation of JIT inventory; for vendor managed inventory (VMI) this sharing is a must. The internet and EDI can be used by the customer to monitor the status of the order placed, request changes in the order and vice-versa, they may be used to inform the customers about the status of their order, besides being used for billing etc. The internet and EDI can be used not only for information sharing/exchange but may also be used for marketing of services, products (especially software) and advertisement etc. The internet is becoming a medium of choice for product marketing, delivery, billing and customer support.

The above was the description of the technology available, below is the description of the supply chain management tools. These tools include supply chain configuration tools (for strategic decision making by determining the number, capacity requirements besides location of facilities etc.); demand planning tools to assist management in understanding the key drivers of demand using sophisticated analytical tools and with provision for interfacing with external data. Supply - planning tools to assist management with decisions such as which products to make, how to make them, what order to make them in and where to source materials from? These tools use interactive production planning, Gantt Charts and simulation and also incorporate advanced constraints such as capacity utilization, customer priority and due dates. Transportation and distribution planning and management tools to assist in the planning of how much to move- which item(s) - where? Using which mode of transportation?, support, carrier preference structure incorporation, consolidation and back-haul opportunity identification; load creation and sequencing, vehicle-scheduling and utilization optimization, operation within a warehouse,  like order allocation, receiving, radio frequency/hand held scanning inventory control (cycle counting, aging, lot control, expiry data tracking etc. And lastly, Enterprise Resource Planning (ERP) software; which provide the transactional data handling support. ERP  grew out of MRP - I and MRP - II by the addition of the more functional domain modules. Generally ERP’s provide tools for  the management of the operational aspects of the supply chain management with a few additional decision support tools. But more and more DSS developers are providing interfacing/integration capabilities with ERP software for advanced tools of decision making support.

Design of  Multiple Criteria DSS for Transportation and Distribution

Transportation and distribution management is one of the major component of SCM. The success or failure of a supply chain depends, to a large extent, on the success of the distribution channels. The solution to the problem of transportation and distribution in a supply chain is usually done through the use of some variant of the classical transportation problem :

Suppose that there are m sources and n destinations. Let ai be the number of supply units available at source i(i=1,2,...,m), and let bj be the number of demand units required at destination j (j=1,2,...,n). Let Cij be the per unit transportation cost on route (i,j) joining source i and destination j. The objective is to determine the number of units transported from source i to destination j such that the total transportation costs are minimised.

Let xij be the number of units shipped from source i to destination j, then the equivalent linear programming model is given as follows:


Subject  to

This model is usually solved by special techniques (called the transportation problem techniques) which are based on the simplex method. The model can be made more general by relaxing the equality constraint 1c. But even with these extensions the present problem can not be considered, as  only the goal of cost minimization is considered in the classical model. In any real life transportation and distribution problem, the number of goals to be achieved (the number of criterion/objective) is more than one. The presence of multiple goals imply that the "classical" transportation model can not be utilized for the solution of the present problem. Owing to the presence of multiple goals the methodology to be used is Multiple‑Criteria Decision Making (MCDM) problem. In the following part of this section we detail a MCDM model for the use in the design of a DSS for the transportation and distribution plan generation of a public sector enterprise.

The model is defined as follows :

Suppose that there are M sources, N destinations, P products and R number of transportation modes. Then let xijkl be the number of units of product k (k=1,2,...,P) transported from the source i (i=1,2,...,M) to the destination j (j=1,2,...,N) by the transportation mode l (l=1,2,...,R). Then we define the following quantities that are available as constraints/goals:

Aik         is the matrix denoting the amount of the product k available at source i (rigid constraint, modelled as a less than equal to type goal).

Djk         is the matrix denoting the amount of the product k required at the destination j (flexible goal, equality type, called the demand goal).

Sijl        is the matrix denoting the distance between the source i and destination j by the transportation mode l.

Tl          is the matrix denoting the transportation tariff per unit weight per unit distance by the transportation mode l.

B          is the transportation budget (flexible goal, less than or equal to type, called the budgetary goal).

Lil         is the matrix denoting the total number of units of all products that can be handled (loaded) at the source i for the transportation mode l (rigid constraint, modelled as a less than or equal to goal).

Ujl         is the matrix denoting the total number of units of all products that can be handled (unloaded) at the destination i for the transportation mode l (rigid constraint, modelled as a less than or equal to goal).

Cijl        is a matrix whose elements are equal to 1 if the mode l is available for transportation between the source i and destination j.

Gijk        is the matrix denoting the amount of product k that the decision maker wants to move from the source i to the destination j (flexible goal, greater than or equal to type, called the movement goal).

Ejk         is the matrix denoting the minimum amount of the product k the decision maker wants to supply to the destination j (flexible goal, greater than or equal to type, called the minimum demand goal).

Wij        is the matrix denoting the maximum amount of all products that decision maker wants to move from the source i to the destination j (flexible goal, less than or equal to type, called the maximum movement goal).

The priorities for all the rigid goals is the highest say P0 (and in the actual implementation, the user is not allowed to set the priorities for the same), thus the rigid goals/constraints are fulfilled first and only then is the other goals fulfilled. For the others let the priorities be as follows:

PD        is the priority for the demand goal.

PB         is the priority for the budgetary goal.

PG        is the priority for the movement goal.

PE         is the priority for the minimum demand goal.

PW        is the priority for the maximum movement goal.

For the sake of exposition/simplicity we take the priorities in the order defined, that is, P0 is the highest priority, PD is the next highest priority, and PW the least preferred.

We also define the following indices and symbols:-

i           is the index for source, i=1,2,...,M.

j           is the index for destination, j=1,2,...,N.

k          is the index for product, k=1,2,...,P.

l           is the index for transportation mode, l=1,2,...,R.

Sindx       denotes that the summation is to be performed over the subscripts indx to the symbol S over the appropriate range.

Using these notations we define the goal programming model as:

lex min {









Sjl Cijl × xijkl + dikA-                       £ Aik

Sil Cijl × xijkl + djkD- - djkD+             = Djk

Sijkl Cijl × xijkl × Sijl × Tl + dB-        £ B

Sjk Cijl × xijkl + dilL-                       £ Lil

Sik Cijl × xijkl + djlU-                       £ Ujl

Sl Cijl × xijkl - dijkG+                        ³ Gijk

Sil Cijl × xijkl - djkE+                       ³ Ejk

Skl Cijl × xijkl + dklW-                     £ Wkl

all d's ³ 0

All the right hand side terms are in general matrices, thus in general all the d's (the deviational terms) are matrices.

This model was used as the backend analytical model to a Transportation and Distribution DSS. The system was designed in the Windows 95/NT environment.[27]


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