# Tactical Planning for Multi-Projects

## An Application to Shipbuilding

### W. Robert Terry

### The University of Toledo

### Ezey M. Dar-El

### Technion - Israel Institute of Technology

Editor's Note—This is the final article in a three part series by Terry and Dar-El concerning planning in the shipbuilding industry. Their earlier articles were “A Multi-Project Planning System—An Application to Shipbuilding” *(PMQ,* June, 1982) and “Strategic Planning for Multi-Projects—An Application to Shipbuilding” *(PMQ,* September, 1982).

A previous paper [18] has observed that: (1) shipyards face multi-resource/multi-project (MRMP) planning problems that are incredibly complex; (2) the prevalence of completion delays and cost overruns indicate that better planning tools are badly needed; and (3) U.S. shipyards are not using any of the existing approaches to MRMP problems. To help alleviate this situation it was proposed that the MRMP problem be decomposed into strategic and tactical components [18]. However, details for solving the strategic and tactical problems were not given there. The purpose of this paper is to describe the approach for solving the tactical problem.

The tactical shipyard planning problem is to specify for each period in the tactical planning horizon how each trade's workforce should be allocated to the various activities necessary for constructing each of the ships. This problem results in a mathematical programming model which is too large and too complex to be solved for existing analytic approaches. This paper has described a heuristic procedure for solving the tactical shipyard planning model.

## Introduction

## Background

The problems of planning for the concurrent execution of several large-scale projects were discussed in an earlier paper by the authors [18] and were shown to be incredibly complex. However, U.S. shipyards which are prototypes of such organizations are not using any of the existing quantitative approaches for planning multiple projects in the face of resource constraints. This occurs in spite of the fact that completion delays and cost overruns indicate that better planning tools are badly needed.

This paper is the third paper in a sequence of articles which is concerned with multi-project planning, applied to shipbuilding. The first paper [18] analyzed the factors responsible for this paradox, revealing that existing multi-resource/multiproject (MRMP) planning approaches fail to differentiate between the strategic and the tactical aspects of such planning problems. The failure to recognize these differences would require a prohibitively large database and would also overload both the strategic and the tactical planners with irrelevant information—the strategic planner would be given too much detail while the planning horizon would be too long for the tactical planner. Also discussed was an approach for decomposing the shipyard planning problem into its strategic and tactical components. The second paper [19] discussed in detail the strategic component, while this paper describes the approach for solving the tactical planning problem.

## Problem Definition

The strategic shipyard plan specifies: (1) target completion dates for the various ships on the shipyard's order book or currently under construction $$$$$ each period during the planning horizon. These target completion dates and workforce levels represent, respectively, the goals for tactical management and the resources which they can use in pursuing these goals.

The tactical planner's task is to develop plans which specify for each period in the tactical planning horizon how each trade's workforce should be allocated to the various activities necessary for constructing each of the ships. In developing these plans it will be necessary for the tactical planner to recognize that: (1) both intertrade and intratrade interference can cause substantial productivity losses; (2) failure to consider strict precedence relationships can cause unnecessary delays and costs to be incurred; and (3) overtime can be used when it is cost effective to do so. Recall that the formulation of the strategic shipyard planning problem assumed that it was not necessary to consider these factors at the strategic planning level. This was done to reduce the size and complexity of the problem so that it could be solved.

The tactical planning process is further complicated by random variation in activity duration times and uncertainty regarding the set of activities which will actually be necessary to construct a particular ship. These factors make it necessary to revise tactical shipyard plans from time to time. The combined effect of the above factors imply that the tactical planning problem is enormously complicated.

Figure 1 Information Flows in Planning Process

The role of tactical planning in the process of developing plans and converting them into action is graphically depicted by Figure 1.

This figure shows that tactical planning is concerned with acquiring the resources specified by the strategic plan and with allocating available resources to the various activities necessary for accomplishing the strategic goals.

This paper describes a method for solving the tactical planning problem for multi-projects. The development of such a procedure is accomplished in three phases. The first phase, described in the following section, specifies a model that describes how the variables under the control of the tactical planner influence cost. The second phase, described in the Literature Review section, analyzes and evaluates existing MRMP planning models. The third phase, described in the fourth section, Tactical Planning Model for MRMP Problems, develops a procedure for solving the problem.

## Economic Impact of Decision Variables

In order to formulate a model for solving the tactical shipyard planning problem it is necessary to identify how each of the variables under the control of the tactical planner influences cost.

If a project is not completed on time, then a lateness penalty cost will be incurred. This cost consists of the opportunity cost of having the shipyard's productive facilities tied up and additional payments of late delivery explicitly specified by the contract. In addition, failure to meet delivery dates can jeopardize the shipyard's ability to get future business.

The tactical planner can influence the amount of lateness penalty cost which a shipyard incurs through the choice of the times at which activities which have a high probability of being on a critical path are started. For example, suppose that each activity on the critical path for a given ship is started at its appropriate “normal” start time. Let the actual time taken to complete each activity vary about its mean. Then the probability density function for project duration times will tend to be normally distributed with mean and variance respectively equal to the sum of the means and variances for each of the activities on the critical path. This will result in a probability of .5 that the project will be completed on time (this assumes that other “near critical” paths do not exist).

The tactical planner can reduce the odds that the project will be late by starting some or perhaps all of the activities on the critical path earlier than their respective start times or else, shortening $$$$$ duration by increasing resource levels. This will cause the expected completion date to be less than the due date which corresponds to shifting the probability density function for project completion times to the left giving a reduced probability of the project being late.

Starting the critical path activities earlier than their scheduled start times can be visualized as insurance against lateness. The cost of resources can be regarded as the premium on the insurance policies for the activities. The tactical planner's task is to determine: (1) how much to spend on insurance; and (2) the best portfolio of policies to purchase.

Overtime can be used to reduce the amount of lateness penalty charges incurred when a project is not completed on time, but the use of overtime involves overtime premium payments. If there are two or more activities on which overtime can be used to reduce the amount of lateness and these activities both use one or more trades which are in short supply, then the problem of determining the optimal allocation of overtime, to the activities arises.

In shipbuilding there are certain situations in which it is not possible to simultaneously perform two or more activities as efficiently as the activities could be performed separately. This can arise when two trades interfere with the work of one another. This phenomenon is referred to as intertrade interference. For example, if electrical cables, pipes, and ventilation ducts all pass through a confined space then electricians, pipefitters, and vent installers could impede each other's progress. It is also possible for members of the same trade to interfere with each other's work. This phenomenon is referred to as intratrade interference. For example, assigning too many floor coverers to a given area could cause each to get in the other's way. The duration of a project can be reduced by judiciously utilizing intertrade and/or intratrade interference. Examples of these are illustrated by Figure 2.

Some activities can be scheduled to start at anytime while others cannot be started until a set of prerequisite activities have been satisfied. These prerequisites can be classified as mandatory or preferred. A given activity cannot be accomplished until all of its mandatory prerequisites have been met. Failure to satisfy all preferred prerequisites will not block the accomplishment of the activity, but will increase the cost of accomplishing it. For example, it might be $$$$$ floor coverings and fixtures since this $$$$$ the need for covering floors and fixtures with drop cloths. Thus, another option for reducing project lateness is to selectively disregard preferred prerequisites.

Random variations in delivery date leadtimes can cause material shortages to arise. Such shortages can delay the start and/or completion of certain activities. When an activity on the critical path is delayed, then project lateness can result. However, this need not necessarily be the case, since there are a variety of actions which can be taken to make up for the time loss caused by the delay. For example, the following methods for reducing project duration: (1) overtime, (2) intertrade interference, (3) intratrade interference, and (4) activity splitting; can be used more extensively than called for in the predelay plan. The risk of material shortages can be reduced by allowing for earlier deliveries when purchasing raw materials, though this should be weighed against increased inventory carrying costs.

Figure 2 Illustration of How Project Duration Can be Reduced by Utilizing Intertrade and Intratrade Interference

Another option for reducing the amount of project lateness is to shift workers from activities whose completion can be postponed without causing their $$$$$ activity. This situation is referred to as activity splitting. However, this will typically cause set-up cost and transportation costs to increase.

The above discussion suggests that the tactical planner can influence the following costs: (1) cost of productivity lost as a result of splitting activities: (2) cost of overtime premiums; (3) cost of productivity lost as a result of intertrade interference; (4) cost of productivity lost as a result of intratrade interference; (5) cost of project lateness; (6) cost of carrying additional material inventories to protect against late deliveries; and (7) cost of disruption as a result of late deliveries of key materials. Thus the tactical shipyard planning problem is to assign each trade's workforce to the various activities in a manner which will minimize the sum of the above costs subject to the following constraints: (1) resource availabilities cannot be exceeded; and (2) no activity can be started until all of its mandatory precedent requirements have been met. However, an idle resource represents an opportunity loss. Therefore, the cost of idle time associated with tactical shipyard plans should be calculated and reported to the strategic shipyard planner. The strategic planner will then compare the cost of laying off and rehiring of workers necessary to eliminate the idle time with the savings in labor cost and revise the strategic shipyard plan if necessary.

## Literature Review

Solution strategies used in solving existing MRMP planning models are found to be either analytic or heuristic. However, only one analytic approach has been published to date. Pritsker, Watters, and Wolfe [16] formulated the problem as a zero-one integer linear programming problem. Unfortunately, such an approach, if applied to problems of the size faced by shipyards, would result in a problem the size of which would far exceed the capacity of present zero-one integer linear programming algorithms [23]. In fact, Elmaghraby [5, p. 217] reported that Lenstra [10] has shown that:

The problem of scheduling activities on multiple resources when the activities are subject to precedence constraints and the availability of the resources is limited is known to be “NP hard” … This implies that, in all probability, there shall be no “efficient” algorithm for solving this $$$$$.

This implies that the tactical shipyard planning problem is “NP hard.” This indicates that a search for an analytic procedure for solving the problem would not likely be fruitful. Thus, it was decided not to consider analytic methods further. This left only two options for attacking the problem: (1) heuristic procedures; or (2) branch and bound theory.

The branch and bound approach is a systematic procedure whereby each point in the entire space of feasible solutions is enumerated either explicitly or implicitly. It is desirable to minimize the number of points that have to be enumerated explicitly. Three techniques are used to accomplish this: (1) dominance; (2) feasibility; and (3) redundancy. However, Baker [1, p. 276-277] has noted:

Unfortunately, all implicit enumeration approaches to the determination of an optimal schedule appear to be susceptible to the combinatorial nature of these problems when they are tested on the large versions typically found in practice. . .there is no evidence that such techniques can reliably handle multi-resource versions of a problem that contains more than 50 activities.

Thus it was decided not to consider branch and bound methods as candidates for the solution procedure. This left heuristic procedures as the only viable option for attacking the tactical shipyard planning problem.

A number of authors [6], [7], [14] and [15] have reported simulation experiments that were designed to evaluate the effectiveness of a number of scheduling rules. The effectiveness of these scheduling rules were evaluated in terms of the following characteristics: (1) on time completion; and (2) efficient resource utilization. None of the scheduling rules were found to be best for all performance measures. In general, due date oriented scheduling rules performed better than resource oriented rules when the performance was measured in terms of on time completion. The reverse was true, in general, when performance was measured in terms of efficient resource utilization.

None of the heuristic approaches to the MRMP scheduling problems, with the exception of a model developed by Dar-El, Behmoaram, and Tur [2], utilized a cost based objective function. The heuristic used by Levy, Thompson and Wiest [11] was designed to reduce peak resource requirements and smooth out period-to-period assignments subject to a constraint on project duration. SPAR-1 developed by Wiest [21] utilized a heuristic which places primary emphasis on completing the job as early as possible. RAMPS, which was developed for proprietary use, utilizes a heuristic which involves minimizing a weighted function of variables such as total slack, idle resources, project delay cost, and number of successors to a given job [9], [13], [20] and [22]. Jennett [8] lists a number of other network analysis programs which are commercially available but for which the scheduling heuristic are kept secret.

The Dar-El, Behmoaram, and Tur (DBT) model [2] was found to provide an excellent conceptual foundation for developing a model for solving the tactical shipyard planning problem. Their approach is based on the assumption that only scarce and expensive resources are likely to influence project completion date and cost. The objective function to this model is equal to the sum of the following terms: (1) total cost of idle resources; (2) total activity splitting penalties; (3) incremental total cost of overtime work; and (4) total project lateness penalties.

The Dar-El, et al. total project lateness penalty is comprised of two elements: (1) some penalty scale which increases with the lateness; and (2) the time at which the penalty begins to apply. Researchers in this field have all applied lateness penalties when activities are scheduled after their “late start” times, but Dar-El et al. argue that the project is already late when the lateness penalty applies and, therefore, to avoid this, the lateness penalty should apply before the activity's “late start” (LS) time. The time period that the penalty applies before its LS time, is called the critical slack and varying its value can have a significant influence on the schedule of a particular project.

Dar-El, et al. introduce another innovative feature into their program. The program enters into an interactive mode when actual costs deviate substantially from the target value—indicating that some resources are not fully utilized. The scheduler is then required to explore the possibility of using alternative methods of executing certain activities which utilize a different combination of resources (Called ARC's; See Dar-El and Tur [4]) in order to increase resource utilization. This feature is employed in the present algorithm.

## Tactical Planning Model for MRMP Problems

## Need for An Improved Approach

The characteristics of the tactical MRMP planning problem were discussed in the section, Economic Impact of Decision Variables. This discussion indicated that the tactical MRMP planning problem can be formulated as a mathematical programming problem. This problem is concerned with the allocation of a number of scarce resources to the various activities necessary for the accomplishment of a number of projects. The objective is to allocate the resources to the projects during each of a number of future time periods in a manner which will minimize the sum of the following costs:

(1) penalty cost as a result of splitting activities;

(2) cost of overtime premiums;

(3) cost of project lateness;

(4) penalty cost as a result of intertrade interference;

(5) penalty cost as a result of intratrade interference;

(6) cost of disruption resulting from late deliveries of materials.

In making this allocation the following constraints must be satisfied:

(1) resource availabilities cannot be exceeded;

(2) no activity can be started until all of its precedent requirements have been met.

As noted in the literature review, none of the existing MRMP models consider the following costs:

(1) cost of intertrade interference;

(2) cost of intratrade interference;

(3) cost of carrying additional material inventories to protect against late deliveries.

The tactical MRMP planning problem is further complicated by the fact that the amount of time required to accomplish a given activity will be dependent on the work method used to complete it. For example, tandem-arc automatic welding equipment requires about 25 percent less time than single-arc automatic welding [12]. Another example is the amount of time required to fabricate a panel will depend on the extent to which assembly line methods are used, the degree to which the assembly line has been balanced, and the priority rules which determine the order in which different types of panels enter the line. The use of a particular work method for a given activity affects not only the time required to complete that activity but the resources which will be available to the other activities. Therefore, it would appear that the tactical shipyard planner should consider alternative work methods.

The literature review also noted that the tactical MRMP planning problem is “NP hard,” which implies it is not likely that an efficient optimization procedure can be developed for solving it. This suggested that the only sensible way to attack the problem would be to develop a heuristic procedure. The next subsection describes a prototype heuristic procedure for solving real world sized MRMP planning problems.

## Heuristic Procedure

The flow chart for the heuristic procedure for solving the tactical MRMP planning problem is shown in Figure 3 and is based on the Dar-El, et al. model [2]. Note that each block in this flow diagram has a number in the upper right hand corner. These numbers will be used to identify the blocks when they are discussed in further detail below.

Figure 3 Basic Flow Chart for the MRMP Tactical Planning Model.

## Block 1

Block 1 reads in the following data: (1) arrival and due dates of every project; (2) lateness penalty and critical slack (a user fixed parameter which can be used to specify the priority associated with starting critical activities prior to their late start dates); (3) total number of activities in each project; (4) set of immediate followers for each activity; (5) the levels of the various resources required to accomplish each activity; (6) unit cost of each resource; (7) amount of each resource available; (8) maximum amount of overtime permitted for each resource; (9) overtime cost for each resource; and (10) splitting penalties associated with each resource.

## Block 2

Block 2 calculates the critical path completion date for each project and the early start, early finish, late start, late finish, and total slack for each activity. These items are stored in matrices for further use in the scheduling process.

## Block 3

Block 3 initializes the calendar to day 1.

## Block 4

Block 4 schedules the activities based on the following logic. At the start of each period every activity associated with any of the projects to be scheduled must be in one of the following mutually exclusive sets: (1) activities which have been completed; (2) activities which have been started but have not yet been completed; (3) activities not yet started that have no precedent activities; (4) activities not yet started that have no precedent activities yet to be completed; and (5) activities that cannot be started due to the fact that one or more precedent activities have not yet been completed.

In a given period, activities in sets 2, 3, and 4 are eligible for the assignment of resources. Such activities are referred to as assignable activities. However, resource availabilities will not usually be sufficient to allow all assignable activities to be assigned simultaneously in a given period. Furthermore, intertrade interference could arise if certain activities are performed together, while intratrade interference could arise if too many persons are assigned to a given activity in a confined area. This creates the problem of determining which activities should be assigned in any given period.

**Intertrade Interference Procedure.** The strategy for determining which activities should be assigned to a given period utilizes the following procedure for handling intertrade interference:

(1) Put all activities in the following categories into a scheduling pool.

(a) Activities which have been started but have not yet been completed;

(b) Activities not yet started that have no precedents; and

(c) Activities not yet started for which all precedents have been completed.

(2) Partition the scheduling pool into:

(a) An interference free subset which contains those activities which can be performed with any other member of the subset without creating interference;

(b) A number of interference subsets such that the activities within a given interference subset interfere with one another, while the activities in different subsets do not.

This is illustrated by Figure 4 which depicts the partitioning of the scheduling pool into an interference free subset and n interference subsets. The activities in each interference subset cannot be performed simultaneously with another member of the same set without creating interference. However, any combination of activities consisting of all activities in the interference free subset and at most one activity from each of the n interference can be performed simultaneously without creating interference.

Figure 4 Partitioning of Scheduling Pool into Interference Free Subset and n Interference Subsets.

(3) Find all possible combinations of activities which consist of one member from each of the mutually exclusive interference subsets and all of the activities in the interference free subset;

This is illustrated by Figure 5 which shows all possible combinations of activities which can be formed when the interference free subset consists of activities A, B, and C; the first interference subset consists of D and E; and the second interference subset consists of F and G.

Figure 5 Formation of All Possible Interference Free Combinations of Activities Where D and E Interfere and F and G Interfere.

(4) Calculate resources required for each combination of activities; compare resource requirements with resource availability levels, and eliminate infeasible combinations;

(5) Calculate value of objective function for each feasible resource combination and keep combinations which minimize value of objective function for that period.

The logic for dealing with intertrade interference is to first attempt to schedule the activities without intertrade interference. If all projects can be completed by their respective due dates, then the interference free solution will dominate any solution which has interference. However, if lateness results, then the scheduler should selectively permit intertrade interference to arise. This is accomplished by removing the constraint that specifies that two activities cannot be performed together and increasing the time required to perform the activities to reflect the effect of intertrade interference.

**Intratrade Interference Procedure.** Intratrade interference can be handled by rerunning the model for each crew size which can be used to perform an activity. In doing this it will be necessary to adjust the activity duration times to account for the effects of diminishing returns which result from intratrade interference and the cost of performing the activity to reflect the size of the crew.

**Cost Minimization Procedure.** The assignments made in a given period determines the set of assignable activities for future periods. The minimum cost attainable in any given period will be dependent on the set of assignable activities for that period. This implies that in order to minimize total system cost it will be necessary to account for the impact that the assignments made in a given period have on the costs of future periods. Unfortunately, this will result in a combinatorial problem which is far too large to be solved in a cost effective manner.

The strategy for avoiding this combinatorial explosion consists of decomposing the multi-period optimization problem into a series of subproblems, one for each period in the planning horizon, and solving these subproblems independently. Admittedly, such an approach cannot guarantee to minimize total system cost. Nevertheless, this appears to be a reasonable approach for dealing with a problem which would otherwise be unsolvable. The subproblem for each period is solved as follows: (1) A “binary enumeration technique” developed by Dar-El and Tur [4] is used to generate all possible combinations of assignable activities. This binary enumeration technique uses bit level storage in order to exploit the fact that a computer word in FORTRAN on the IBM 360 contains 32 bits. In addition, the time required to perform logic operations is much less at the bit level than at the word level; (2) The resource requirements for each assignable activity combination is computed and compared to the amounts of the various resources which are available during the period of interest. These combinations which require more of any given resources than is available are classified as infeasible and discarded. The remainder of the combinations are feasible; (3) The value of the objective function is computed for each feasible combination of assignable activities; and (4) The minimum value of the objective function for the single period problem is determined by comparing each value computed with the previous minimum value. The new value replaces the previous minimum when the new value is less than the previous minimum. Otherwise, the new value is discarded.

The following notation is used to specify the formulas for calculating cost referred to in the discussion.

CD | = | calendar date |

CF(i) | = | expediting factor for resource type i |

CL(j) | = | lateness penalty on project j |

CS(i) | = | unit cost for resource type i |

CSLK(j,k) | = | critical slack for activity k on project j (a user fixed parameter which can be used to specify the priority associated with starting critical activities prior to their late start date) |

DUR(j,k) | = | remaining duration of the activity k on project j |

HRA(i) | = | higher resource availability level for resource type i |

LF(j,k) | = | late finish date of the activity k on project j |

LRA(i) | = | lower resource availability level for resource type i |

RA(i) | = | number of units available of resource type i |

RR(i) | = | number of units required by resource type i |

SPEN(i) | = | splitting penalty for resource type i |

TCIR | = | total cost for idle resources |

TCOT | = | incremental total cost of overtime work |

TASP | = | total activity splitting penalties |

TPLP | = | total project lateness penalties |

The formulas for the constituent terms in the objective function are as follows:

**Late Delivery Costing Procedure.** The cost of late deliveries of materials can be handled by regarding the arrival of a material as an activity and calculating the lateness penalty for that activity. In this case the penalty function will be the expected cost of lateness caused by material shortages which corresponds to the various values of delivery lead time.

The following example is presented to illustrate the logic for specifying the lateness penalty function and the associated incremental inventory carrying cost function.

The main engine is to be built by a subcontractor. If the engine is not delivered in time to be installed prior to launch, then an additional cost of installation equal to C_{D} will be incurred. Let t represent the latest date for delivery of the engine which will permit it to be installed prior to launch. If the shipyard specifies a delivery date of t then there will be a 50 percent chance of receiving delivery on time, 33.3 percent chance of being one period late, and a 16.7 percent chance of being two periods late. The shipyard has to pay M dollars for the engine on delivery. If the delivery date denoted by t_{d} is specified to equal t, then there will be a 50 percent chance that delivery will be at least one period late. The expected cost of late delivery will be .5 C_{D}. If t_{d} is specified to equal t_{-1}, then the expected cost of late delivery will be .167 C_{D}. However, there will be a 50 percent chance that the engine wil be delivered one period early. If this occurs, then the shipyard will incur an opportunity cost of iM dollars where i presents the shipyard's cost of capital. The expected opportunity loss will be .5 iM. If t_{d} is specified to be equal to t-_{2}, then there will be a zero percent chance of late delivery. However, there will be a 50 percent chance of receiving delivery two periods early and a 33.3 percent chance of receiving delivery one period early. The expected opportunity loss will be equal to .5 (2i + i^{2}) M + .337 iM.

The lateness penalty function and the inventory carrying cost function for the above situation is shown in Figure 6.

Figure 6 Lateness Penalty Function and Inventory Carrying Cost Function.

## Block 5

Block 5 is an updating routine which is called when the scheduling process for a given day has been completed. It performs the following operations: (1) reduces the duration for the activities scheduled; (2) decreases the total slack of unscheduled activities by one unit; (3) increases the calendar date by one unit; and (4) removes activities that have been completed from the set of available activities.

## Block 6

Block 6 checks whether or not all activities have been scheduled.

## Interactive Mode

Blocks 7, 8, and 9 form a subsystem which permits the model to be used in an interactive mode, the need for which is explained below.

The amount of time required to accomplish a given activity will be dependent on the work method used to complete it. The use of a particular work method for a given activity affects not only the time required to complete that activity but the resources which will be available to the other activities. Therefore, if the objective of the tactical shipyard planner is to minimize total system cost, then all alternative work methods should be considered. However, this will increase the size of a combinatorial problem which is already too large to be solved analytically.

One way to attack this problem is to incorporate a monitor subsystem into the model which: (1) compares DELTA, the minimum cost value of the objective function to an upper control limit; and (2) sets off an alarm when the value of DELTA exceeds its upper control limit. When the alarm occurs the scheduler should attempt to identify new work methods. The model will be rerun with each of these new work methods and the one with the minimum value of DELTA will be utilized.

The strategy of generating alternative work methods is likely to require nontrivial amounts of engineering time. Therefore, it is likely to be expensive. Therefore, postponing the generating alternative work methods until an alarm occurs is appealing from the standpoint of saving engineering time. This advantage is offset somewhat by the necessity of having to make multiple runs which will increase computation cost.

The process of comparing the minimum cost value of the objective function to an upper control limit is accomplished by Block 7. When the upper control limit is exceeded, Block 8 sets off an alarm. Block 9 represents the interactive process.

## Conclusions

The tactical shipyard planning problem is to specify for each period in the tactical planning horizon how each trade's workforce should be allocated to the various activities necessary for constructing each of the ships. This problem results in a mathematical programming model which is too large and too complex to be solved for existing analytic approaches. This paper has described a heuristic procedure for solving the tactical shipyard planning model.

## References

1. Baker, K.R., *Introduction to Sequencing and Scheduling,* New York: John Wiley and Sons, Inc., 1974.

2. Dar-El, E.M., A. Behmoaram, and Y. Tur, “SCREAM-Scarce Resource Allocation to Multi-Projects, *Project Management Quarterly,* December, 1978.

3. Dar-El, E.M. and Y. Tur, “A Multi-Resource Project Scheduling Algorithm,” *AIIE Transactions,* 1977.

4. Dar-El E M & Tur Y (1978) Resource allocation of a multi-resource project for variable resource availabilities,” *AIIE Trans,. 10*(3), 299-306.

5. Elmaghraby, S.E., *Activity Networks: Project Planning and Control by Network Models* New York: John Wiley and Sons, Inc., 1977.

6. Fendley, L., “Towards the Development of a Complete Multi-Project Scheduling System, *Journal of Industrial Engineering,* 1968.

7. Gonguet, L., “Comparison of Three Heuristic Procedures for Allocating Resources and Producing Schedule,” in *Project Planning by Network Analysis,* Lombaers, H.J.M. (Ed), Amsterdam: North Holland, 1969.

8. Jennett, E., “Availability of CPM Programs, *Project Management Quarterly,* 1970.

9. Lambourne, S., “Resource Allocation and Multiproject Scheduling (RAMPS)—A New Tool in Planning and Control,” *Computer Journal,* 1963, *5*, 300-304.

10. Lenstra, J.K., *Sequencing by Enumerative Methods.* Unpublished Ph.D. dissertation, University of Amsterdam, Holland, 1976.

11. Levy, F.K., G.L. Thompson, and J.J. Wiest, “Multi-Ship, Multi-Shop, Workload Smoothing Program, *Naval Research Logistics Quarterly,* 1962.

12. Mack-Forlist, D.M. and A. Newman, *The Conversion of Shipbuilding from Military to Civilian Markets.* New York: Praeger Publishers, 1970.

13. Moshman, M.I. Johnson, and M. Larsen, “RAMPS—A Technique for Resource Allocation and Multi-Project Scheduling,” *Proceedings Spring Joint Computer Conference,* 1963.

14. Pascoe, T.L., *Heuristic Method for Allocating Resources,* Ph.D. dissertation, University of Cambridge, 1965.

15. Patterson, J.H., “Alternate Methods of Project Scheduling with Limited Resources, *Naval Research Logistics Quarterly,* 1973.

16. Pritsker, A.A.B., L.J. Watters, and P.M. Wolfe, “Multi-Project Scheduling with Limited Resources: A Zero-One Programming Approach,” *Management Science,* January, 1969.

17. Terry W.R., F. Green, and A. Magnuson, Confidential survey of U.S. shipyards conducted by the Industrial Engineering/Operations Research Department at Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1979.

18. Terry, W.R., and E.M. Dar-El, Multi-Project Planning System—An Application to Shipbuilding, Technical Report UTEC ME 80-143, Department of Mechanical and Industrial Engineering, University of Utah, Salt Lake City, Utah, and 1980A, and *Project Management Quarterly,* June, 1982.

19. Terry W.R., and E.M. Dar-El, Strategic Planning for Multi-Projects,” Technical Report UTEC ME 80-148, Department of Mechanical and Industrial Engineering, University of Utah, Salt Lake City, Utah, 1980B and *Project Management Quarterly,* September, 1982.

20. Wiest, J.D., *The Scheduling of Large Projects with Limited Resources,* Ph.D. dissertation, Carnegie Institute of Technology, 1963.

21. Wiest, J.D., “A Heuristic Model for Scheduling Large Projects with Limited Resources,” *Management Science,* June, 1967.

22. Wiest, J.D., “A Heuristic Scheduling and Resource Allocation Model for Evaluating Alternative Weapon System Programs,” RAND Corporation Report No. RM-5769-RP, 1969.

23. Wiest, J. D. and F.K. Levy, *A Management Guide to PERT/CPM with GERT/PDM/DCPM and Other Networks,* Englewood Cliffs, N.J.: Prentice Hall, Inc., 1977.

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