a review of the literature on the missile-allocation problem

A Review of the Literature on the Missile-Allocation ProblemAuthor(s): Samuel MatlinSource: Operations Research, Vol. 18, No. 2 (Mar. - Apr., 1970), pp. 334-373Published by: INFORMSStable URL: http://www.jstor.org/stable/168691 .

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A REVIEW OF THE LITERATURE ON THE

MISSILE-ALLOCATION PROBLEM

Samuel Matlin

General Electric Company, Philadelphia, Pennsylvania

(Received August 22, 1968)

This paper reviews the missile-allocation-problem literature. The problem considered is: given an existing weapon force and a set of targets, what is the optimal allocation of weapons to targets? References are organized by type, characterized by submodel, discussed, and annotated. It is proposed that this review methodology by applied to other appropriate areas.

THIS REPORT has two purposes: to review in an organized fashion the unclassified literature on the missile-allocation problem (MAP), and

to suggest a method for similar literature surveys in other fields. The problem considered is: given an existing weapon force and a set of

targets, what is the optimal allocation of weapons to targets? This is not to be confused with the force-mix problem, which asks: what weapon mix should be developed, under constraints of time and money, to maximize the damage to the enemy? Target allocation is actually a submodel of the latter problem, since alternative force mixes should be compared on effectiveness only after each has been optimally targeted.

There are many elements of the MAP. A partial list of considerations appears in Table I. No model can deal with the problem in full detail. Various simplified assumptions are made, of necessity, and the differences between models reside essentially in the particular sets of simplifying assumptions adopted. The elements of the MAP may be synthesized as illustrated in Fig. 1. Various references treat the submodels in different degrees of detail. The five major submodels (weapon system, target complex, engagement, damage, algorithm) are characterized in Section I.

Sensitivity questions are rarely broached in the literature. They are included in Fig. 1 to suggest that they should be of interest. Two kinds of sensitivity are identified: whether the entire allocation is drastically changed by adding or deleting a weapon or a target, and whether the total expected damage is drastically changed by perturbations in offense and defense requirements, strategies, inventories, and other parameters.

It is suggested that this approach to model characterization for the MAP be applied to other areas of interest to the defense community. A central- ized attempt (say, by the Military Applications Section of the OPERATIONS

334



TABLE I

THE MISSILE ALLOCATION PROBLEM: A PARTIAL PARAMETER LISTING

ATTACKER Booster Characteristics (for each weapon type)

Location Availability Range Survivability Accuracy Reliability Memory Bomb damage assessment Penetration aid mix and effectiveness Etc.

RV Characteristics Accuracy Signature Yield Indirect bomb damage assessment A range, A crossrange capability Survivability

Etc. Polaris/Poseidon and bombers Command and control

Target sharing Redundancy and survivability Reprogramming Reaction time Weapon commitment Etc.

OBJECTIVES Offense

Damage criterion per target Total criteria

Defense Damage criterion per target Over-all criteria

INTELLIGENCE What the defense knows about the offense

Number of attacking missiles Number of salvos Salvo composition Missile payload mix Etc.

What the offense knows about the defense Terminal ABM inventories, distribution among targets Area ABM inventories Preferential defense strategy Etc.

DEFENDER Target complex

Locations Location uncertainties Configurations G&G Values Damage function Hardness Etc.

Defenses Terminal inventories Terminal effectiveness Area coverages Area characteristics (inventory, choice, effectiveness) Radar capabilities (tracking, impact prediction, capability) Civilian defense program

SCENARIO (strategic environment) Enemy first strike

Assured destruction mission for US US attrition US first strike

Damage limiting mission for US (Counterforce, countervalue, retained) mix

Selective threat targeting philosophy Progressive confrontation (tit-for-tat) targeting Collateral damage restrictions

335



336 Saimuel Mcitin

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Review of the Literature on the Missile-Allocation Problem 337

RESEARCH SOCIETY OF AMERICA) should be made to identify the areas that lend themselves to this kind of review, to analyze each such area as in this paper, and to provide the community with as many surveys as possible. Once a survey approach has been developed in an area, it is further suggested that either a reviewer or the author of a new paper in the field characterize the contribution in this more detailed and codified manner, and use this characterization as an abstract or review of the paper.

I. MODEL CHARACTERIZATION

THE REFERENCES in the missile-allocation-problem (MAP) literature are reviewed here by summarizing their submodel characteristics. The submodels common to all references are: the weapon system, the target complex, the engagement, the damage model, and the algorithm. The weapon system is further categorized into scope, reach, and commitment strategy. The target complex is further partitioned into scope, value system, and defenses.

Each submodel (with the exception of the algorithm) is characterized by a letter of the alphabet that serves not only to identify the assumptions but also to give some indication of the naivete or sophistication of these assumptions. Thus an 'A' characterization suggests a very simple set of assumptions, a 'D' implies a complex submodel, with 'B' and 'C' inter- mediate. No reference is complex in all submodels; generally if the weapon system is sophisticated, the target complex is not, or else the engagement is simplified. Several of the references combine the defense, engagement, and damage models by specifying a single parameter, P., the probability that the ith weapon penetrates and kills the jth target. This is characterized as an a-level submodel and is described in subsection 4. This approach permits the reader to assess the value of any single reference for his particular purposes, and also expedites a comparison of references pinpointing similarities and differences. Although it might be tempting to characterize a reference by a summary complexity rating, this would be incorrect, since the features are incommensurable.

Finally, to include features that are not common to several references, a subsection is reserved for notes, special features, and results.

The categories selected for this review are described in the sequel. A summary appears as Figure 2.

1. The Weapon System

The weapon system is characterized in three categories: the scope of the system (number of different weapon types considered, and penetration aids); weapon 'reach' (which weapons can reach which targets, with what degradations in payload and accuracy); and weapon commitment policy



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(number of waves launched, quality of bomb-damage assessment, weapon availability uncertainties).

The scope of the weapon system is described by two dichotomies; whether a single type of weapon or more than one type is considered, and whether penetration aids are employed. ' single weapon type implies that every weapon appears the same to the target complex, each having the same yield, accuracy, radar cross section, etc. Furthermore, each booster carries the same payload. Different weapon types are distinguished by differing yields, accuracies, or payloads. The penetration aids usually modeled are terminal decoys and area decoys or chaff. No other penetration aid (e.g., precursor or ECM) is explicitly modeled in the literature, although it may be implicit if the engagement model employs a probability-of-penetration parameter. In this characterization, however, if the number of terminal or area decoys is not explicitly provided in the analysis, it is considered to be a no-penetration-aids model. Thus, four descriptive categories exist: one weapon type or more than one, each combined with penetration aids or no penetration aids. The simplest models consist of one weapon type and no penetration aids; these are designated level A. The next level of complexity involves one weapon type with penetration aids, and is designated level B. The next, level C, represents more than one weapon type but no penetration aids, while the most complex scope, D, would be more than one weapon type and with penetration aids.

Weapon 'reach' is usually provided in the literature by means of an incidence matrix. The' column headings are the targets, and the rows are the boosters or weapons. A zero-one matrix has the entry 0 or 1 in the ith row and jth column, according to whether the ith weapon cannot or can reach the jth target, respectively. For the simplest models, level A, all weapons can reach all targets without degradation in either payload or accuracy. In this case, the incidence matrix is not even given explicitly, since all its entries would be ones. At the next level of model sophistication, B, an explicit zero-one matrix is either given or easily obtainable from such data as booster location and range capability and target location but, again, no weapon degradation is considered. The third level of sophistication, C, employs (or can easily employ) an incidence matrix whose entries represent the payload deliverable from booster i to target j. Here, range from booster to target is what usually determines the payload (numbers of warheads and decoys), but accuracy does not degrade with range. A level C model does not occur in a pure missile allocation problem, since in that problem it is assumed that a booster payload is out in the field and cannot be so varied (i.e., the RV-decoy mix is not reduced to reach distant targets, but rather another weapon type is employed, where the other type may consist of the same booster with fewer decoys). A level C model has



340 Samruel Matlin

implications for the force-mix problem (of which the missile allocation problem is a submodel), since the analysis associated with it produces a near-optimal force composition (under the constraints of booster and payload alternatives being examined) in addition to an optimal or near- optimal targeting.

The final level of sophistication (complexity of weapon 'reach'), submodel D, lies beyond incidence matrix characterization, as the matrix is neither explicitly provided nor is it easily obtainable. Instead, booster and warhead footprint vs. deliverable payloads may be given, together with booster and target locations. In such a case, no single incidence matrix suffices. For example, if a booster contained a maneuvering RV, that RV could reach different subsets of targets with different accuracies depending on range from booster location to target region. Again, a booster with multiple warheads can reach different groups of targets depending upon range and launch azimuth, with varying degradations. The unclassified literature does not contain any analyses at level D.

The weapon commitment strategies available to the offense are organized in two dichotomies. The first is concerned with whether the commitment is deterministic (all weapons available and launched reliably, and the bomb-damage-assessment system works perfectly), or probabilistic (random booster attrition experienced, either through enemy action or random launch unreliability). The second dichotomy is concerned with whether a single simultaneous and total launch strategy is utilized, i.e., a single attack wave, which would correspond to the absence of a bomb-damage-assessment (BDA) system, or whether the attack is delivered in more than one wave, the latter being associated either with the presence of a BDA system, or a preliminary attack on defenses or control centers (having indirect value) followed by attacks against directly valued targets.

The simplest weapon commitment model, level A, assumes a single attack wave (no BDA) with a perfectly known force. In level B, the single wave probabilistic case, a random subset of boosters would not be suc- cessfully launched, either because a first strike would have been absorbed and it would not be known which specific boosters had survived, or because some percent of the boosters would have been unreliable and failed to launch, the specific failures being unpredictable. This level of analysis is required if target sharing among squadrons is considered, e.g., if some boosters from two different wings are to be targeted to the same city.

In level C, the attack force is known and is committed in more than one wave. If BDA is present its reports are perfect; attrition at the engagement end may be probabilistic, but all available boosters are known and launched perfectly, and BDA reports are accurate. Finally, model D as-




sumes the attack is delivered in several waves after preliminary random weapon attrition is experienced.

2. The Target Complex

The target complex submodel is characterized by the types of targets considered (point, area, collateral), the values assigned to the targets (which are appropriately accumulated to provide a measure of effectiveness of a given missile allocation), and the defenses associated with the target complex (terminal, area, preferential).

The types of targets usually considered in the MAP literature are either point targets exclusively, or area as well as point targets. Examples of point targets are missile silos, radar installations, bridges, small cities, etc. Whenever the weapon lethal radius is large enough in comparison to the target so that a single penetrating well-placed weapon is capable of killing the target, it is considered to be a point target, even though it may actually be areal. If more than one weapon is required to cover the target it is considered to be an area target, such as a large city, an extensive airfield or harbor, etc. Note that the determination of whether a target is point or area cannot be made by consideration of the target alone; weapon lethal radius must also be considered. Furthermore, if a large city is broken down into individual aim points, with values associated with each aim point rather than with the city as a whole, then it is considered to be a collection of point targets rather than an area target. Independent point targets are such that no single weapon can kill more than one point at a time. By contrast, several collateral targets may be killed with one weapon.

Three levels of analyses associated with the target-type submodel are identified for the purpose of this survey. The simplest level A considers independent point targets only. Level B extends to independent area targets with or without point targets included. In level C, collateral point targets are examined. Level D would describe collateral area targets (of which no examples appear in the MAP literature).

Expected target value killed is the usual measure of effectiveness by which different missile allocations are compared. Invariably in the literature it is assumed that the value or worth associated with a target is the same for the offense as for the defense, though this need not be the case, and in reality, probably is not. In the unclassified literature, a single value scale is ubiquitously employed, but again this is not a necessity; one easily envisions an analysis in which two or more scales are appropriate (e.g., population fatality, in which case city target value would be proportional to population; and industrial capacity). The value scales usually do not vary with time, and are generally linear (the latter assumption implying



342 Samuel Matlin

that it is twice as valuable to extract 100 million fatalities as to extract 50 million fatalities, an assumption that may very well be inappropriate).

Each of the target-value submodels falls into one of four categories. In the simplest submodels, level A, all targets are of equal value (or weight, worth, importance), or no values are assigned, which implies (by default) equal values. Level B models assume that the targets may be ranked in order of priority, but no numerical values are assigned (ordinal value scale). In level C models, numerical value assignments are made on a single cardinal scale, usually normalized so that the sum of all target values is unity. In the case of city targets, value is usually proportional to population. The last category of target valuation, level D, involves targets having indirect or intrinsic value as well as targets with direct or extrinsic value. By this is meant that some targets have the property that no value is assigned -for killing them, but if they are eliminated, then targets with assigned values can be accumulated either automatically or more easily.

The target defense submodel may be characterized according to the type of defense (none, terminal only, area with or without terminal defenses, either preferential or not), and the treatment in the analysis. The treatment is dichotomized into whether the defense parameters are explicitly provided (numbers of AM1M's are given) or implicit (in terms of specified target penetration probabilities). The explicit case is further dichotomized according to whether the defense allocation is known to the offense.

The simplest model, level A, assumes there are no defenses. Level B models consider the case where only terminal defenses exist,

and the number of terminal defenders at each target is known to the offense, which has the effect of adding a known 'price of admission' to a target which the offense must surmount before it can buy value.

Level C models deal with area defenses (with or without terminal defenses) where the number of antimissile missiles (AMM's) in the area- defense inventory is known, and where it is assumed that the defense is uniform rather than preferential; intercepts will be attempted against all attackers in the area. The effect is to place an admission price on the area that must be paid before even a single target can be bought.

The next model in complexity, level D, indicates defensive capability implicitly by providing either a penetration probability at each individual target (terminal defense) or two penetration probabilities, one for the area and a second for each target within the area (combined area and terminal defense). This defense model is popular if analytic damage models are to be utilized. It is usually assumed that the offense knows the penetration probabilities. Where several independent defense islands are considered (a defense island consists of a subset of targets defended by a single area defense; there may be other subsets of targets defended by




other area defenses, but a defense in one area cannot be used in another area), probabilities of penetrating each defense island may be specified separately. Preferential defense strategy is not considered except for the trivial case in which some targets may be assigned penetration probabilities of unity (undefended); however, the offense is presumed to know these targets.

A level E model is one in which the total terminal-defense inventory is explicitly known to the offense, but the defense allocation of AMM's to specific targets is unknown. Such analyses are usually performed using game theory (or linear programming, which is equivalent to a game). A special feature of such investigations is that the optimal defense allocation is derived as well as the optimum offense allocation. The defense makes the first move, allocating defenders to targets in a manner unknown to the offense, after which it cannot change its allocations.

The most complex model, level F, considers the preferential area-defense case in which the total area-defense inventory is known to the offense, but the defense has the option of defending some targets while not defending others, with various numbers of interceptors. The targets being defended and the number of interceptors allocated to each defended target are unknown to the offense. It is customarily assumed in level F models that the defense has the capability of predicting which offense weapons are destined to impact particular targets. Here, as in level E, the optimal defense allocation usually is derived.

In addition, an a-level characterization is made, as described in subsection 4.

3. The Engagement

The engagement submodel of the MAP derives the probability that a weapon penetrates the target defenses (for target defense submodels at levels B, C, E, or F). Subsequently, the damage submodel will determine the damage attributable to a set of penetrating weapons. The engagement may be characterized according to whether the offense is deterministic (i.e., the weapons land where they were aimed and the penetration aids work perfectly), or probabilistic (i.e., the weapons have probabilities of not impacting at their intended burst points and the penetration aids are not completely effective), and whether the defense is deterministic (radars and interceptors are completely effective) or probabilistic (the probability of identifying and then negating a warhead is less than unity, since dis- crimination and AMM reliability and effectiveness are less than perfect). When the defense is deterministic (perfect), the kill mode is exhaustion. When the defense is probabilistic (less than perfect), the kill mode is either by leakthrough (if the defense is far from perfect), or by both leakthrough



344 Samuel Matlin

and exhaustion (experience has shown that if AMM effectiveness is high and more than one weapon is required to kill a target, the exhaustion mode dominates).

The simplest model, level A, assumes that both offense and defense are deterministic (perfect). This may be thought of as the 'idealized' situation. The number of penetrating weapons in this case is simply the differ- ence between the number of attackers and number of defenders, diminished by the number of penetrating decoys. The next level of sophistication, B, is characterized by perfect defense but imperfect offense, an 'offense-conservative' model since it gives the benefit of the doubt to the defense. The number of penetrators is diminished (as in A) by the offense ineffectiveness. In level C, the offense is perfect while the defense is not; a 'defense-conservative' model. Finally, level D examines the more 'realistic' case in which offense and defense are probabilistic. The probability of penetration or number of penetrators is usually computed as a function of the number of attackers and defenders. In the simplest cases the probability of penetration is merely their ratio.

In addition, an a-level characterization is made, as described in the next subsection.

4. The Damage Submodel

In this subsection, the way target damage or value accumulates as a function of the number and types of penetrating weapons is characterized. Two dichotomies are employed: damage may be either deterministic or probabilistic and be either partial (fine-grained) or total (the entire target either survives or is killed). Of the four cases, the simplest, level A, assumes that the damage granularity is either zero or one, and is deterministic. For example, total target value will be obtained by one penetrating weapon (or n penetrating weapons). Allowance may be provided to vary n from target to target, so that target j is completely killed by nj penetrating warheads.

A more sophisticated analysis, level B, considers that a target is either killed or not with some probability, which may either be given or computed as a function of the number of penetrating weapons.

A finer assessment of damage accumulation is afforded by a level C analysis in which the opportunity to accrue partial target value is provided but is deterministic. The partial damage may accumulate discretely (e.g., a table of damage vs. number of penetrating weapons; either the same for all targets or a different table for each target), or continuously (in which case analytic expressions for damage as a function of the number of penetrating weapons are given). The continuous (or at least piece-wise continuous in regions) expression may range from being simply linear




[which assumes that the nth penetrator does as much damage as the (n- 1)st penetrator] to convex [usually monotonically increasing but with decreasing marginal returns, i.e., the nth penetrator does less damage than the (n- 1)st] to some completely arbitrary function.

The more sophisticated damage analysis level D combines the partial damage option with the probabilistic model; there is a probability of doing less than total damage. Value accumulates as a function of the number and type of penetrating weapons, but probabilistically.

If the weapon-scope submodel involves several different weapon types, and the weapon commitment strategy allows for a mixed allocation in which targets are attacked by several types of weapons, then the damage model may become quite complicated, particularly if it is partial and probabilistic. In this case it would probably be necessary to resort to a Monte Carlo approach. Such confusion is characterized as level E.

A special characterization has been made to accommodate several references in which the defense, engagement, and damage submodels have been merged by giving the single parameter Pij, the probability that the ith weapon kills the jth target. This has the effect of combining considerations of target defense, engagement results, and probability of target kill, given successful penetration. Such a model is identified as an a-level characterization, and is ranked greater in complexity than a target defense level A, engagement model level A, damage model level A type of analysis, but it is simpler than any other trio of characterizations. An a-level model is, however, easily generalizable to a D-D-B level model, as Pij may be further analyzed as:

Pij= 1 -REjA j,

where R =the reliability of the ith weapon, Ej== the ineffectiveness of the defense at the jth target, and A j =the probability that the weapon is delivered accurately enough to kill the jth target, given successful penetration and weapon reliability. It is assumed in an a-level characterization that a single weapon is capable of killing the target completely (i.e., a zero/one damage model where one penetrator will kill the target).

5. The Algorithm

The algorithms or computational procedures most commonly employed to determine the optimum allocation of weapons to targets are: analysis (the expected damage equations are differentiated with respect to parameters of interest, set to zero and solved), game theory, graphical or manual techniques (the analyst tries to equalize marginal returns from several targets), graph theory, linear programming, dynamic programming, ex- haustive searches, Monte Carlo techniques, or combinations of these.



346 Samuel Matlin

Lagrange-multiplier techniques are frequently incorporated with these methods since they allow constraints to be easily handled in the optimization, thereby reducing the computational effort. Double Lagrange multipliers are employed in two-sided allocations or games.

The properties of the MAP solution that are of interest and that are intimately related to the algorithm employed are (a) whether the solution is optimal as opposed to near-optimal, and whether the optimality is established, either by proof or by the nature of the: algorithm (e.g., a dynamic programming algorithm operating on integer inputs will automatically produce an optimal solution by virtue of having considered every possibility); (b) whether the solution is integral or continuous; (c) whether the optimal defensive allocation is also determined; and (d) the computational complexity and capability. The ideal algorithm should provide integer solutions, yield a proven optimal solution, derive the optimum defensive allocation as well as the optimum offensive allocation, be capable of handling large weapon and target complexes, run rapidly, be insensitive to small variations in- weapon and target numbers and associated parameters, and provide a global rather than a local or restricted solution.

For the purposes of this survey, the algorithm employed will be identified by name, and as many of the above characteristics as are identifiable from the report will be furnished in the notes and special features. No attempt will be made to rank the techniques in order of complexity.

6. Notes, Special Features, and Results

The characterizations already described may not completely capture the essence of a report. Allowance has been made, therefore, to add any other significant information that might have been bypassed. Included here are such items as:

(a) The possibility of easily generalizing or expanding a capability that was described in the source in a limited or restricted context.

(b) Special features such as those usually associated with a dynamic programming approach. Due to the nature of the search, not only is the optimal allocation of N weapons derived, but that of N-1, N -2, as well.

(c) General results such as, under particular assumptions, targets should be attacked and defended by weapon inventories proportional to target value.

A summary of the characterizations described in subsections 1-5 appears as Fig. 2. This may be employed by the reader to classify other reports on the MAP as they become available, and to serve as a handy reference key in reviewing the reports in this survey.

II. THE LITERATURE SURVEY

THIS DISCUSSION categorizes the references as follows: Type I: Allocation models: single-weapon type




Type II: Allocation models: multiple-weapon types Type III: Game models Type IV: Special-feature models This section discusses the literature in the framework of this categoriza-

tion. The next section provides an annotated listing of all the references surveyed together with the characterizations described in Section I. Figure 3 summarizes this categorization of the literature.

Although all the models surveyed are allocation models, this name is applied here only to the one-sided allocation problems, and in particular where the offense is performing the allocation. The two-sided allocation investigations are covered under game models. Models in which the defense allocation problem is of primary interest are discussed under defense-oriented special-feature models.

Type I: Allocation Models-Single-Weapon Type

Only two references employ an ordinal target value scale: [MIT] and [AUS] (see the annotated references in Section III). These also happen to be the simplest models. They are both assignment models in which a 0/1 incidence matrix is specified against undefended targets, which are ranked in order of priority. Identical weapons are committed simultaneously from an intact force against independent point targets in [MIT], where it is assumed that a single weapon will kill the target, the weapons being perfectly reliable and equally and completely effective against all targets. Integral solutions are obtained by a search technique, and optimality is proved by induction. [AUS] extends this slightly by specifying the number of weapons required to kill each target. (However, the [MIT] algorithm appears to be applicable to this case as well.) This has the effect of permitting area targets to come under consideration; however, 0/1 damage is still assumed. Integer solutions are again obtained, and a proof of optimality by graph theoretic methods is included.

The influence of priority-ranked targets is to require assignment of weapons to higher ranked targets before lower-priority targets are considered. The advantage of a cardinal value model is that total assignment effectiveness can be calculated by summing over targets; the cardinal values permit commensurable effectiveness measures at different targets. A deficiency of the ordinal value scale models is that expected value killed at several lower-priority targets cannot be accumulated in preference to value at a single higher-priority target, even if the lower targets require fewer weapons.

The remaining references all employ a cardinal value scale. The simplest among these are the two deterministic models [BRA] and [WAR]. In [BRA], weapon reach is unconstrained (all weapons reach all targets



348 Samuel Mautlin

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without degradation); but arbitrary damage vs. number of weapons is tabled for each target, thus allowing area targets, partial damage, and terminal defenses to be considered (as admission costs). A dynamic programming technique provides integer solutions.

[WAR] generalizes [BRA]'s model by constraining the weapon reach; a 0/1 incidence matrix is given that refers to launch areas vs. targets. Limits on the number of weapons that may be deployed at each launch area may also be specified. The tabled damage functions are required to be concave. An integer linear programming algorithm is employed.

The next three models, [DEN], [MAN], and [WAL], introduce probabilistic considerations in a simple manner; the probability that a weapon kills a target is given (a-level models). In [DEN], the incidence matrix entries are Pj, the probability that any weapon kills the jth target. A probabilistic argument provides integer solutions, the optimal strategy being that an added missile should be assigned to the target for which the expectation of value destroyed is largest; that is, the targets which provide the greatest marginal return should be attacked. This model is identical to the linear programming approach of [MAN] in its assumptions, but computational improvement is claimed. Finally, [WAL] extends these by providing an arbitrary damage curve as a function of the number of penetrating weapons. Graph-theoretic methods are employed.

Several of these models are historically related. In 1957 (at the Princeton University conference on linear programming, March 13-15), FLOOD described a target-assignment model similar to the 'personnel assignment' problem, in which the expected value surviving to be minimized was:

Zj-=1 Vj57i=1 ( 1-PijXij))

where vj is the value of the jth target, Pij is the probability that the ith weapon kills the jth target, and Xxj is the probability that the ith weapon is assigned to the jth target. Because the minimand involved probabilities and was nonlinear in form, Flood concluded that linear programming was inapplicable. MANNE [MAN] reformulated Flood's problem as a transpor- tation-type linear-programming-under-uncertainty problem by making the simplifying assumptions that: (a) the jth target is killed with probability Pj for any weapon assigned against it (Pij=Pj for all i); and (b) the approximate survival probabilities are to coincide with the original ones for integral assignments, i.e., for Xxj values of zero and one. This reduces the minimand to

Ej=1 v -p)Y

where Yj is the number of weapons assigned to the jth target. Although the minimand is still nonlinear, the method of separable convex functions is invoked to permit a linear programming formulation.



350 Samuel Matlin

DENBROEDER, ET. AL. [DEN] then developed an iterative procedure to solve Manne's formulation, which enjoys computational advantages over the linear programming approach. The algorithm reduces to assigning the next weapon to where the expected marginal return is largest.

Finally, WALKUP AND MACLAREN [WAL] use a graph-theoretic derived algorithm to solve Manne's formulation. They claim computational, advantages over the linear programming method and also question one of Manne's assumptions.

Type II: Allocation Models-Multiple-Weapon Types

The allocation problem is more complicated when several weapon types and/or penetration aids are considered. If decoys are modeled, weapon penetration probability is usually found by one of three techniques: (a) decoys and interceptors are assumed to be perfect and the exhaust-then-kill mode is employed so that interceptors serve only to set an admission price to the target and weapon penetration probability is zero or one, according to whether the defense has been exhausted; (b) an a-type model is employed in which the probability of weapon penetration is given [an implicit way of handling penetrations, simplified to the extent that varying numbers of decoys (and boosters required) against terminal defenses cannot be examined]; and (c) penetration probability is taken to be a function of the ratio of attackers to defenders, as in [BEL] and [PER].

Target damage models may be greatly complicated if multiple weapon types are considered as in [DAY]. A simplifying dodge is illustrated in [LEM], where different weapons are converted to equivalent numbers of a single standardized weapon, the allocation then being made on this single- weapon type, the results then converted back to the appropriate numbers of the original weapons. At the very least, introduction of multiple weapons can significantly increase the time required to compute an optimal allocation.

[JAC] and [LEM] are similar analyses in which multiple weapon types without penetration aids are considered. Weapon reach in both cases is provided by an incidence matrix whose entries fij may be interpreted in two ways: (a) fij is the kill probability of weapon i against point target j, or (b) fij is the fraction of the value remaining at target j, which weapon i is expected to extract. These models implicitly confound the effects of defenses, reliabilities, delivery accuracies, weapon-reach degradations, etc. Expected damage accumulates monotonically as a function of the number of weapons. In the single-weapon-type case, the law of diminishing returns prevails. For area targets this generally occurs because several weapons are required to cover the target, and the (n+ 1)st weapon does less expected damage than the nth weapon; for point targets this occurs because the




probability of killing the target also builds up in this fashion, with expected value killed a concave function of the number of weapons and the single- shot kill probabilities, with decreasing marginal returns. With several weapon types the damage builds up monotonically but not necessarily convexly. In [JAC], an intact force is simultaneously committed, while [LEM] provides for launch attrition. Both models employ cardinal value scales assume no defenses, and can accommodate area as well as point targets because of the ambiguous nature of the incidence matrix. [JAC] assumes a perfect deterministic engagement that, in the absence of defenses, means that the weapons penetrate and extract the given expected fractions of target values. [LEM] allows for delivery inaccuracy by computing the probability that the ith weapon kills the jth target, given a successful launch and penetration, as a function of target hardness, weapon yield, and CEP. The damage model in [LEM] is simpler since the different weapon types are converted to a single standardized type: partial damage accumulates as a function of the number of standard penetrators. Since such a conversion is not made in [JAC], the damage must be computed according to the particular mix of weapons assigned. [JAC] employs a modified dynamic programming algorithm to obtain integer solutions; however, only a small number of targets may be considered. [LEM] employs analysis to obtain continuous optimal solutions; however, LEMUS observes that the analytic solution, after rounding off, is almost identical to the integer solution in [DEN].

The next two models, [BEL] and [PER], are also similar in their assumptions. Both consider a single weapon type with decoys explicitly provided for (although [BEL] points out the possibility of extending to several weapon types). Both assume all weapons can reach all targets without degradation, the weapons being simultaneously committed; intact in [BEL], but launch and flight attrition a possibility in [PER]. Known terminal defenses at targets with cardinal values are assumed in both models. [BEL] considers point and area targets, while [PER] is limited to point targets. Both models assume the damage to be exponentially related to the ratio of attackers to defenders at a target. [BEL] utilizes dynamic programming together with Lagrange multipliers, which reduce the computational effort to obtain not only the optimum weapon allocation, but the optimum defense allocation and least-cost mix of attackers and decoys as well (the latter as a function of the Lagrange multiplier, which represents the cost of a decoy), thus providing implications for the force-mix problem. In addition, the optimum allocation policy for any number of weapons ? n is a by-product of the dynamic programming approach. [PER] employs analysis and an extremely illuminating graphical approach to obtain an integral near- optimum solution. The optimal defense allocation is also developed. In



352 Samuel Matlin

both analyses damage accumulates monotonically with decreasing returns at each target as a function of the number of attackers. Optimum strategy in such cases consists of attacking the targets at the most effective levels, eventually equalizing expected value per attacker (adjusting for discrete- ness). The defense should seek to equalize maximum possible ratio of value killed per attacker at each target. If some targets or weapons are removed, then except perhaps for minor readjustments the original allocation is still best, since the value per attacker has not been disturbed. This explains why [BEL] exhibits linear damage with number of attackers.

The next two references are instructive, one in theory and the other in practice. [EVE] is concerned with the general 'cell problem,' that of optimally allocating multiple resources (weapons) to a number of independent ventures (targets), where the total payoff is the sum of the payoffs that accrue from each venture. Lagrange multipliers are employed, a device that reduces the problem to a series of independent unconstrained maximization problems, one for each cell. EVERETT observes that the Lagrange multiplier method of solving constrained maximum problems is not limited to differentiable functions, but may be applied in situations involving maximization of any type of function over any set of strategies, discrete or continuous, numerical or nonnumerical, with constraints that can be represented as bounds on real-valued functions over the same strategy set. The MAP is briefly formulated as a cell problem, the author noting that the solution is quite simple and straightforward.

The approach in [DAY] is unique in that a three-stage method of assigning weapons to targets is employed that decomposes the general allocation problem into a set of targeting problems in the small and a targeting problem in the large. The former are solved by a sequential optimizing method (such as [FIR]) that provides information for solving the latter by nonlinear programming. Near-optimal designated ground zeroes and heights of burst are derived for an undefended target complex in which points within a subcomplex are dependent, while each subcomplex is independent.

The final multiple-weapon allocation model, [FUR], provides a computer program for optimally allocating a mixed offense force opposed by a defense with overlapping area battery coverages. Two interesting simplifi- cations are employed. The multiple-weapon attack is handled by assigning only one payload type to each target, but allowing a mix of payloads delivered over the target set, thus facilitating the damage calculations while permitting 'mixed' attacks. The problem of penetration aids and imperfect weapons is approximated by specifying the number of interceptors required to destroy the payload at high and low altitudes, as well as the number of payloads of each type required to exhaust the terminal defense at each target.




Type III: Game Models

Game models consider the two-sided allocation problem, and examine the situation in which neither side knows the other's weapon assignment at each target, although total inventories may be known to both sides. That is, the offense knows that the defense has a total inventory of D interceptors, but it does not know how these are distributed among the targets. If there are terminal interceptors, the offense is ignorant of how many will defend each target (although it may know whether a target is scheduled for defense since presumably it has information about terminal radar emplace- ments). If there are area interceptors, the offense does not know which targets are to be defended, nor how many interceptors will be assigned to each target chosen to be defended. On the other hand, the defense is presumed to be ignorant of which targets the offense will choose to attack, and of the number of offensive weapons assigned to each target. The one-move game investigated is played by each side choosing an allocation that then does not change during the course of the engagement.

The terminal defense case is explored by [COH], [MAT 67], [SCH], [MCE], [PEN], and [OWE]. Preferential area defenses are examined in [EIS] and [GAL]. Other gaming approaches, [FOX] and [MAT 67], are concerned with a special feature of the MAP, the force structuring aspect, and are described in Section Type IV.

[MCE] examines the highly simplified game in which T targets are attacked by A identical missiles and defended by D identical interceptors. Missiles and interceptors are perfect (Pk= 1.0). The targets are of equal value, and a single excess missile kills a target. Optimum offense and defense deployments are derived for the offense-dominant case (A > D) and the defense-dominant case (A < D).

[PEN] extends this model by allowing targets to differ in value. A further extension is considered in which A more missiles than interceptors are required at a target to kill it. A final extension combines the possibil- ities of different values and the A requirements.

[IMAT 67] relaxes the perfect weapon requirement by allowing both missile and interceptor kill probabilities less than unity. [SCH] analyzes the same case by other methods.

[COH] extends the game by considering the case in which the offense has several types of weapons that are less than perfect, the defense has different weapon types as well, and the targets vary in value. The attacker is assumed to be at least as strong as the defender (A> D). DRESHER'S

'no-soft-spot' principle is invoked, which claims that an optimal defense strategy is to defend only those targets that, under a concentrated attack, would yield the attacker the value of the game and, conversely, the attacker should attack only those targets that are chosen to be defended.



354 Samuel Matlin

[OWE] considers the case in which the defense has a fixed budget to split between active (terminal defenders) and passive (shelters) defense.

[EIS], [GAL 1], [PEN 69], [GAL 2], and [GOO] extend the game to the consideration of preferential area defenses working in conjunction with terminal defenses. In these games, as in the foregoing, the optimum defense deployment is derived concomitantly with the optimum offense deployment. GALIANO claims that "the optimal attack is proportional to target value in the face of area defense, as is the optimal terminal deployment itself. Furthermore, on targets chosen to be defended by area defense, the area defense allocation is equivalent in amount to the extra attack on that target [over the price], hence also proportional to target value. . ."

The method of Lagrange multipliers as used by [EIS] is dangerous, as observed by [EVE]: "As a final note, the reader is cautioned against indis- criminately applying this method to min-max problems (where there are two sides allocating resources, an attacker and a defender, for example, with opposing interests). It is tempting in such cases to introduce multipliers for both sides and then carry out a min-max operation on the resulting Lagrangian, the analogy of the pure maximization case. However ... the procedure is not 'fail-safe,' but can and does in many instances produce erroneous results." The seriousness of this warning is ameliorated by [PUG], where it is admitted that the method is not justified and may lead to erroneous allocations, but that usually correct results are obtained. Furthermore, PUGH gives techniques for verifying results and computing bounds for the error.

Type IV: Special-Feature Models

The references in this category are more concerned with special features of the MAP than with the over-all classical allocation problem.

Targeting in the small. Almost all of the previous analyses assumed target independence; a missile allocated to one target could not simultaneously damage some other target as well. [FIR] and [GRO] consider the dependent target case. These models are not concerned with the global allocation problem, but rather with a small region of the target complex. In [DAY], one of these targeting-in-the-small algorithms is employed as a submodel.

[FIR] provides a search procedure to determine where weapons should be aimed to maximize damage against a group of quasi-separate point targets. The technique is computerized to handle twenty targets of varying value, location, expected defense, and hardness; twenty weapons of varying reliability, yield, and delivery accuracy; and 250 aimpoints, which may be either actual targets or points between targets.

[GRO] offers a method that determines whether a single given nuclear




weapon, when delivered with a given CEP, is capable of defeating a group of area targets of different sizes, vulnerabilities, and target-location uncertainties, with a single shot.

Weapon attrition. The previous references considered weapon attrition (if at all) only from the point of view of reliability. The major concern of [BOE] is primarily with the target-sharing-among-squadrons aspect of the problem. The concern here is that if two missiles with decoys share the same terminally defended target and one of the missiles fails, the other might not have enough decoys to defeat the defense and hence should be assigned to some other target in preference to 'feeding' the defense.

[MAC] treats the problem: what is an optimal weapon assignment under the condition that only a reduced force will be available for combat? Re- dundant targeting (but not across squadrons) is then the major concern. It is explored Monte-Carlo style by randomly attriting an original incidence matrix.

Force structuring. In the MAP, a weapon force is given and the problem is to target it optimally. This is a subproblem of a larger consideration: what should the weapon mix or force structure be, so that when optimally targeted, it will do the most damage (for a fixed cost or budget)? The force-structure aspect of the MAP is significant in the following models.

A slight restructuring of the missile force is permitted in [MOR], in that a trade-off of booster payload vs. range is considered. This is a force- requirement model that minimizes the number of boosters required to meet the target demands, where boosters carry varying numbers of RV's of a single type, depending on the range to which they are delivered. The number of RV's to be placed on each booster as well as their optimal targeting is derived using a small-scale desk-side linear program (number of targets times launch sites is approximately thirty).

[TAY] is a resource-allocation analysis in which the problem is to distribute the total available resources among combinations of target and type of already developed weapons in such a manner that the value of the targets surviving the attacks is minimized.

In [MAT 65], a two-sided resource allocation game is presented in which one player allocates his resources between two offensive weapon systems while his opponent allocates his resources between two defense weapon systems.

[BAS] is a weapon-requirements model (to achieve a desired level of damage) performed in two phases-a computer phase, followed by a manual phase. The computer yields the number of boosters required as a function of fatalities per attacker, for various levels of percent value attacked. A manual graphical search is then required that considers damage desired and total area-defense inventory to determine the number of boosters required.



356 Samuel Matlin

The percent value to be attacked and the subset of cities to be attacked in order of priority (via marginal returns) also results.

Indirect values. [PIC] examines the situation in which the target complex consists of N launch sites and C redundant control centers, 1 <C <N, such that any one surviving control center is capable of launching all N missiles. The N sites have equal values. The control centers have no value in themselves; however, if and only if all C centers are killed, total value N is obtained. Results in the continuous case indicate that optimum offense strategy consists of allocating all weapons to either sites or centers. In the discrete case, a mixed strategy is best unless the total number of attackers is divisible by both N and C.

Defense orientation. [PUG] is primarily concerned with the methodologi- cal aspects of the double-Lagrange-multiplier method applied to the problem of allocating defense resources. This report extends [EVE] by providing "a technique for verifying results and computing bounds for the error, a technique that is necessary because the uncritical application of the method can occasionally give results that are far from optimum."

[FOX] is concerned with the defense-allocation problem, and provides a double-Lagrange-multiplier model that is an extension of [PUG] and [EVE]. The defense optimizes its allocation by choosing how many cities to defend as well as how to distribute its terminal inventory over these cities. The offense is assumed to know the number of terminal defenders at each city, while the defense knows only the total offense inventory.

[DEW] considers the problem of allocating interceptors to targets protected by several overlapping area defense isles. The inventory at each defense island is given (a natural extension would be to derive this), and the attack size of each target is known. The number of missiles from overlapping isles to be used in defense of each target is determined via integer linear programming. Mention is made of a related method used in a cost- effectiveness algorithm for MIRV target partitioning.

Optimization criteria. The usual measure of effectiveness optimized in allocation studies is the expected value killed or survived. [THO] compares two different defense criteria: minimize the expected number of penetrators, and maximize the probability of no penetrators. Attack situations are discussed in which the two criteria result in identical defensive missile allocations, and in which the allocations are different.

[WEI] examines the criteria question in the general context of stochastic assignment models and suggests a criterion that, under certain conditions, might be preferable to the traditional one.

It should be noted that the criteria employed in gaming models are different still; it is not total expected value that is optimized, but, rather, the offense desires to maximize the minimum expected return (max-min),




while the defense seeks to minimize the maximum offense return (min- max).

III. ANNOTATED REFERENCES

THIS SECTION lists alphabetically the references reviewed, together with brief annotations (usually consisting of the authors' abstracts). The model characterizations are gathered in Table II; however, the notes associated with these characterizations are included with the annotations.

[AUS] AUSTIN, T. L., "MEEG Operations Analysis for CFR Study (U)" Appendix C: Autonetics Targeting Algorithm, 67-SMQA-17, TRW, Sep- tember 11, 1967, SRD.

Appendix C: Autonetics Targeting Model, pp. 97-115 (U). "An algorithm is described for assigning weapons to targets under the following conditions: (a) each weapon is constrained from attacking more than some specified subset of the target list; (b) each target requires some specified number of weapons to attack it for optimal destruction; and (c) the target list has a priority ordering. The algorithm accomplishes a maximal assignment of weapons and an attack on a maximal priority set of targets."

[BAS] BASICKES, S., "Booster Allocation Model for Combined Terminal and Preferential Area Defenses," 8025-029, General Electric Co., Valley Forge, Pa. pp. 42, November 6, 1967.

Model is used to evaluate ICBM force structuring for attacks on a large number of defended urban targets. Candidate re-entry systems are allocated to targets subject to booster range constraints and defense levels. The allocation is optimized on a target-by-target basis. The resultant system is then evaluated for a specific damage level to determine level of attack and required number of boosters.

Notes: (1) This is a weapon-requirements model (to do a certain level of damage), rather than a MAP. (2) Computer output is number of boosters vs. fatalities per attacker, for various percent value attacked. A manual graphical search is then required that considers damage desired and total area defense inventory to determine the number of boosters required. The percent value to be attacked and the subset of cities to be attacked in order of priority (marginal returns) also results.

Results: The solutions are very sensitive to the input parameters (particularly area interceptors and damage required).

[BEL] BELLMAN, R. E., DREYFUS, S. E., GROSS, 0. A., AND JOHNSON, S. M.,

"On the Computational Solution of Dynamic-Programming Processes- XIV: Missile Allocation Problems," RM 2282, Rand, Santa Monica, Calif., November 13, 1959.

Problems involving the optimum allocation of attack against a target system and



TABLE II

CHARACTERIZATION OF THE MISSILE-ALLOCATION-LITERATURE

MODEL CHARACTERIZATION

Weapon system Target complex _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ E n g a g e - D m g

ment Damage Algorithm

Scope Reach Coit- Scope Value Defense Model model mit.

GR [AUS] A B A B B A A A IS,-OP

is, OP

SE, MA [BAS] D C A' B C B, F A C CS;NO;-LS

CS, NO, LS

DP, LM

[BEL] B' A A B C B B D TS

TS

2

[BOE] A B B B A' B A C'

DP [BRA] A A A B C Al A C IS,-OP

is, OP

GA [COHI C A A A C EB A E2 ----------

IS, OP, TS

NP [DAY] C A B B' C A A E BS

CS

A(Model I) AN [DEN] A C A A and I a' a-

C (Model II) IS, OP

LP [DEW] A (I) A A C (2) A A IS

(N=i) is

LM [EVE] D A A B C B D B__

SB, LP [FIR] A A A C C a a a IS,-NO

IS, NO

SB, LM [FOX] A A A B C' B2 A C3 ----------

LP, LM [FUR] D' B A B C B, C2 A3 D4 NO5

N05

A B A(N=i) AN [GALi] A A A & C & A & CS,-IS

B I!F C CS, is

AN, GA [GAL2] A A A, C' A A B, C A A

E, F2 (N=I) TS

358




TABLE II-Continued

Weapon system Target complex _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ E n g ag e- D am ag e A l o i h

ment model Algorithm Scope Reach Cmi Scope Value Defense Model

mit.

GA, LM [GOO] B' A A B C C, E, F D' D ----

is, Cs, NO, TS

AN, MA [GRO] A' A A D A A B D

DP, MA [JAC] C C A B C A A E IS;-SS

is, Ss

AN [LEM] C1 C B B C A B B CS,---

Cs, OP ___ - ____----- ----- -_ __ ___ -__ ___ _----- MC

Mc [MAC] A B B* A C a a a-IS,-

IS, NO

LP [MAN] A C A A C a a a-I5,-

is, OP

GA [MAT6s] C A Al A C A2 A E CS;-TS

(N =2) CS, TS

GA [MAT67] A A A A A E D B IS,-OP;TS

IS, OP, TS

AN, GA [MCE] A A A A A E A A IS,-OP;TS

IS, OP, TS

SE [MIT] A B A A B A A Al IS,-OP

(N~i is, OP

LP [MOR] Al C A B A A2 A A3 CS,-SS

Cs, Ss

AN, LM [OWE] A A A B C B A C1 NO,-TS

NO, TS

A(Models I &3) GA, LM

[PEN67] A A A A & E A A C(Models iS, CS

2 &4) CP, TS



360 Samuel Matlin

TABLE II-Continued

Weapon system Target complex

__ _______________|__ ___________________ _ ____ Engage-____

ment Damage Algorithm

Scope Reach mi.Scope Value Def enseMoe mdl

AN, GA IPEN69] A A C A A EF A A -----

(N =E) CS, TS

AN, MA [PER] B A B A C B D B --

IS, NO, TS

AN [PIC] A A A Al D2 a-a--

CS, OP

LM [PUG] C A A B C B D C

NO, TS

GA, AN [SCH] A A A Al A E D2 C2 ----------

CS, NO, TS

SE TuAY] C Bi A B C a---- an a

a a a ~~~GR [W AL] A C A B C -- -- - -- - - -- -- -- - - - - -

B D C

LP [WAR] A Bi A B C A A C2 -----

is

the optimal allocation of defense against this attack are considered using dynamic programming (expedited by Lagrange multipliers in special cases).

Notes: (1) May be extended to D. (2) Outputs optimal policy for any number of weapons ?N.

Results: Model provides least-cost mix of attackers and decoys vs. the Lagrange multiplier (which represents the cost of a decoy), and thus has implications for the force-mix problem.

[BOE] MEEG, "Operations Analysis for CFR Study (U)" Appendix B: Boeing's Squadron Targeting Model, TRW, Report 67-SMQA-17, pp. 161, September 11, 1967. SRD. [Appendix B: Squadron Targeting Model (Boeing), pp. 85-95 (U).]

An intuitive 'near-optimum' guide for targeting across squadrons in two ways: (1) a low-intelligence-confidence model wherein 'adequate' system effectiveness is desired for a wide gamut of levels of surviving missiles; and (2) a high-intelligence-




confidence model wherein good performance (approximate equivalent to force re- planning, retargeting) is desired in the region of the expected threat.

Notes: (1) Missile requirements per target are specified. (2) Incidence matrix manipulation.

[BRA] BRADFORD, J. C., "Determination, of the Optimal Assignment of a Weapon System to Several Targets," AER-EITM-9, Vought Aeronautics, Dallas, Texas, pp. 14, May 12, 1961.

A dynamic programming model is used to solve the problem: "Suppose a single weapon system is being considered, of which there are N units available for assignment. There are M targets to which the units may be assigned. Some of the targets may or may not be of the same type. Suppose further that for each target i, there is a known effectiveness function or payoff function Ei(ni) representing the value of assigning ni units of the weapon to target i. Then the problem is that of determining l *, Am subject to the constraint (1) hi+ --. +im =N such that (2) El(,i) +E2(A 2)+ - +Em(Qm) ==E(N) is a maximum." Note that there are no defenses; arbitrary payoff functions vs. number of attackers.

Notes: (1) The tabled damage function would permit a level B defense submodel.

[COH] COHEN, N. D., "An Attack-Defense Game with Matrix Strategies," RM-4274-1-PR, Rand, Santa Monica, Calif., pp. 28, August, 1966.

Author's abstract: "This memorandum presents the results and the method of analysis for an attack-defense game involving allocation of resources. Each of the two players is assumed to have several different types of resources to be divided in an optimal fashion among a fixed set of targets. The payoff function of the game is convex. The 'no-soft-spot' principle of M. DRESHER and the concept of the generalized inverse of a matrix are used to determine optimal strategies for each player and the value of the game."

[DAY] DAY, RICHARD H., (University of Wisconsin), "Allocating Weapons to Target Complexes by Means of Nonlinear Programming," Operations Research 14, 992-1013 (1966).

Author's abstract: "The paper presents a three-stage method of assigning weapons to targets and illustrates its application with a small sample problem. The method decomposes the general allocation problem into a set of targeting problems in the small and a targeting problem in the large. The former are solved by a sequential optimizing method that provides information for solving the latter by nonlinear programming. The discussion is not advanced and illustrates a basic feature of applied operations research: the combination of existing operational methods to achieve approximate solutions to complicated problems. The study may have some added interest because of its relation to problems of decentralized planning."

Notes: (1) Each complex is independent, although points within a complex are dependent.



362 Samuel Macitlin

Special feature: Three-stage optimization combining 'targeting-in-the-small' (within complex) with 'targeting-in-the-large' (between complexes).

[DEN] DENBROEDER, G. G. JR., ELLISON, R. E., AND EMERLING, L. (Lock- heed), "On Optimum Target Assignments," Operations Research 7, 322-326 (1959).

Author's abstract: "This note is concerned with two target assignment models. An optimum assignment is one which maximizes the expected value of targets destroyed. The first model, which admits an explicit solution, associates values only with the number of targets destroyed. An algorithm which enjoys a computational nicety is established when the values of the individual targets are assumed known. This latter model is a special case of FLOOD'S target-assignment model."

Notes: (1) Pj, the probability that any weapon kills the jth target, is given. Model II is Flood's model as investigated by [MAN], where it is reformulated as a trans- portation problem.

[DEW] DEWITTE, LEENDERT, "Optimum Defense Against Given Threat," TR-669(S6230-37)-4, Aerospace Corporation, San Bernardino, California, pp. 18, May 1966.

Author's abstract: "In order to study the effects of various offensive strategies, it is often assumed that the enemy's defensive response will always be optimal. In many cases the determination of this optimal defense is not trivial. The present report solves the optimal defense problem for targets of specified individual values defended by overlapping groups of antimissiles. The solution is obtained by reducing the problem to a linear programming problem with zero-one variables for which known solutions exist in the literature. The assignment of the defensive missiles among the various launch sites also allows an integer linear programming formulation."

Assumptions: Offensive and ABM kill probabilities are unity. Each of N targets is defended by missiles of at least one defensive launch complex. Each target has a given value.

Notes: (1) Specified attack, 13-level defense reach. (2) Inventory at each defense island known; allocation from overlapping islands to targets sought.

[EIS] EISEN, DENNIS, "Defense Models 1. Basic Analytical Approaches to Ballistic Missile Defense Models," Lambda 3, Lambda Corp. Arlington, Va., pp. 24, September 1967.

Author's introduction: "This paper presents several simple models of ballistic missile defense effectiveness, for which an analytic approach leads to fairly tractable rela- tions for optimal attack and defense levels against a set of targets of fixed or varying values and vulnerabilities.... The following section discusses the approach for analytic solution of two-sided optimization problems. In all cases, the objective function has been formulated in terms consistent with the Generalized La-




grange Multiplier technique.... Three terminal defense models are next discussed.... The final section discusses the addition of area defense to one of the terminal defense models."

[EVE] EVERETT, HUGH, III (IDA), "Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources," Operations Research JQ, 399-417 (1963).

Author's abstract: "The usefulness of Lagrange multipliers for optimization in the presence of constraints is not limited to differentiable functions. They can be applied to problems of maximizing an arbitrary real valued objective function over any set whatever, subject to bounds on the values of any other finite collection of real valued functions defined on the same set. While the use of the Lagrange multipliers does not guarantee that a solution will necessarily be found for all problems, it is 'fail-safe' in the sense that any solution found by their use is a true solution. Since the method is so simple compared to other available methods, it is often worth trying first, and succeeds in a surprising fraction of cases. They are particularly well suited to the solution of problems of allocating limited resources among a set of independent activities."

[FIR] FIRSTMAN, SIDNEY I., "An Approximating Algorithm for an Opti- mum Aim-Points Problems," P-1678, Rand Corp., Santa Monica, Calif., pp. 28, 22, April 1959.

If the offense has a given number of weapons to direct against a target complex that is composed of several or many quasi-separate targets, where should each weapon be aimed so that the resultant expected damage is maximized? Weapons can be aimed directly at targets, or at some chosen point in the space between targets. Missile reliability, expected enemy missile defense, missile inaccuracy, and lethal radius for each target are considered. Each target is characterized by its relative value and hardness. The weapons are delivered independently according to some given impact distribution, and no information of results is obtained between shots.

[FOX] Fox, PETER D., WEINSTEIN, IRAM J., "Lagrange Multiplier Models for Allocating Defense Resources," Stanford Research Institute, Menlo Park, Calif., pp. 15.

This report reviews PUGH'S double Lagrange multiplier model for solving the offense-defense deployment game and then offers an alternative but related procedure.

Notes: (1) Values are population fatalities. (2) The offense knows the number of terminal defenders at each city. However, since this is a defense-allocation problem, the defense must find its optimal allocation knowing only the total offense inventory, not the attack size at each city. (3) At each city, there is a concave payoff to the offense.

[FUR] FURMAN, G. G., "Program OPSTRA: Optimal Allocation of a



364 Samuel Matlin

Mixed Offense Force Opposed by a Defense with Overlapping Area Battery Coverages," TR-5205-34, Stanford Research Institute, Menlo Park, Calif., January 1969.

IDA symposium abstract: "The goal of the program is the allocation of a reliable offense force, consisting of a variety of missile payloads, over a defended target set in a damage-maximizing way. The defense consists of area and terminal defenses, each having a number of reliable interceptors known to the offense. The coverage patterns of area batteries overlap freely. The program is not intended to itself determine defense-allocation strategies, as the allocation of interceptors to batteries is a user input. For this defense-conservative scenario, the program utilizes a new form of EVERETT'S Lagrange multiplier method to generate offense allocations that bound or give the maximum expected target damage."

Notes: (1) Each target receives only one payload type, even though a mix of payloads is delivered over the target set. (2) Terminal defense at each target specified; area interceptors at each of several overlapping batteries specified; both defenses are subtractive. (3) The number of perfect interceptors required to destroy the payload at high and low altitudes is specified, as is the number of payloads of each type required to exhaust the terminal defense at each target. (4) Square-root-law and power-law damage functions. (5) Typical running times per case are two minutes on the CDC 6400.

[GALl] GALIANO, ROBERT J., "Defense Models II. Simple Analytic Models for Terminal and Preferential Area Defense," Lambda 4, Lambda Corp., Arlington, Va., pp. 23, September 1967.

Author's abstract: "This paper briefly describes certain antiballistic missile defense models currently realized in a computer program called EXCHG, developed to replace, for certain applications, a more detailed computer model called IVAN 2, separately described. IVAN 2 was developed to evaluate force postures against terminal and nationwide preferential area defense, treating as well random area defenses, convertible terminal defenses, and multiple nonoverlapping islands of area defense coverage...."

[GAL2] GALIANO, ROBERT J., "Defense Models III. Preferential Defense," Lambda Paper 5, Lambda Corporation, Arlington, Va., pp. 41, March 1967.

Author's abstract: "Randomized allocations of attack and defense weapons, not dependent on knowledge of the opponent's deployment are much invoked in cur- rent missile defense studies. The implied strategies are often termed preallocater or precommit strategies, since the allocations can be determined in advance and applied or revealed only at engagement time at the targets, giving the effect of simultaneous decisions on employment of forces. Thus, typical game theoretical techniques are useful to determine optimal strategies on either side. It is observed that preferential area defense, which depends upon impact point prediction and longer range intercept can select the most threatening of the attackers among many targets for priority engagement and achieve a factor of two greater effectiveness



Review of the Literature on the Missle-Allocation Problem 365

relative to such simultaneous randomized allocations. In fact, the higher level of effectiveness does not require the area defense to know the exact attack levels at each target, only the total force to arrive over all defended targets. Thus, the same level of full preferential effectiveness can be achieved against simultaneous or sequential attacks. When area defense is limited (by restricted geographic coverage) to protection of many independent islands or tragets, then a randomized attack strategy can deny knowledge of the intended attack level to the area defense at any island and force the defense back to the preallocation level of effectiveness. Of course, the limit of fixed known local defense (island of one target each) is still worse for the defense. The attacker can then concentrate on an optimal subset of the defended targets and gain a further factor of two in effectiveness by design- ing his last move after observing the opposing deployment."

Notes: (1) With and without damage assessment capability for attacker. (2) In- cludes nonoverlapping defense islands.

[GOO] GOODRICH, R. L., "Defense Models V. IVAN 2. A Detailed Model for Preferential Defense," Lambda Paper 7, Lambda Corporation, Arl- ington, Va., pp. 73, June 1968.

Author's abstract: "IVAN 2 (a computer program) treats the problem of calculating blast damage to a discrete set of targets with a spectrum of values, under a combination of area and ballistic missile defenses. A variety of deployment assumptions may be tested. The model is capable of optimal and near-optimal deployments for all three elements-attack, area defense (either preferential or random), and terminal defense."

Notes: (1) Penetration aids implicit, as number of defenders required to nullify a single booster is given. (2) A computer program is fully described.

[GRO] GROVES, ARTHUR D., "A Method for Determining the Effectiveness of a Single Nuclear Weapon Against a Group of Targets," BRL MR 1689, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland, pp. 20, August 1965.

Author's abstract: "A method is given whereby it can be determined whether a given nuclear weapon, when delivered with a given circular probable error, is capable of defeating a group of targets with a single shot. There is no restriction on the number of targets, and they may be of different sizes and vulnerabilities; each may have a circular probable error of target location."

Notes: (1) A single weapon is considered.

A 'targeting-in-the-small' analysis, in which the analysis determines the smallest- yield weapon to defeat the target set, and manual overlays aid in selecting the aim- point.

[JAC] JACOBSON, A. E., AND CRABTREE, D., "An Optimal Reassignment Algorithm," ESD-TR-67-380, Mitre Corp., Bedford, Mass., pp. 25. Decem- ber 1965.



366 Samuel Mcatlin

Given a set of JIl targets T1, , Tm having values V1, *.*, Vm, a total of N weapons W,, *.., VVN, and the effectiveness of each weapon on each target in terms of the fraction of the target value killed (where the effect of one weapon against a particular target is assumed to be independent of other weapons that may be assigned to the same target), an algorithm is developed to produce an optimal assignment of weapons to targets that maximizes value destroyed. The algorithm is a discrete comparison procedure that adds a target at the nth stage and compares the previously best allocation to the n-1 targets and all possible reassignments with the added nth target.

[LIEM] LEMUS, F., AND DAVID, K. H. (SHAPE), "An Optimum Allocation of Different Weapons to a Target Complex," Operations Research 11, 787-794 (1963).

Author's abstract: .. . The object of this paper is to generalize an analytic solution to the case where the attacker has more than one type of weapon available for assignment to an undefended or virtually undefended target complex. Defense strategies are not included. The paper presents a computational example where three types of weapons, each with a different number of attacking vehicles, are allocated to twenty targets. This example also permits a comparison between the fractional results of an analytical solution and the integral values obtained by a computer-oriented algorithm."

Notes: (1) Different weapon types are made equivalent to a 'standard' type.

Results: The analytic solution is almost identical (after rounding off) to the solution in [DEN].

[MAC] MACLAREN, lM. D., AND WALKUP, D. W., "A Missile Targeting Problem," Boeing Sci. Res. Labs., Seattle, Washington, pp. 36, June 1964.

Author's abstract: "The following missile assignment problem is considered. Mis- siles are to be assigned to targets in two distinct steps. First, each missile is programmed so that it can be fired at any one of a small number of targets, the number of targets being the missile capability. The programming of the missiles is represented by a qualification matrix Q. Second, if battle occurs, all missiles are to be assigned to targets and launched. Each missile must be assigned to a target for which it is programmed. It is assumed that only a random subset X of the missiles will actually be available for battle, and so the assignment must be made for a reduced qualification matrix Q(X). The questions considered are 'what is an optimal assignment given the reduced qualification matrix Q(X),' and 'what can be expected from this assignment.' Use of a damage function is proposed. An optimal assignment is one which maximizes the value of the damage function. The damage function may be chosen to represent a wide variety of optimization requirements. The main part of the paper describes Monte Carlo procedures for estimating the expected damage and the probability that the damage will be at least c for any number c."

Notes: Special feature: An original incidence matrix is randomly attrited (probability a weapon is available is p).




[MAN] MANNE, ALAN S. (Yale U.), "A Target Assignment Problem," Operations Research 346-351 (1958).

Author's abstract: "This paper is concerned with a target-assignment model of a probabilistic and nonlinear nature, but nevertheless one which is closely related to the 'personnel-assignment' problem. It is shown here that, despite the apparent nonlinearities, it is possible to devise a linear programming formulation that will ordinarily provide a close approximation to the original problem."

Note: (1) The method reduces to attacking the targets providing the greatest marginal return.

[MAT65] M\ATHESON, JOHN D., "An Application of Game Theory to an Offense-Defense Resource Allocation Problem," AR 65-7, ANSER, Falls Church, Va., pp. 36, October 1965.

Author's abstract: "This report presents a two-sided resource allocation game in which one player allocates his resources between two offensive weapon systems to maximize target destruction. The opponent allocates his resources between two defensive weapon systems to minimize target destruction. The game is continuous in the sense that all variables are assumed to be continuous, and static in the sense that time is not a variable. Explicit solutions are obtained in terms of the parameters of the problem."

Notes: (1) The defense moves first; then the offense attacks knowing the defense

deployment. (2) Defense weapon type {a} is only effective against offense weapon

l be~~~~~~~~~ type {b}

Feature: This is a force-structuring problem since it advises what the mix of weapon types should be.

[MAT67] MATHESON, J. D., ENDRISS, S., CHRISTIE, D., AND LAKE, D., "Preferential Strategies with Imperfect Weapons," AR-67-1, Analytic Services, Inc., Falls Church, Va., pp. 221, April 1967.

Author's abstract: "The report discusses targeting strategies for attack and defense when the weapons on both sides have a probability of kill less than unity. Canon- ical minimax solutions are formulated for all probabilities of kill and for all den- sities of attack and defense. The application of those solutions to integral problems is discussed and illustrated. Computer programs are provided to determine average target survival for variable attack density with constant defense density and for variable defense density with constant attack density."

FMCE] McEWEN, W. R., "The Attack and Defense of Targets by Missiles," DRA-62-9, Air Force Office of Scientific Research, Holloman AFB, N.M., pp. 43, July 1962.

Author's abstract: "This paper derives optimum methods of attacking and defending targets with missiles under certain assumptions as to the number of missiles,



368 Samuel Matlin

antimissiles, and targets. Both probability and game theory are used in the deri- vations, and the results obtained by both methods are shown to be substantially the same."

Summary: t targets are attacked by N missiles and defended by M antimissiles with Pk=1.0. The targets are of equal value, and one surviving missile kills a target. For the ith target, ni and mi are derived.

Notes: (1) Limiting assumption: no target is attacked or defended with more than twice the average number of weapons per target in the whole inventory.

[MIT] MEEG, "Operations Analysis for CFR Study (U)," Appendix D: The Mitre Algorithm, pp. 117-129 (U).

Targets are ranked in decreasing value. Missiles are assigned so as to maximize total value. The next-valued target need not be a separate target, but might be the next portion of a previous target. Defenses are not considered. Algorithm begins: starting with a rank list and the targets stored in each missile, seek a missile which contains the first target on the rank list. A proof (by induction) is then provided.

Notes: (1) The algorithm can just as easily be applied to the case (as in [AUS]) where weapon requirements of each target are given.

[MOR] MORGAN, E. M., AND FLEMING, R., "Status of Force Structuring- Linear Programming Allocation Model," PIR 8194-444, General Electric Co., Valley Forge, Pa., 15 January 1968.

This report describes a 605 desk-side program to minimize the number of boosters required to meet the target demands. Linear programming is used to find the optimal allocation of boosters from launch site i to target j (Xij), given the number of launch sites (L), number of target sites (T), range from launch site i to target j (Dij), effective payload deliverable from launch site i to target j(Pxj), maximum booster range capability (R), maximum number of boosters at launch site i(b,), effective payload required at target j(Nj), and total available boosters (B). The force-structure aspect of the model occurs since the number of RV's deliverable vs. range is tabled.

Notes: (1) A single RV type, but boosters may carry different numbers of RV's. (2) Although targets are undefended, since RV requirements per target are tabled, this is easily extended to a level B defense. (3) Nj specified for each target. Special feature: The number of RV's placed on each booster is derived.

[OWE] OWEN, GUILLERMO, "Minimization of Fatalities in a Nuclear Attack Model," Operations Research 17, 489-505 (1969).

Author's abstract: "This paper considers a two-sided war game in which one side (the defender) must first deploy its defenses, consisting of both a passive defense (shelters), and an active defense (antimissile missiles). The other side (the attacker) then decides how to aim its missiles. The defender is constrained by budget




limitations, while the attacker is constrained by the number of missiles available. The payoff is in terms of fatalities. The paper uses a convex duality theorem to change the min-max problem to a pure minimization problem and obtains a solution that obeys the no-soft-spot rule. An example shows the effects of attack and budget sizes, as well as of the costs of ABM defense."

Notes: (1) Damage is also a function of passive defense investment.

[PEN 67] PENN, ALAN I., "Defense Models VIII. A Lagrangian Approach to Constrained Matrix Games," Lambda 11, Lambda Corp., Arlington, Va., September 1967, pp. 40.

Author's introduction: "This paper treats the problem of the multicellular matrix game, constrained in expected resource consumption. Four formulations are reviewed: the discrete matrix game, the discrete matrix game with generalized La- grange multipliers, the continuous infinite matrix game, and the continuous infinite matrix game with generalized Lagrange multipliers. The following illustra- tive example is solved: If T targets are simultaneously attacked and defended by a total of A and D perfect weapons, respectively, what should be the optimum allocation of weapons per target for each side . . ."

Notes: Four models are examined: (1) Discrete matrix game. (2) Discrete matrix game with generalized Lagrange multipliers. (3) Continuous infinite matrix game. (4) Continuous infinite matrix game with generalized Lagrange multipliers. One-penetrator kills in Models 1 and 2; A-penetrators kill in Models 3 and 4.

[PEN 69] PENN, ALAN I., "Defense Models XV. Preferential Area Defense in Support of Preallocated Defense," Lambda Paper 27, Lambda Cor- poration, Arlington, Va., pp. 28, January 1969.

Author's abstract: "The interaction of a system of preallocated defense weapons and a system of nationwide area defenders is examined. The attack is assumed to occur in a multiwave, time-phased mode in order to withhold attack density information from the area defenders. Within the class of 'defend-to-the-death' defenses, in which a target once defended by the area defenders will essentially continue to be so defended, an optimal deployment of the composite defense is derived. It is found that the defense is able to ensure the same survivability by playing other than defend-to-the-death strategies."

[PER] PERKINS, F. M., "Optimum Weapon Deployment for Nuclear Attack," Operations Research 9, 77-94 (1961).

Author's abstract: "Presented herein are solutions for the optimum deployment of both offensive and defensive weapons for a nuclear missile or aircraft attack against point targets. These solutions make it possible to determine rapidly the optimum deployment for both offense and defense and to estimate the expected level of destruction. The equations give some insight as to the interrelations between such factors as surviving worth, target hardness, warhead yield, accuracy, and defensive and offensive vehicle inventories."



370 Samuel Matlin

Notes: (1) Optimizes marginal return via a search. (2) For a special case of constant yield, CEP, and target hardness, it is best for the defense to divide his total worth equally among all targets.

[PIC] PICCARIELLO, HARRY J., "A Missile Allocation Problem," Operations Research l0, 795-798 (1962).

Author's abstract: "The problem of finding the allocation of weapons to a target system that results in the largest payoff is present in almost every war-gaming exercise. A technique often employed to find a solution to such a problem is to consider the problem as a continuous allocation process. However, the continuous solution may not furnish the solution to the actual problem, which is a discrete allocation process. This paper presents a solution to a missile allocation problem when the problem is considered in its continuous form. The same problem is investigated in its discrete form and it is shown that the solution in the continuous case is not, in general, a solution for the discrete case. Further, a necessary condition is derived for the existence of a solution for the discrete problem that differs from the solution for the corresponding continuous problem."

Assumptions: A missile complex consists of N missiles and C redundant control centers, 1 <C <N, such that a single surviving control center can launch all the missiles. The problem is to find that allocation of T identical missiles that, when attacking the N +C complex, will minimize the expected number of missiles launched. Pi =probability of survival of a missile launch pad, P2 =probability of survival of a control center.

Notes: (1) There are N independent point targets but also C other point targets, 1 <C <N, such that killing all C targets is equivalent to killing all N. (2) N targets have equal values; C targets have no value in themselves unless all C are killed, which has total value N. (3) Necessary condition for optimality in integer case is derived.

Results: In the continuous case, maximum value is killed using one of the two extreme strategies: allocate either all weapons to missile sites or to control centers.

In the discrete case, the extreme strategies are best only if the total number of attackers (T) is divisible by both N and C. A necessary condition for a mixed strategy to be optimal is that T -X be divisible by N, or X be divisible by C (where X =total number of weapons allocated to control centers).

[PUG] PUGH, G. E., "Lagrange Multipliers and the Optimal Allocation of Defense Resources," Operations Research 12, 543-567 (1964).

Author's abstract: "A simple extension of the Lagrange optimization method is valuable in allocating defense resources among a large number of independent defense locations. This kind of allocation problem is complicated by the fact that the attacker can optimize his attack against whatever defense is chosen. This paper describes an extended or double Lagrange method, which provides strictly optimal allocations for the attacker, but not necessarily for the defender. Never- theless, by careful use of the method, it is possible to obtain defense allocations




that are at least approximately optimum. The major part of the paper provides a technique for verifying results and computing bounds for the error-a technique that is necessary because the uncritical application of the method can occasionally give results that are far from optimum."

[SCH] SCHREIBER, THOMAS S., "Optimum Strategies in a Battle Involving Many Identical Targets (Defense Models XIV)," Lambda Paper 23, Lambda Corp., Arlington, Va., pp. 22, November, 1968.

Author's abstract: "Under certain restrictive conditions, the optimum allocations of attacking weapons and defending interceptors to a number of identical targets are derived, assuming each adversary is ignorant of the other's allocation, but does know the force sizes and the target survivability function. It is shown that, in general, the optimum allocations fall into four canonical cases and the characteristics of these are described. The number of targets, weapons, and interceptors are regarded as continuous quantities which enables the problem of determining optimum allocations to be formulated as a problem in continuous game theory. The solution is obtained by solving a pair of integral equations from which the characteristics of the allocations are obtained, and which can be solved explicitly in certain cases of practical interest." NOTE: This is the problem treated in [MAT 67] by a different method, and for integral values of weapons and interceptors.

Notes: (1) May also be interpreted as B. (2) 'One-on-one' engagement postulated (only one defender is launched against each attacker until exhaustion); P(A, D), the probability that a target is killed when attacked by A and defended by D weapons, is given.

[STR] STROM, S. C., AND SNYDER, J., "Weapons Application and Damage Assessment Program WANDA I (Interim Report)," GE COMPUTER Dept., CVN-66-1-104, RADC TR-67-77, Falls Church, Va., March 1967, pp. 112, (C).

A computer program based on [DAY], with extensions.

[TAY] TAYLOR. JAMES L., AND WALSH, JOHN, "Planning by Resource Allocation Methods Illustrated by Military Applications," SP48000003, System Development Corp., Santa Monica, Calif., January 1964, pp. 27.

Author's abstract: "This planning refers to the production and use of types of weapons that have already been developed. All combinations of targets and type of weapon are considered. The problem is to distribute the total available resources among these combinations in such a manner that the value of the targets surviving the attacks is minimized. A deterministic model of a flexible nature is developed. An elementary iterative method for solving the allocation of resources problem is obtained. These results are computer oriented and should be feasible for application even when the model is of an exceedingly complicated nature."

Notes: (1) Implicitly provided incidence matrix.

This model has force-structuring implications.



372 Samuel Matlin

[THO] THOMAS, R. E., "The Equivalence of Two Objective Functions for Optimal Missile Allocation," Bell Telephone Laboratories, Whippany, N.J., 1967. Presented at 32nd National ORSA Meeting, Chicago, No- vember 1967.

Author's abstract (in ORSA Bulletin): "In this paper we consider the allocation of missiles to defend a region from attackers. Two simple criteria for finding optimal defensive strategies are: (1) minimize the expected number of penetrators; and (2) maximize the probability of no penetrators. We discuss attack situations in which the two criteria result in identical defensive missile allocations. The proofs of such equivalence contain exact allocation formulas which are easily computed. We also demonstrate attack situations for which the allocations are different under these two optimality criteria."

[WAL] WALKUP, DAVID W., AND MACLAREN, M. DONALD, "A Multiple- Assignment Problem," Math. Note 347, Boeing, April 1964, pp. 14.

Author's summary: "A generalization of KUHN'S simple assignment problem is considered: there are m men and n tasks given with each man qualified for certain of the tasks. The output from each task is given as a concave function of the number of qualified men assigned to it. Find an assignment of men to tasks, perhaps more than one man to a task, so as to maximize total output. An algorithm for solving this general problem is given in which transfers like those used by Kuhn on the simple problem are selected using a node-labeling procedure on a related network. The algorithm yields for every k, 1 <k <m, an optimal assignment of the first k men only, employing a simple transfer to increase k by one. Several special forms of the generalized problem are considered including a target-assignment problem which A. S. MANNE has formulated as a linear program."

Note: Although the targeting application is not mentioned in this report, the tabled characterization may be made by obvious interpretations of the parameters of the model.

[WAR] WARD, L. E., JR., "An Application of Linear Programming to Weapons Strategy: The Problem of Massive Attack," NAVORD Rpt. 5455 NOTS, U. S. Naval Ordnance Test Station, China Lake, Calif., 25 September 1957, pp. 33.

Author's abstract: "The problem of massive attack is that of selecting an optimum strategy, given a large number of enemy targets and a limited weapon capacity. Mathematically, the problem is characterized as an exercise in concave programming, i.e., it is required to maximize a concave functional, given a variety of constraints, both linear and nonlinear. It is seen that the problem is amenable to linear programming techniques, and a solution involving a sequence of applications of the simplex method is described. Finally, an example is developed in consider- able detail."

Notes: (1) 0/1 incidence matrix refers to launch area vs. targets. Reduction of computational effort ensues by reducing this to a launch area vs. target region in-




cidence matrix. Part of the problem is to allocate boosters to launch areas under certain constraints (no more than b boosters per launch area). (2) Tabled discrete damage function Di vs. ni is concave.

[WEI] WEIL, ROMAN L., JR., "Functional Selection for the Stochastic Assignment Model," Operations Research 15, 1063-1067 (1967).

Author's abstract: "The choice of a function to be optimized is unambiguous in the ordinary assignment model. In the stochastic assignment model, there is no clear choice of functional. Six different criteria for the stochastic assignment model are discussed and one is suggested as best."



a review of the literature on the missile-allocation problem

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