Capacitated Transportation Research Articles

Multi-location transshipment problem with capacitated transportation

We consider coordination among stocking locations through replenishment strategies that take explicitly into consideration transshipments, transfer of a product among locations at the same echelon level. We incorporate transportation capacity such that transshipment quantities between stocking locations are bounded due to transportation media or the location’s transshipment policy. We model different cases of transshipment capacity as a capacitated network flow problem embedded in a stochastic optimization problem. Under the assumption of instantaneous transshipments, we develop a solution procedure based on infinitesimal perturbation analysis to solve the stochastic optimization problem, where the objective is to find the policy that minimizes the expected total cost of inventory, shortage, and transshipments. Such a numerical approach provides the flexibility to solve complex problems. Investigating two problem settings, we show the impact of transshipment capacity between stocking locations on system behavior. We observe that transportation capacity constraints not only increase total cost, they also modify the inventory distribution throughout the network.

On the integrated production, inventory, and distribution routing problem

The integrated Production, Inventory, and Distribution Routing Problem (PIDRP) is concerned with coordinating production, inventory, and delivery operations to meet customer demand with the objective of minimizing the cost. The PIDRP considered in this paper also involves heterogeneous transporters with noninstantaneous traveling times and multiple customer demand centers each with its own inventory capacity. Optimally solving such an integrated problem is in general difficult due to its combinatorial nature, especially when transporter routing is involved. We propose a two-phase solution approach to this problem. Phase I solves a mixed-integer program which includes all the constraints in the original model but with the transporter routings being restricted to direct shipments between facilities and customer demand centers. The resulting optimal solution to phase I is always feasible to the original model. Phase II solves an associated consolidation problem to handle the potential inefficiency of direct shipment. The delivery consolidation problem is formulated as a capacitated transportation problem with additional constraints and is solved heuristically. Unlike the classical decoupled approach, this two-phase approach does not separate the optimization for the production lot sizes and the transportation schedules. Its main advantage lies in its ability to simultaneously coordinate the production, inventory, and transportation operations of the entire planning horizon, without the need to aggregate the demand or relax the constraints on transportation capacities. This enables us to quickly identify a quality suboptimal solution to the original complex problem. The suboptimality is, however, due to the simplified assumption that in phase I only direct shipment is deployed, which is then partially corrected for by the effort of phase II. We evaluate the performance of this proposed two-phase approach and report on its application to a real-life supply network.

Paradox in a non-linear capacitated transportation problem

This paper discusses a paradox in fixed charge capacitated transportation problem where the objective function is the sum of two linear fractional functions consisting of variables costs and fixed charges respectively. A paradox arises when the transportation problem admits of an objective function value which is lower than the optimal objective function value, by transporting larger quantities of goods over the same route. A sufficient condition for the existence of a paradox is established. Paradoxical range of flow is obtained for any given flow in which the corresponding objective function value is less than the optimum value of the given transportation problem. Numerical illustration is included in support of theory.

Solving a capacitated four-index transportation problem

Solving a capacitated four-index transportation problem

Fuzzy programming with hyperbolic membership functions for multiobjective capacitated transportation problem

The linear multiobjective capacitated transportation problem in which the supply and demand constraints are all equality type, capacity restrictions on each route are specified and the objectives are non commensurable and conflicting in nature. The fuzzy programming with hyperbolic membership functions for obtaining efficient solutions as well as the best compromise solution of a multiobjective capacitated transportation problem has been presented in this paper. A Fortran program has been developed based on this algorithm. An example is included to illustrate the methodology. Also this method is compared with one existing fuzzy programming with linear membership functions using same example.

Strategic capacity planning in supply chain design for a new market opportunity

This paper addresses the problem of supply chain design at the strategic level when production/distribution of a new market opportunity has to be launched in an existing supply chain. The new market opportunity is characterized by a deterministic forecast expected to occur per period. The product (or service) is assumed to be produced (or provided) in a three-stage capacitated supply chain where the first stage concerns suppliers, the second stage producers and the final stage customers. There could be multiple alternatives at each stage which are defined as nodes. Nodes in each stage are connected to the next stage through capacitated transportation systems. Production capacity at the second stage (i.e. producers) are also limited since they may already be involved in other existing activities. The objective is to perform strategic capacity planning in the supply chain in order to meet the demand of the new opportunity at minimal cost. A linear running cost is associated with each node. If the decision is to increase the capacity of a node, then a fixed cost applies, followed by a cost that is proportional to the additional capacity. The overall problem can be modelled as a large-scale mixed integer linear programming problem. A solution algorithm is developed to overcome difficulties associated with the size of the problem and is tested on empirical data sets. The overall contribution is an analytical tool that can be employed by managers responding to the new market opportunity at the strategic level for supply chain design.

Modeling of capacitated transportation systems for integral scheduling

Motivated by a planned automated cargo transportation network, we consider transportation problems in which the finite capacity of resources (such as vehicles, docks, parking places) has to be taken into account. For such problems, it is often even difficult to construct a good feasible solution. We present a flexible modeling methodology which allows to construct, evaluate, and improve feasible solutions. This new modeling approach is evaluated on instances stemming from a simulation model of the planned cargo transportation system.

Resolution of a capacitated transportation problem with four subscripts

The classical transportation problem is actually well known both in theory and numerical resolution. We are interested in the multi‐subscripts capacitated transportation problem of axial sum launched by specialists some years ago. Our work deals with the capacitated problem with four subscripts for which we have established an existence criterion, an optimality condition and an algorithm of resolution.

Optimization of the transportation expense of a firm with contractual supplies

We consider nonlinear nonconvex capacitated transportation problems where the nonlinearity occurs only in the last row of the transportation tableau. This transportation model can be successfully applied to economics representing the nonlinearity caused by penalties for unsatisfied contractual quantities or by changing in price. An algorithm for local optimization, based on the algorithms for solving linear transportation problems, is suggested. It consists of three phases – initial, linear and nonlinear. The nonlinear phase uses auxiliary linear transportation problems. The algorithm is illustrated by proper examples. Sufficient conditions for satisfying contractual supplies are given.

Optimal political districting

For the political districting problem, I propose the following solution methodology: (a) use Lagrangian relaxation to determine the centres of the districts, then, (b) use the transportation technique to assign population units to centres, and finally, (c) resolve the splitting problem by solving a sequence of capacitated transportation problems. This method is applied to the problem of determining the provincial districts for the City of Saskatoon, Canada, and the results are compared with the actual districting done in 1993 by the Electoral Boundaries Commission.

Scheduling to maximize customer satisfaction: A project for the shad valley program

This paper describes a project that was done for the Shad Valley Program, where it was required to assign students to seminars so as to maximize the satisfaction of the students with their assignments. We begin by describing the problem, its inputs and constraints. Two models are proposed to determine optimal assignments. The first model is based on the Capacitated Transportation Problem and a network formulation is proposed to solve it. The second model is a two phase model whose first phase involves solving a Bottleneck Capacitated Transportation Problem and the second phase solving a Capacitated Transportation Problem. A simple search algorithm is proposed that solves the second model. Implementation of these models is described and the results obtained are discussed. Extensions to the two models are also proposed.

An Algorithm for Solving Capacitated Multicommodity p-median Transportation Problems

An algorithm for solving a special capacitated multicommodity p-median transportation problem (CMPMTP), which arises in container terminal management, is presented. There are some algorithms to solve similar kinds of problems. The formulation here is different from the existing modelling of the p-median or some related location problems. We extend the existing work by applying a Lagrangean relaxation to the CMPMTP. In order to obtain a satisfactory solution, a heuristic branch-and-bound algorithm is designed to search for a better solution, if one is possible. A comparison is also made with different algorithms.

An Algorithm for Solving Capacitated Multicommodity p-median Transportation Problems

An Algorithm for Solving Capacitated Multicommodity p-median Transportation Problems

Probabilistic Analysis of Network Flow Algorithms

This paper is concerned with the design and probabilistic analysis of algorithms for the maximum-flow problem and capacitated transportation problems. These algorithms run in linear time and, under certain assumptions about the probability distribution of edge capacities, obtain an optimal solution with high probability. The design of our algorithms is based on the following general method, which we call the mimicking method, for solving problems in which some of the input data are deterministic and some are random with a known distribution: 1. Replace each random variable in the problem by its expectation; this gives a deterministic problem instance that has a special form, making it particularly easy to solve; 2. Solve the resulting deterministic problem instance; 3. Taking into account the actual values of the random variables, mimic the solution of the deterministic instance to obtain a near-optimal solution to the original problem; 4. Fine-tune this suboptimal solution to obtain an optimal solution. We present linear time algorithms to compute a feasible flow in directed and undirected capacitated transportation problem instances. The algorithms are shown to be successful with high probability when the probability distribution of the input data satisfies certain assumptions. We also consider the maximum flow problem with multiple sources and sinks. We show that with high probability the minimum cut isolates either the sources or the sinks, and we give a linear-time algorithm that produces a maximum flow with high probability.

A logistics problem: a non-markovian queueing network model and algorithm

In this paper, a stochastic version of the classical deterministic balanced single commodity capacitated transportation network problem is presented. In this model, each arc of the network connects a supply node to a demand node and the flow of units forming along each arc of the network forms a stochastic process (i.e.G/M/1 queueing system with generally distributed interarrival time, a Markovian server, a single server, infinite capacity, and the first come first served queueing discipline). In this model, the total transportation cost is minimized such that the total supply rate is equal to the total demand rate, and the resulting probability of finding excessive congestion along each arc (i.e., the resulting probability of finding congestion inside the queueing system formed along each arc in excess of a fixed number) is equal to a desirable value

On the computational behavior of a polynomial-time network flow algorithm

A variation on the Edmonds-Karp scaling approach to the minimum cost network flow problem is examined. This algorithm, which scales costs rather than right-hand sides, also runs in polynomial time. Large-scale computational experiments indicate that the computational behavior of such scaling algorithms may be much better than had been presumed. Within several distributions of square, dense, capacitated transportation problems, a cost scaling code, SCALE, exhibits linear growth in average execution time with the number of edges, while two network simplex codes, RNET and GNET, exhibit greater than linear growth.Our experiments reveal that median and mean execution times are predictable with surprising accuracy for all of the three codes and all three distributions from which test problems were generated. Moreover, for fixed problem size, individual execution times appear to behave as though they are approximately lognormally distributed with constant variance. The experiments also reveal sensitivity of the parameters in the models, and in the models themselves, to variations in the distribution of problems. This argues for caution in the interpretation of such computational studies beyond the realm in which the computations were performed.

Equitable Demand Adjustment for Infeasible Transportation Problems

We present a modelling/solution procedure for adjusting demands to obtain an ‘equitably infeasible’ solution for an infeasible transportation problem. The infeasibility may be due to total supply not being equal to total demand, or inadmissible routes (arcs). We show that the problem can be modelled and solved as a pre-emptive, multicriteria, capacitated transportation problem, whose objective is to minimize the maximum deviation between the fractional undersupply to the demand nodes or, equivalently, to minimize the fractional undersupply of the demands. Further, we show that an optimal solution can be obtained by solving either a single-criterion transportation problem (by choosing sufficiently large weights to aggregate the criteria) or a sequence of single-criterion transportation problems.

Probabilistic Analysis of the Capacitated Transportation Problem

We consider the capacitated transportation problem defined by sets of supplies ai, i ∈ I, demands bj, j ∈ J, and capacities cij, i ∈ I, j ∈ J. Assuming that the capacities are random variables, we prove asymptotic conditions on the supplies and demands which assure that a feasible solution exists almost surely. The proof is constructive and supplies an algorithm whose running time is O(| I| |J|). We then apply the results to the maximum flow problem.

MODIFIED STEPPING‐STONE METHOD AS A TEACHING AID FOR CAPACITATED TRANSPORTATION PROBLEMS

ABSTRACTThis paper extends the standard stepping‐stone method to the case of capacitated transportation problems. The purpose is to offer an alternative to network‐oriented or simplex‐based methods (such as the out‐of‐kilter and upper‐bounding techniques) often used in handling the capacitated transportation problem but generally beyond the scope of introductory decision sciences courses. The proposed procedure is based on the simple stepping‐stone method and requires only a few modifications in the cost/assignment matrix and some alterations in the pivoting rules. As such, it is much easier to understand and apply than the above‐mentioned solution algorithms. It therefore can serve as an effective tool for teaching capacitated transportation problems. Two numerical examples illustrate applications of the proposed method, procedures for sensitivity analysis, and comparisons with linear programming.

Solving network manpower problems with side constraints

A number of authors have used goal-programming to solve manpower problems. Some of these authors have also formulated goal-programming manpower models in a network so as to take advantage of the rapid codes now available for the solution of capacitated transportation problems. In this paper, a number of methods of dealing with side constraints (those not included in the network formulation) are explored. Of particular interest are constraints for dealing with attrition and for expressing budgetary restrictions.