Netform Modeling and Applications

FRED GLOVER School of BusinessUniversity of ColoradoBoulder, Colorado 80309-0419

DARWIN KLINGMAN Graduate School of BusinessThe University of TexasAustin. Texas 78712

NANCY PHILLIPS Graduate school of BusinessThe University of Texas

Many real-world applications have profited from netform inno-vations in both modeling and solution strategies. Practical expe-rience shoves that advances in netform modeling and solutionstrategies overcome many of the difficulties in conceptual de-sign and problem solving of previous approaches to system op-timization. Moreover, they provide the type of technologies re-quired of truly useful decision-planning tools, technologies thatfacilitate modeling, solution, and implementation. The ultimatetest and worth of computer-based planning models, however,depends on their use by practitioners. In this tutorial, we showhow certain algebraic models can be viewed graphically usingnetform modeling and describe several large practical problemswe have solved. Some of our insights can make it easier forpractitioners to take advantage of these technologies.

The growth of the computer industry computer can record and manipulate ex-

has profoundly influenced many tremely large amounts of data. Without

areas, affecting none more dramatically this capability, many of the tools of man-

than management science. Knowledge agement science would be mere theoretical

about ways to solve optimization problems niceties.

in industry and government has exploded The techniques for building, solving, re-

since World War II, largely because the fining, and analyzing computer-based

Copyright SI 1990, The Inslitiite of ManaRi'monl Sciences NETWORKS/GRAPHS—APPUCATIONSD091.2102/'JO/2004/O0t)7$01.35

This paper was refereed.

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GLOVER, KLINGMAN, PHILLIPS

planning models have evolved steadily ascomputer hardware has changed. Thisevolution has spawned two new and im-portant technologies, computer implemen-tation technology and problem representa-tion technology. Both have been stronglyaffected by developments in the field ofoptimization. Within the optimization field,perhaps no domain has had a greater im-pact, as the source of innovations in imple-mentation and representation technology,than network optimization. These innova-tions are changing our notions about how

Knowledge about ways tosolve optimization problemshas exploded since WorldWar IL

to conceptualize and exploit optimizationproblems. The technologies of computerimplementation and problem representa-tion have profited from network optimiza-tion chiefly because advances in this fieldhave intimately related problem solving tothe identification and exploitation of struc-ture. Over the last decade and a half, thedevelopment of models based on charac-terizing structure for the purpose of insightand more effective solution has resulted inthe field of netforms—or network-relatedformulations. This field has (1) expandedawareness among OR theoreticians andpractitioners of the range of problems thatcan be formulated as networks and gener-alized networks; (2) identified ways to ex-press problems—and key components ofproblems—in a network format, especiallyin contexts that do not immediately sug-gest this possibility; (3) introduced flexible

notations to identify additional constrain-ing conditions, such as "all-or-none" and"multiple choice" flow restrictions (and in-teger restrictions in generalized networks),which make it possible to represent prob-lems outside the usual network category ina network-related framework; (4) devel-oped a repertoire of model constructs todocument which netforms are best forcommunication and which are best for so-lution {not always the same); and (5)evolved computer implementation strate-gies to take fullest advantage of netformrepresentations.

The fertile interaction between computerimplementation technology and problemrepresentation technology associated withthese developments has greatly profitedfrom research on pure and generalized net-work problems, subsequently adapted tobroader contexts. These innovations in-clude those of Barr, Glover, and Klingman[1977, 1979]; Bertsekas and Tseng [1988];Bradley, Brown, and Graves [1977];Cunningham [1979]; Galil and Tardos[1986]; Gilsinn and Witzgall [1973];Glover, Karney, and Klingman [1972,1974]; Glover, Karney, Klingman, andNapier [1974]; Glover and Klingman[1988b]; Glover, Klingman, and Phillips[1988]; Glover, Klingman, and Stutz[1974]; Helgason, Kennington, and Lall[1976]; Hung [1983]; Ikura and Nemhauser[1986]; Jensen and Barnes [1980]; Kameyand Klingman [1976]; Kennington andHelgason [1980]; Orlin [1984, 1988]; Rossand Soland [1975]; Srinivasan andThompson [1972, 1973]; and Tardos[1985]. This research has dramatically re-duced the cost of solving problems in thelinear and mixed integer network domain.

INTERFACES 20:4 8

NETFORM MODELING

beyond the reductions due to improve-ments in computer hardware or software.For example, the cost of solving networkproblems witb 2,400 equations and500,000 arcs has been reduced from thou-sands of dollars in the late '60s to less than$100 in the late '80s.

Using specialized network algorithms tosolve network problems provides signifi-cant gains over using commercial linearprogramming (LP) codes. Tbe best solutionalgorithm for pure network problems inthe late '60s was the out-of-kilter algo-rithm [Dantzig 1963; Ford and Fulkerson1962]. Testing in the late '60s through the'70s, Glover, Karney, and Klingman [1974];Glover, Karney, Klingman, and Napier[1974]; Karney and Klingman [1976]; andSrinivasan and Thompson [1973], showed

The cost of solving networkproblems with 2,400 equationsand 500,000 arcs has beenreduced from thousands ofdollars in the late '60s to lessthan $100 in the late'80s.

that tbe primal simplex algorithm was bestif one devised specialized data structuresand pivot and start procedures. In tbe '80s,the testing of nonextreme point algorithms[Bertsekas and Tseng 1988; Glover andKlingman 1988b] began to challenge theprimal simplex algorithm; it is becomingless clear which algorithm is best for purenetwork problems. For generalized net-work problems, the primal simplex algo-rithm with specialized data structures isstill fastest.

These advances in solution technologyhave stimulated the development of mod-eling techniques for handling a multitudeof problems that arise in applications ofscheduling, routing, resource allocation,production, inventory management, facili-ties location, distribution planning, andotber areas. These new modeling tech-niques [Glover, Hultz, and Klingman 1978,1979; Glover and Klingman 1977; Glover,Klingman, and McMillan 1977; Glover andMulvey 1980] are mathematically andsymbolically linked to network and aug-mented network structures and constitutethe central focus of netform technology.This technology allows users to concep-tualize formulations of their problemsgraphically. Its pictorial aspect has provento be extremely valuable in communicatingand refining problem interrelationshipswithout the use of mathematics and com-puter jargon. Thus it protects the nontech-nical person against technical legerdemainand exaggerated claims of model "real-ism." Tbis tecbnology also often yields amodel that can be solved as a sequence oflinear network problems or by merging so-lutions to linear networks in progressivelyrefined stages.Pure Network Models

Pure network problems embody a groupof distinct model types, including shortestpath, assignment, transportation, andtransshipment problems. Any of these net-work problems can be characterized by acoefficient matrix that bas at most one +1and one —1 entry in each column. Themost general of these model types is thetransshipment problem.

Tbe transshipment model appears inmany applications, either directly or as a

July-August 1990

GLOVER, KLINGMAN, PHILLIPS

subproblem. An illustration of this modelfor a cash-flow problem is depicted in Fig-ure 1. The arrows in network models arecalled arcs and tbe circles are called nodes.Triangles leading into or out of nodes rep-resent supplies and demands, respectively.Arcs indicate allowable flow paths be-tween nodes and bave lower and upperbounds, shown in parentheses, and costs,shown in rectangles.

The objective in the transshipment prob-lem is to determine how much to shipalong eacb arc within tbe limits stipulatedby tbe bounds in order to satisfy all sup-plies and demands and to minimize totalcost. By satisfying supplies and demandswe mean that the total flow into the nodeminus the total flow out must equal its de-mand, and the total flow out of the nodeminus the total flow in must equal supply.For all other nodes, the flow into the nodemust equal the flow out.

It is important to understand how atransshipment problem may be statedmathematically to appreciate the connec-tions between graphical and algebraicstructures. To sTate a network problem al-gebraically, we define a variable for eacbarc. For example, let X,, denote the flow onthe arc from node / to node / and c,, denotethe unit cost on tbis arc. Henceforth, anarc will be denoted as an ordered pair {(,;)where the first component specifies thefrom node and the other component the totiode. Next we create the objective functionfor the problem as an expression involvingthe costs and variables. For the problem inFigure 1, the objective function would be

Upon identifying the objective function,we create a constraint for each node thatexpresses the restriction on flow into andout of the node. To do this, it is convenientto view a supply as an inflow and a de-mand as an outflow. Then the requirementat each node can be expressed as Total In-flow - Total Outfiow, or equivalently.Total Inflow - Total Outflow = 0.

The customary transposition of any con-stant term of this equation to the rightband side causes supplies to be denoted asnegative quantities and demands to be de-noted as positive quantities. To see tbis,consider constructing the "Total Inflow- Total Outflow = 0" equation for node Cin Figure 1. The total fiow into node Cconsists of tbe five units of supply and(KAC + Xoc)- The total flow out of node C isXcF + CG + CD- Thus we have 5 + X^c

+ Xoc - i^cf + Xcc + Xcn) = 0, or aftertransposing the constant term, X^c + X^c- Xch - Xcc - ^cn = - 5 . Thus the supplyof five becomes expressed as a negativequantity because of its movement to therigbt band side of tbe equality sign. On tbeother hand, a demand, (whicb is includedin the outfiow that is subtracted in the "in-flow-outfiow" equation), appears as a posi-tive quantity when moved to the righthand side. This also discloses, incidentally,that a supply may be viewed as a negativedemand, and vice versa. The entire alge-braic statement of the capacitated trans-shipment cash fiow problem is also shownin Figure 1.

Each X,, appears in exactly two nodeequations, (the equation for node / and theequation for node ;'). Further, since X,, con-tributes to the outfiow of tbe node ( equa-tion and to the inflow of the node; equa-

INTHRFACES 20:4 10

NETFORM MODELING

Minimize

Subject to

+ 5X^0 +

- X , = -10

= -12

+Xr

= 15

- 12

0 < ^ 7;

0 < 0 <

0 < X B D < 1 0 ; 0 < XSE < 8; 0 < X

0 < XCG ^ 10; 0 < XQG < 1 5; 0 < X^G < 9

10;

Figure 1: Capacitated transshipment cash-flow model. The nodes correspond to subsidiaries ofa central company that operates in different locations. The supplies and demands represent ex-cess or deficit cash, respectively. Thus, nodes A, B, and C have excess funds, nodes D and £have no funds, and nodes f and G have deficit funds.

The arc from node A to node C indicates that it is possible to transfer funds from subsidiaryA to subsidiary C. The absence of an arc indicates that it is not possible to transfer funds di-rectly between the corresponding pair of subsidiaries (though it may be possible to transferfunds indirectly by means of a sequence of arcs through intermediate subsidiaries). The arcfrom node A to node C has a lower bound of 0, an upper bound of 7, and a cost of 2. Thenumbers in the semi-circles on the arcs illustrate a solution which satisfies the node equationsand the bound requirements.

July-August 1990 11

GLOVER, KLINGMAN, PHILLIPS

tion, it appears with a coefficient of -1 inthe former and with a coefficient of +1 inthe latter. Thus, each column of the coeffi-cient matrix has one -1 and one +1 entry.If the restrictions at the nodes are stated asinequalities rather than equations (that is.Total Inflow - Total Outflow < 0 or TotalInflow - Total Outflow > 0), then slack orsurplus variables are added to convertthese constraints to equations, and it is ad-missible for columns to contain a singlenonzero entry. Inequality restrictions atnodes generally arise by stipulating thatsupplies or demands must be "at least" or"at most" a certain amount.

By reversing the above steps, it is possi-ble to represent any LP problem whosecoefficient matrix has at most one +1 andat most one - 1 entry per column as atransshipment problem. That is, to con-struct a graph that corresponds to such aproblem, create a node for each constraintand an arc for each variable, affixing sup-plies, demands and bounds in the mannerindicated.Applications of Pure Network Problems

Pure network problems provide modelsfor many mathematical optimization prob-lems and for major components of manyadditional problems. Inventory mainte-nance problems [Evans 1978; Taha 1971;Wagner 1969] for example, typically ex-hibit an underlying network structure. Acousin of the inventory maintenance prob-lem is the PERT/CPM problem, whichseeks the best way to sequence a complexset of interdependent activities. The PERT/CPM framework, which constitutes one ofthe simplest network model forms, hasbeen used in a variety of practical applica-tions {including construction of the Polaris

submarine) and has been reported to savesubstantial costs and greatly speedcompletion of complex projects.

Other types of problems involving the ef-fective management of resources also fre-quently exhibit network structures. Suchproblems are becoming increasingly impor-tant in government and industry. Directnetwork formulations of water resourcemanagement problems, for example, arefinding use in a number of states. In these,canals, river reaches, and pipelines takethe role of arcs, while reservoirs andpumping stations take the role of nodes.Planning over time frequently looms as amajor consideration in these applications.

The Texas Water Development Boardand the government of Poland introducewhat-if analyses into water resource man-agement by using a succession of simula-tions having alternative supply and de-mand configurations and solve the result-ing network for each simulation run. Thestep of finding the optimum solution toeach network problem is used to determinethe best response to meet demands for wa-ter use, given a particular supply and-demand configuration. (This use of simula-tion, in which parameters are varied toachieve what-if analyses by means of rig-orous solution techniques, is to be con-trasted with the common "quasi-optimization" use of simulation.) To ana-lyze the full range of relevant configura-tions, roughly 500 such runs must be madeeach month. The feasibility and cost-effectiveness of such runs is due to the effi-ciency with which the underlyingnetworks are solved.

The problem of determining flows andheads in a general pipeline system (such as

INTERFACES 20:4 12

NETFORM MODELING

in municipal water systems) with reser-voirs, pumps, gate and check valves, givenfixed inputs and withdrawals has beenshown [Collins et al. 1978] to be equivalentto a convex transshipment problem underthe assumption of convex head losses.Such problems are easily solved as ordi-nary transshipment problems using apiecewise linear approximation of the con-vex function. Since the convexity require-ments are usually satisfied for real pipenetworks, this is an example of anotherclass of real-world problems that can nowbe handled by network procedures withfar greater effectiveness than by theprocedures used in the past.

Another important instance of the use ofnetwork models occurs in manpower pro-motion and assignment problems. AT&Thas developed such models to guaranteeacceptable hiring and promotion policies inaccordance with HEW rules and regula-tions. A number of cash-managementproblems can also be modeled as trans-shipment problems. These models includesources of funds in addition to cash (suchas maturing accounts and notes receivable,sales of securities, and borrowing) and usesof funds other than a single investment.The generalized network model we willdiscuss later makes it possible to furtherincorporate discount, interest, and otherfinancial considerations directly into themodel.

Many nonlinear problems involve net-work subproblems. One of the most basicand prevalent forms of nonlinear problemsis the fixed-charge network problem,whose major offshoots include the genreknown as location problems. The nonlin-ear element of a fixed-charge network is

the fixed-charge arc, which has the follow-ing special property: whenever the arc is"used" (that is, permitted to transmitflow), a charge is incurred that is indepen-dent of the amount of fiow across the arc.Fixed-charge networks have been used tomodel problems of plant and warehouselocation, equipment purchasing and leas-ing, personnel hiring, and offshore oil-drilling platform location, among others[Taha 1971; Wagner 1969].

To better utilize forecasting data and tosupport economically rational operationaldecisions, Klingman, Phillips, Steiger,Wirth, Padman, and Krishnan [1987] andKlingman, Phillips, Steiger, Wirth, andYoung [1986] developed an optimization-based decision support system for planningsupply, distribution, and marketing, calledthe SDM system, for Citgo Petroleum Cor-poration. The SDM system integrates Cit-go's key economic and physical supply,distribution, and marketing characteristicsover a short-term (11-week) planning hori-zon. This time horizon incorporates inven-tory planning and time lags for manufac-turing and distribution. To model this tim-ing problem, they partitioned the basicmodel into time zones and employed repli-cations of the basic model to accommodatethe distinct time periods (weeks), as shownin Figure 2.

Top management uses the system tomake such decisions as where to sell prod-ucts, what price to charge, where to buy ortrade products, how much to buy or trade,how much of a product to hold in inven-tory, and how much product to ship byeach mode of transportation. All informa-tion is provided by location, by line ofbusiness, and by week. The model incor-

July-August 1990 13

GLOVER, KLINGMAN, PHILLIPS

TIME ZONE 1

Initial Storage Invemory

TIME ZONE 2 TIME ZONE 3

WEEK 1 liiilial In-transitInventiiry

WEEK 2

in-iransit Inventoiy+ Trani-portation+ Handlitig Ciist+ Markciing Overhead

Figure 2: Network representation of time zones and lime periods. Single circles representproduct storage terminals, specifically a distribution center terminal at Lake Charles and otherterminals at Meridian, Birmingham, Atlanta, Sparlanburg, Charlotte, and Richmond. Doublecircles represent locations in a distribution pipeline called Colonial in each time zone. For ex-ample, "Col A" represents the segment of the Colonial pipeline in time zone 1 which containsthe product storage terminals thai are within one week's travel time from the refinery. "Col B"represents the segment of the Colonial pipeline in time zone 2 which contains Ihose terminalsbetween one and two weeks away, etc. (Double circles are used for pipeline nodes to make thenetwork easier to "read." Due to the large size of the problem and the number of differentcomponents which nodes were used to represent, the network diagram for the entire problemalso employs additional node symbols, such as half circles and ellipses.) The vertical arcs be-tween product storage terminal nodes represent inventory held at storage terminals betweentime periods. The diagonal arcs between pipeline nodes represent in-transit inventory. For ex-ample, product which is in the "A" segment of the Colonial pipeline in week 1 will travel tothe "B" segment in week 2 (if it is not lifted into the Meridian or Birmingham terminals).Thus week 1 demand in lime zones 2 and 3 can only be satisfied by initial in-transit or storageinventory, modeled by the supply triangles shown, since product cannot travel from LakeCharles to Col B or Col C in one week. (Demand triangles, not shown, would be attached toeach product terminal node.)

porates the critical timing considerations

associated with all of these decisions.

The SDM system contains a separate

minimum cost (or as an option, maximum

profit) flow network model for each prod-

uct (corresponding to the product manag-

ers who work with the supply, distribu-

tion, and marketing personnel associated

with each product). Each model is gener-

ated using input data from a corporate

data base, and the optimal solution is ag-

gregated into output reports tailored for

use by the individual operational manag-

ers. The objective function for the profit

INTERFACES 20:4 14

NETFORM MODELING

maximization model is calculated as totalsales revenue less expenses, such as trans-portation costs, terminal handling costs,and the time value of money associatedwith in-transit and terminal inventory. Theconstraints of the model include forecastedprice and demand functions, inventory ca-pacities, exchange agreement options, spotmarket purchase and sale volume limits,purchase, sale, and trade agreement op-tions, and retail demand. Management canalso include preliminary shipment sched-ules in the model as finalized decisions. Fi-nally, the SDM system employs rule-basedartificial intelligence systems to arrive atexchange agreement payback rules formodel constraints, and it uses an expertsystem approach to screen output reportsto provide management with exceptionreports.

Due to the complex combination of mul-tiple product sources, distribution timing,trading alternatives, and price sensitive de-mands, the authors used a number ofmodeling techniques to control the size ofthe model and to keep the model in net-work format and linear. These greatly ex-tended the model shown in Figure 2. Thenetwork framework was important for twomain reasons. First, the pictorial aspect ofnetwork models was extremely valuable tothe authors in working with a broad spec-trum of personnel to develop, validate, andimplement this integrated model. Second,network models permit rapid solutionspeed, which allowed management to re-spond quickly to the dynamics of a com-modity market industry by using the SDMsystem extensively in what-if sessions. Forexample, on a medium-sized computer, anIBM 4381 mod 2, each product model con-

sisting of approximately 3,000 equationsand 15,000 variables can be solved usingan efficient network optimization code[Glover and Klingman 1982] in approxi-mately 30 seconds. (To run the entire SDMsystem, model generation requires abouttwo minutes and report processing aboutseven minutes.) The SDM system allowedCitgo to reduce product inventories by$116 million (based on historical inven-tory-to-sales ratios) resulting in a $14 mil-lion per year reduction in working capitalcosts. The system also provided many in-sights into supply, distribution, and mar-keting that contributed an estimated addi-tional $2.5 million per year to bottom lineprofits.

In another application of pure networktransshipment modeling, Klingman andPhillips [1984] developed a new modelingand solution approach to make enlistedpersonnel assignment decisions for theMarine Corps. Our military services seek toaccommodate three primary types of goalsor criteria in assigning enlisted personnelto positions. Each goal type may have sev-eral subgoals, leading to a model that hasmany criteria or objective functions. Thesecriteria express the desire of the military tofill as many positions as possible in somepriority fashion, the cost of assigning aperson to a position, the utility of such anassignment to the military organization,and the desirability of the assignment tothat person. In addition, the Marine Corpsalso specifies proportionate or dispropor-tionate percentage fill policies for handlingshortages within a job type. The criteriaare preemptive and are handled in order ofdecreasing importance of the three goaltypes: maximum fill, maximum priority

July-August 1990 15

GLOVER, KLINGMAN, PHILLIPS

distribution fill, and maximum fit.To evaluate alternative modeling and so-

lution approaches to the Marine Corps'problem, Klingman and Phillips used arandom problem parameter generator pro-vided by the USMC to generate a singletest problem. The generator produced aproblem that had 10,000 persons, 240 jobtypes, and three job groups. Two of thegroups have proportional fill criteria andthe other a disproportional distribution fillcriteria. The highest priority group requires5i persons to fill its ith job type and theother groups require a constant number ofpersons to fill their job types. The parame-ter generator randomly generates eligibleperson/job arcs and associates four fit cri-teria coefficients with each one. Using thisparameter generator, Klingman andPhillips developed a large-scale multicri-teria pure network transshipment modelwith 10,245 nodes, 748,930 arcs, and 11criteria. (There are four fit criteria, threepriority-fill criteria— two proportional andone disproportional, three job-group-fillcriteria, and an overall fill criterion to max-imize the total number of people assignedto jobs.) The model correctly handles theproportionate and disproportionate per-centage fill policies and simultaneouslyyields integer solutions for large-scaleproblems. Klingman and Phiilips solvedthe problem using six different weightingstrategies for aggregating the 11 criteria inthe model. The best of these strategies re-quired about 127 minutes of CPU time ona PRIME 550 mini-computer. It is signifi-cant that this large-scale multicriteria testproblem could be solved on such a low-cost computer (less than $200,000 in 1980)in a reasonable period of time.

Generalized NetworksThe generalized network (GN) problem

represents a class of LP problems that hasreceived much less attention than it de-serves. Recently, as many new generalizednetwork applications have been identifiedand computer codes able to solve theseproblems efficiently have emerged, gener-alized networks are coming to be appre-ciated as rivaling or even surpassing purenetworks in their practical significance.Generalized networks include pure net-works as a special case. The GN problem,by allowing nonzero column entries otherthan ±1, is actually the broadest classifica-tion of linear network-related problems.GN problems arise in resource allocation,production, distribution, scheduling,capital budgeting, and other settings.

The most effective procedures for mod-eling and communicating pure networkproblems are based on viewing these prob-lems as directed graphs. Generalized net-work problems can also be represented asdirected graphs by means of appropriateconventions. In particular, the coefficientmatrix can be transformed (if the variablesare bounded) so that at least one entry inany column with two nonzero entries is- 1 . In this way, a directed arc is createdthat leads from the node associated withthe —1 to the node associated with theother nonzero entry. (If both entries are- I , the arc may be directed either way.) Asingle nonzero entry in a column is repre-sented by an arc that touches only onenode.

There is an important distinction be-tween arcs in pure network problems andarcs in GN problems. An arc of a general-ized network has a multiplier. This mulh-

INTERFACES 20:4 16

NETFORM MODELING

plier is the nonzero coefficient associatedwith the node at the arrowhead of the arc(tbat is, the terminal node of tbe directedarc). In pure networks, this multiplier isalways 4-1.

Tbese ideas are illustrated by tbe GNproblem sbown in Figure 3 along with tbeassociated network. As with pure networkproblems, each row of the coefficient ma-trix is associated with a node and each col-umn with an arc. That is, a node corre-sponds to a problem equation and an arccorresponds to a problem variable. Conse-quently, each arc has a cost, lower bound.

and upper bound. Costs are shown withinrectangles, and bounds are shown withinparentheses. Arc multipliers are shownwithin triangles.

The multiplier of a generalized networkproblem acts upon the flow across an arc;the amount of flow starting out on the arcwill not necessarily be the amount arrivingat the opposite end. Specifically, the flowentering the arc is multiplied by the valueof the multiplier to produce the quanhty offlow leaving the arc. The arc's cost, lowerbound, and upper bound refer only to theunits of flow entering the arc.

Minimize

Subject to

5X13 + 3X23 + 1X54 " 9X,

- 1 X 2 3

+ 1X23

0 < X, < 5,

=0

-1X24 +1/3X32 = 0

-1X32 -1X34 = 0

-1/5X24 +3X34 - X4 =0

0 < X,2 < 3, 0 < Xi3 < 4, 0 < X23 < 6,

Figure 3; Generalized network. Supply into node 1 and demand oul of node 4 are modeledusing bounded arcs, rather than supply and demand triangles. Thus node 1 has a supply of atmost 5 while node 4 has a demand of exactly 10. (The arc out of node 4 could have been mod-eled alternatively as a demand triangle with a demand of 10. In that case the variable X4would be omitted, and the last equation would be -l/SXj^ + 3X3, - 10.) If 2 units start on thearc from node 1 to node 2, the multiplier of 2 will cause 4 units to arrive al node 2. Likewise,10 units starting on the arc from node 2 to node 4 will result in -1 units arriving at node 4 (or,equivalently, 2 units leaving node 4 on that arc) since the multiplier in this case is -1 /5 .

July-August 1990 17

GLOVER, KLINGMAN, PHILLIPS

Applications of Generalized NetworksGeneralized networks can successfully

model many problems that have no purenetwork equivalent. This is made possibleby two useful interpretations of arc multi-pliers. First, multipliers can be viewed asmodifying the amount of flow of someparticular item. By means of flow modifica-tion, generalized networks can model suchsituations as evaporation, seepage, deterio-ration, breeding, interest rates, sewagetreatment, purification processes withvarying efficiencies, machine efficiency,and structural strength design. However, itis also possible to interpret the multiplica-tion process as transforming one type ofitem into another. This interpretation pro-vides a way to model such processes asmanufacturing, production, fuel to energyconversion, blending, crew scheduling,manpower to job requirements, and cur-rency exchanges. The following applica-tions are examples of possible uses ofgeneralized networks.

Bhaumik [1973] modeled a water distri-bution system with losses as a generalizednetwork problem. This model is primarilyconcerned with the movement of waterthrough canals to various reservoirs. How-ever, it also considers the retention of wa-ter over several time periods. The multi-pliers in this case represent the losses fromevaporation and seepage.

Gilliam and Turner [1974] proposed afile reduction model that has the form of ageneralized transportation model with asingle extra constraint. This model is de-signed to facilitate the reduction of ex-tremely large microdata files to smaller,statistically representative files. The objec-tive is to minimize the amount of informa-

tion lost in the reduction process. The arcsrepresent paths from the original recordsto the reduced records. A nonzero flow onan arc implies that the original record is tobe represented by the reduced record con-nected to it by that arc. The multipliers onthe arcs are used to insure that the reducedfile is truly representative of all of theoriginal records.

Kim [1972] utilized generalized networksto represent copper refining processes andmodeled the electrolytic refining procedureby a large d-c electrical network. The arcsare current paths with the multipliers rep-resenting the appropriate resistances. Inthis way, Kim analyzes the effect of shortcircuits in the refining process.

Charnes and Cooper [1961] identifiedapplications of generalized networks forboth plastic-limit analysis and warehousefunds-flow models. In plastic-limit analy-sis, the network is generated by formingthe equations for horizontal and verticalequilibria and by employing a couplingtechnique. The warehouse funds-flowmodel is actually a multi-time periodmodel. The arcs are used to representsales, production, and inventory of bothproducts and cash. The multipliers are in-troduced to facilitate the conversionsbetween cash and products.

Crum [1976] modeled a cash manage-ment problem for a multi-national firm asa generalized network. This model incor-porated transfer pricing, receivables andpayables, collections, dividend payments,interest payments, royalties, and manage-ment fees. The arcs represented possiblecash-fiow patterns and the multipliers,costs, savings, liquidity changes, andexchange rates.

INTERFACES 20:4 18

NETFORM MODELING

While the Alaskan Pipeline was beingbuilt, Congress funded the development ofa model for the distribution of natural gasacross the United States. The model in-cludes the largest 240 distributors and thelargest 100 pipelines in the United States.The model used multiple objective func-tions to represent the hierarchical way gasis distributed (residential customers beforecommercial customers, and so forth). Usinga commercial LP code to solve this modelcost over $1,000 for each solution run.However the problem can be modeled as ageneralized network; because gas in thepipeline is used to drive the pipelinepumps, gas is lost as it moves along thepipeline. The generalized network codeNET-G [Glover, Hultz, Klingman, andStutz 1978] solved the model for about $12per run.

Glover, Glover, and Martinson [1984]use a netform model somewhat differently;they formulated a large-scale linear-programming problem for resource plan-ning as a generalized network with sideconstraints. These constraints had the spe-cial form of compelling flows on certainarcs to satisfy proportionality restrictionsbased on flows of other related arcs, Signif-icantly, all linear programming problemscan be represented as generalized net-works with these types of side constraints,although in this setting these constraintsarose naturally and were not created as amodeling device.

The goal of this model was to determinethe optimal management of public range-lands by the US Bureau of Land Manage-ment (BLM) relative to uses for grazingand for supporting wildlife. This requiredidentifying optimal numbers of animals of

various types, in consideration of their di-etary requirements, that would be allowedto populate specific regions. The large sizeof the problem arose from the massivearea of rangelands involved and the multi-ple nutritional and geographic factors af-fecting the possible allocations of differentwildlife types to these regions. A simpiifiedcomponent of the model is shown inFigure 4.

To carry out necessary analysis, the BLMwanted to solve multiple versions of theseproblems daily. The amount of timeneeded to solve them using standard linearprogramming made it impractical. A spe-cialized procedure for progressively settingand revising arc bounds in the netform,however, made it possible to solve theseproblems 20 to 300 times faster than bythe previous LP system, obtaining solu-tions within 95 percent of optimality (usu-ally 98 percent or better). The speed im-provements were greatest for the largerproblems where the time factors were in-creasingly critical. The Bureau of LandManagement routinely uses the netform-based system an average of 700 times amonth throughout the US.

Another generalized transshipment net-work was the basic component of anoptimization-based logistics planning sys-tem that Klingman, Mote, and Phillips[1988] developed for W. R. Grace Com-pany, one of the nation's largest suppliersof phosphate-based chemical products.The mathematical model underlying thissystem includes production, distribution,multiple time periods, and multiple com-modities. W. R. Grace initially formulatedthe model as a linear programming prob-lem with 3,696 constraints and 21,564

July-August 1990 19

GLOVER, KLINGMAN, PHILLIPS

Animal Nodes Plant Nodes

Figure 4: US Bureau of Land Management model. In the geographical region represented bythis diagram, there are two types of animals, represented by the nodes labeled 1 and 2, each ofwhich can consume varying amounts of three types of plants, represented by the nodes labeledA, B and C. The volume of food consumed by each animal of types 1 and 2 are represented bythe multipliers M, and M;, respectively, shown associated with the arcs entering nodes 1 and2. These multipliers transform numbers of animals flowing on these arcs, denoted by F^ and fjand determined by allocations in other parts of the network, into total volume of food con-sumed. Each of the arcs from the animal nodes to the plant nodes has a lower and upperbound of the special form illustrated within parentheses for the first and last of these arcs. Thedistinguishing feature of these bounds is that they are variable, and depend on the amounts offlow F, and f; entering nodes 1 and 2. Specifically, two fractions / ^ and gi^ are applicable tothe arc from node 1 to node A, indicating that the amount of flow on this arc must be at least/",*rjVf]*/i and at most f, *Afi*^,^ (which identifies the least and greatest quantities of plant Athat type 1 animals will consume in an appropriate diet). Similar variable bounds bracket theflows on the other arcs from animal nodes to plant nodes, as illustrated on the arc from node 2to node C. The total available quantity of each of the three plant types in the given region isrepresented by the bounds on the arcs out of nodes A, B and C. The BLM objective was toutilize the total plant food available to the fullest extent possible, subject to the dietary re-quirements of the various animal species. This was accomplished by placing a cost of -1 (thatis, a profit of 1) on the arc leaving node R and using a cost-minimizing algorithm to solve theproblem.

variables. An integrated approach to mod-

eling and solving the problem enhanced

top management's understanding of the

model and made the problem more tracta-

ble for the company's DEC 20/60 com-

puter. The key features of this approach

are decomposition of the problem into a

generalized network component and a

small linear nonnetwork component, trans

formation of the generalized network com-

ponent into a pure network component.

incorporation of most of the nonnetwork

component into the pure network via an

innovative relaxation approach, and incor-

poration of the remainder into the objec-

tive function via Lagrangian procedures.

The resulting model relaxation was solved

using efficient pure network solution tech-

niques to obtain an advanced starting basis

for a basis partitioning algorithm. This ap-

proach reduced solution time approxi-

mately 10-fold, In addition, W. R. Grace

INTERFACES 20:4 20

NETFORM MODELING

used insights obtained from the solutionsto make multimillion dollar decisions.

Additional applications of generalizednetworks inciude machine loading prob-lems [Charnes and Cooper 1961; Wagner1969], blending problems [Charnes andCooper 1961; Wagner 1969], the catererproblem [Dantzig 1963; Wagner 1969], andscheduling problems such as productionand distribution problems, crew schedul-ing, aircraft scheduling, and manpowertraining [Charnes and Cooper 1961;Dantzig 1963; Wagner 1969].Integer Generalized Networks

An especially important realm of appli-cation extends the use of generalized net-works by requiring that flows on particulargeneralized arcs must occur in integer(whole number) amounts. Introducing theinteger requirement into the GN problemenables it to model an unexpected diversityof additional applications, including suchproblems as scheduling variable-lengthtelevision commercials into time slots, as-signing jobs to computers in computer net-works, scheduling payments on accountswhere contractual agreements specifylump sum payments, and designing com-munication networks with capacityconstraints.

Using integer requirements in GN prob-lems enables any 0-1 LP problem to bemodeled as an integer GN problem[Glover, Klingman, and McMillan 1977;Glover and Mulvey 1980]. These tech-niques can also accommodate mixed inte-ger 0-1 LP problems where the continuouspart of the problem is a transportation,transshipment, or generalized networkproblem itself. An illustration in Crum,Klingman, and Tavis [1979] shows how

contemporary financial capital allocationmodels can be modeled as integer GNproblems, important real-world applica-tions with such a "mixed" structure alsoinclude a variety of plant location models,energy models, and physical distributionmodels.

Integer generalized networks are alsoused to model requirements that flows oncertain arcs must equal their upper orlower bounds, or that at least (or at most) aspecified number of arcs from particularsets must have zero flows and other similarlogical requirements. These netform repre-sentations consisting of networks with eas-ily specified side conditions are able to rig-orously express all problem elements thatwould ordinarily require expression in analgebraic form (as by customary mathe-matical programming formulation tech-niques). Consequently, they effectively re-place the obscure and unilluminating alge-braic representation by an equivalent, butmuch easier to understand, pictorial repre-sentation. We will show how this comesabout by several examples, beginning witha fundamental technique that reappears invarious guises in a variety of practical set-tings. The applications also demonstratethat the underlying network-related struc-tures of the netform approach can often beexploited by special solution methods thatare far more efficient than the methodspreviously developed for the algebraicrepresentations.

Figure 5 illustrates a useful modeling de-vice based on integer constrained general-ized arcs commonly employed in the net-form approach. An extension of this deviceto handle an even more useful set of con-ditions is illustrated in the model compo-

July-August 1990 21

GLOVER, KLINGMAN, PHILLIPS

nent shown in Figure 6. The combination

of arc multipliers and 0-1 integer restric-

tions gives rise to what generally is called

an integer network or a 0-1 generalized

network. The following real-world applica

tions demonstrate more fully uses of this

netform modeling tool.

Applications of Integer GeneralizedNetworks

The netform concept has also improved

the solution of a mixed integer program-

ming problem for determining the mini-

mum cost refueling schedule for nuclear

reactors. Kazmersky [1974] initially mod-

eled this problem as a mixed integer pro-

10.1) A

Figure 5: Generalized network with integerflow restriction. This figure represents only acomponent of a model, and thus no suppliesor demands are shown. The bounds, costs,and multipliers are depicted by the same con-ventions employed before. In addition, theasterisk on the arc from node 0 to node A in-dicates that its flow must be an integer. If theflow is 0, then 3-0 = 0 and no flow gets trans-mitted to node A. But if the flow is 1, then 3units are transmitted to node A. Further, be-cause of the upper bounds of 1 on each of thethree arcs leaving node A, the only possibleway to distribute the three units flowing intonode A is to send exactly one unit to each ofthe nodes i, 2, and 3. Thus, by giving all arcsbounds of 0 and 1 and introducing a general-ized arc, the following effect has beenachieved: when the flow on the arc from node0 to node A is 0, the flow on each of the threearcs out of node A is 0; when the flow on thearc from node 0 to A is 1, the flow on each ofthe three arcs out of node A is 1.

gramming problem with no apparent con-

nection to networks. However, after work-

ing closely with Dr. Kazmersky, we

discovered a way to express the problem

by a 0-1 GN that was not only equivalent

to the original formulation but that also

succeeded in reducing the size of the prob-

lem [Glover, Klingman, and Phillips 1988].

(0,1

lnvF!(tmenl Decision

n l Types

Figure 6: An equipment investment. This fig-ure is the same as Figure 5 except that multi-pliers have now been added to the three arcsleaving node A. For concreteness, we maysuppose this diagram represents an invest-ment decision: to invest in project A (if theflow on the arc from 0 to A is 1) or not to in-vest in project A (if the flow on the arc from 0to A is 0). Then the nodes 1, 2, and 3 identifydifferent components of this investment proj-ect. In this example, the investment project isset in an equipment purchase context, andnodes 1, 2, and 3 represent different equip-ment types. In other contexts, these nodesmight represent different types of aircraft ina fleet, different parcels of land in a real es-tate venture, different types of stock in aportfolio, etc. The multipliers on the arcs intonodes 1, 2, and 3 represent the quantities ofeach component of project A (each type ofequipment) that would be acquired if in factthe decision is made to invest in that project.By the conventions and flow relationshipspreviously described, the diagram of Figure 6transforms the investment into its compo-nents in precisely the manner desired; that is,a flow of 1 on the arc from node 0 to node A(representing the decision to invest) translatesinto six units of equipment 1 at node 1, eightunits of equipment 2 at node 2, and five unitsof equipment 3 at node 3.

INTERFACES 20:4 22

NETFORM MODELING

While the original formulation by itselfconsumes more than 20 pages [Kazmersky1974], we described the full 0-1 GN modelin a handful of pages without usingcomplex mathematical notation.

In addition, making use of the 0-1 GNformulation, we were able to develop abranch-and-bound solution procedure thatsolved GN subproblems, yielding signifi-cant gains over previous solution efforts.We solved four versions of this problemusing data supplied by the Tennessee Val-ley Authority. The first three versions,which required half an hour to two hoursto solve on an IBM 4381 computer usingthe MPSX solution system, took less than20 minutes using the 0-1 GN formulationand the specialized solution approach. Thefourth version was by far the most diffi-cult, involving 173 constraints, 126 zero-one variables, and 511 continuous vari-ables. The original mixed integer formula-tion was run for seven hours on an IBM4381 using MPSX and then taken off themachine to avoid further computer-runcosts. At the end of the seven hours, thebest (minimum cost) solution obtained hadan objective function value of$136,173,440. We imposed a time limit of30 minutes on the 0-1 GN solution effortand obtained a solution that had an objec-tive function value of $125,174,727: animprovement of more than $10,000,000.As this application shows, using the net-form approach can improve the solutionsfor problems too complex to be solved op-timally (within practical time limits) bystandard approaches.

In another application of integer gener-alized networks. Glover, Klingman, andMote [1989] addressed the problem of de-

veloping optimization models and proce-dures for maximizing the collective profi-ciency {or "readiness") levels of US Armyunits. The models they proposed addressboth equipment procurement decisions andallocation decisions. The mathematical for-mulations of these models constitute large-scale, mixed-integer programs that poten-tially contain over 8,500 constraints and150,000 discrete variables. The new ap-proach developed for this problem may beviewed as an integration of optimizationstrategies and intelligent search strategies.The integer programming formulation canbe given a netform representation as a dis-crete generalized network (Figure 7), pro-viding a basis for identifying structure ca-pable of being exploited bothmathematically and heuristically.

The preliminary results that the authorsobtained from empirical testing on ran-domly generated problem data are veryencouraging. Prescriptions derived fromthe solutions indicate that the US Armymay be able to improve total unit profi-ciency levels by as much as 25 percent,while dramatically reducing solution time(compared to the Army's current expertsystem, solution time will go from 17hours to less than four minutes for a stan-dard sized problem). Based on these re-sults, the US Army is launching a long-range development project to implementthis approach.Future Directions

We are entering a new age of computerapplications. The evolution of ideas andperspectives that has led to expert systems,knowledge engineering, object orientedprogramming, and intelligent work stationshas underscored the fundamental role of

July-August 1990 23

GLOVER, KLINGMAN, PHILLIPS

Unit Nodes Equipment Nodes Budget Node

Figure 7: Mixed integer generalized network formulation for Army readiness. The arcs intonodes Uj represent the decision to upgrade unit ( to its desired proficiency rating. The arc be-tween nodes Ui and £, represent whether unit i is allocated equipment type j to eliminate ashortage. The two arcs out of node E, represent the number of items of equipment allocatedfrom inventory or procured, respectively. The coefficients are defined as follows: P, = positiveweight assigned to unit / denoting the unit's relative priority for proficiency upgrading; JV= number of equipment categories in which unit i must eliminate equipment shortages in or-der lo receive the desired proficiency rating; S;, = minimum number of items of equipmenttype ;• which unit i must receive in order to eliminate a shortage; C, = unit cos! lo procure oneitem of equipment type ;; X, = number of items of equipment category ; in inventory; Q,^ upper bound on procurement quantities for equipment category;; and D = total fiscalbudget available for procurement.

representational systems in interactions be-tween human beings and machines. Be-cause we tend to express complex relation-ships by pictures and diagrams, technologyis relying increasingly on pictorial repre-sentations as an indispensable element ofnew advances.

Netforms are being drawn into this pro-cess in a natural fashion and are currentlystimulating the design of intelligent inter-faces on several fronts. Netforms' legacy ofsuccessful applications of visual modelingtechnology, spanning diverse business,government and scientific settings,provides a background for bringing picto-rial representations into human-machine

interfaces to achieve practical, bottom-linebenefits.

Greenberg [1987, 1988] and Murphy,Stohr and Asthana [1988] propose uses ofnetforms as integral parts of system diag-nosis and suggest ways to create intelligentfront and back ends for computer optimi-zation software. Glover and Greenberg[1987] propose using netforms to enhancethe operation of expert systems. A growingbody of research is dedicated to detectingnetwork structures that may be hidden in-side more complex models [Bixby 1984;Bixby and Wagner 1985; Brown, McBrideand Wood 1985; Truemper 1983]. This hasled more recently to research into creating

INTERFACES 20:4 24

NETFORM MODELING

these and other netform structures in prob-

lem settings where they may not otherwise

be found [Glover and Klingman 1988a].

Currently, Freeman et al. [1989] are de-

veloping a computer system for causal

analysis called PRONET in the telecommu-

nications industry that is an artificial intel-

ligence (Ai) tool to facilitate reasoning

about complex interrelationships. The un-

derlying constructs of this system are

highly compatible with those of netforms,

and they envision linking the AI features

with optimization by relying more fully on

netform representations in the next stage.

With these innovative developments,

netforms are being used in a growing

number of realms, and these uses affect

how problems are translated into solvable

forms and how they are conceptualized

and communicated. More important, these

applications are affecting the range of

problems we perceive as susceptible to for-

mulation, enlarging the domains we can

represent for obtaining improved insights

and solutions.

Acknowledgments

This research was supported in part by

the Center for Business Decision Analysis,

the Hugh Roy Cullen Centennial Chair in

Business Administration, and the Office of

Naval Research under contracts N00014-

87-K-0190 and N000U-89-j-n32. Repro-

duction in whole or in part is permitted for

any purpose of the US Government.

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