Sustainable Yield Analysis in a Multicohort Single-species Fishery: A Mathematical Programming Approach

The concept of sustainable yield, i.e. the fish catch that can be maintained in the long run from a fishery in a steady state, is widely used in fisheries management. In traditional methods of sustainable yield analysis, based on the Schaefer model of a fish stock, the age structure of the stock is ignored. Approaches based on the Beverton-Holt multicohort fish population model take account of age structure but assume that instantaneous natural and fishing mortality rates are constant throughout the year. Using a fish population model in which this assumption is not required, a mixed integer programming model is developed for the analysis of a multicohort single-species fishery in a steady state. This new method of sustainable yield analysis is demonstrated using data for the western mackerel fishery. Comparisons with results from other studies of this fishery are presented.     

IS FISHING COMPATIBLE WITH ENVIRONMENTAL CONSERVATION: A STOCHASTIC MODEL WITH AN ELEMENT OF SELF‐PROTECTION

Abstract The purpose of this paper is to introduce the impact of fishing activity on a marine ecosystem. The fishing activity is considered not only through annual harvest but also through a second component, called the degree of protection of the fishery environment. This characterizes the environmental impact of fishing. A stochastic dynamic programming problem is presented in infinite horizon, where a sole owner seeks to maximize a discounted expected profit. The main hypothesis states that the stock–recruitment relationship is stochastic and that both components of the fishing activity have an impact on the probability law of the state of the fishery environment. The optimal fishing policy is obtained and compared with standard models. This optimal policy has the following properties: is not a constant escapement policy and indicates an element of self‐protection by the fishery manager. The paper ends with a discussion on the existence of degrees of protection of the fishery environment that take into account the environmental conservation and preservation of economic activity.     

A comparison of different dispatching policies in two-warehouse inventory systems for deteriorating items over a finite time horizon

This paper considers a two-warehouse inventory problem for deteriorating items with a constant demand rate over a finite time horizon. A modified first-in-first-out dispatching policy is first proposed, and a new two-warehouse inventory model based on this dispatching policy is developed. The results of this model are then compared with those of other models based on classical dispatching policies, such as the last-in-first-out, modified last-in-first-out and first-in-first-out dispatching policies. We also prove the existence and uniqueness of the optimal solutions for the models considered. Finally, a numerical example is presented to illustrate the results, and several key conditions are derived for comparing the general cases of these four models.     

The production–inventory problem of a product with time varying demand,production and deterioration rates

In this paper, we consider the production–inventory problem in which the demand, production and deterioration rates of a product are assumed to vary with time. Shortages of a cycle are allowed to be backlogged partially. Two models are developed for the problem by employing different modeling approaches over an infinite planning horizon. Solution procedures are derived for determining the optimal replenishment policies. A procedure to find the near-optimal operating policy of the problem over a finite time horizon is also suggested.     

THE EFFECTS OF DIFFERENT STRATEGIC VARIABLES IN NONCOOPERATIVE FISHERIES GAMES

Abstract In this paper, we use stock size, harvest quantity, and fishing effort as strategic variables. We model a two‐agent noncooperative fishery game, where the agents (nations) harvest a common fish stock. The planning horizon is infinite. The model is solved successively using one instrument at a time as the strategic variable in the game. The net present values of fishing and the escapement stock level from the three different models are compared to show how the choice of variables affects the results. The choice of strategic variable is not a trivial one, as the results are shown to be sensitive to the discounting, the stock's rate of growth, and the assumptions about the distribution of the fish in response to harvesting.     

Optimal fishery with coastal catch

In many spatial resource models, it is assumed that an agent is able to harvest the resource over the complete spatial domain. However, agents frequently only have access to a resource at particular locations at which a moving biomass, such as fish or game, may be caught or hunted. Here, we analyze an infinite time‐horizon optimal control problem with boundary harvesting and (systems of) parabolic partial differential equations as state dynamics. We formally derive the associated canonical system, consisting of a forward–backward diffusion system with boundary controls, and numerically compute the canonical steady states and the optimal time‐dependent paths, and their dependence on parameters. We start with some one‐species fishing models, and then extend the analysis to a predator–prey model of the Lotka–Volterra type. The models are rather generic, and our methods are quite general, and thus should be applicable to large classes of structurally similar bioeconomic problems with boundary controls. Recommedations for Resource Managers

• Just like ordinary differential equation‐constrained (optimal) control problems and distributed partial differential equation (PDE) constrained control problems, boundary control problems with PDE state dynamics may be formally treated by the Pontryagin's maximum principle or canonical system formalism (state and adjoint PDEs).
• These problems may have multiple (locally) optimal solutions; a first overview of suitable choices can be obtained by identifying canonical steady states.
• The computation of canonical paths toward some optimal steady state yields temporal information about the optimal harvesting, possibly including waiting time behavior for the stock to recover from a low‐stock initial state, and nonmonotonic (in time) harvesting efforts.
• Multispecies fishery models may lead to asymmetric effects; for instance, it may be optimal to capture a predator species to protect the prey, even for high costs and low market values of the predators.

     

Asymptotic properties of constrained Markov Decision Processes

We present in this paper several asymptotic properties of constrained Markov Decision Processes (MDPs) with a countable state space. We treat both the discounted and the expected average cost, with unbounded cost. We are interested in (1) the convergence of finite horizon MDPs to the infinite horizon MDP, (2) convergence of MDPs with a truncated state space to the problem with infinite state space, (3) convergence of MDPs as the discount factor goes to a limit. In all these cases we establish the convergence of optimal values and policies. Moreover, based on the optimal policy for the limiting problem, we construct policies which are almost optimal for the other (approximating) problems. Based on the convergence of MDPs with a truncated state space to the problem with infinite state space, we show that an optimal stationary policy exists such that the number of randomisations it uses is less or equal to the number of constraints plus one. We finally apply the results to a dynamic scheduling problem.This work was partially supported by the Chateaubriand fellowship from the French embassy in Israel and by the European Grant BRA-QMIPS of CEC DG XIII     

The structure of optimal solutions for harvesting a renewable resource

We consider the problem of optimal harvesting of a renewable resource whose dynamics are governed by logistic growth and whose payoff is proportional to the harvest. We consider both the case of a finite and an infinite time horizon and analyse the structure of the optimal solutions and their dependence on the parameters of the model. We show that the optimal policy can only have one of three structures: (1) maximal harvesting effort until the resource is depleted, (2) zero harvesting during an initial time interval followed by a subsequent switch to maximal harvesting effort, or (3) a singular solution, which corresponds to an intermediate level of harvesting, accompanied by the most rapid approach path. All three scenarios emerge, with minor variations, with finite and infinite time horizons, depending on the particular combination of parameters of the system. We characterize the conditions under which the singular solution is optimal and present suggestions for designing an optimal and sustainable harvesting strategy. Recommendations for Resource Managers :

• We have rigorously explored a standard optimal harvesting model and its steady states.
• We show that three different types of solutions may emerge: (i) maximal harvesting eventually leading to a complete depletion of the stock; (ii) maximal harvesting with a potential period of idleness leading to a positive stock; (iii) an initial phase of either no or full harvesting followed by a period of intermediate harvesting intensity leading to a positive stock (singular solution).
• With some modifications, similar results hold for a finite planning horizon.
• Which of these three scenarios emerges in the finite horizon case depends not only on the parameter values but also on the length of the planning horizon.

     

Optimal EOQ Models for Deteriorating Items with Time-Varying Demand

In this paper, optimal inventory lot-sizing models are developed for deteriorating items with general continuous time-varying demand over a finite planning horizon and under three replenishment policies. The deterioration rate is assumed to be a constant fraction of the on-hand inventory. Shortages are permitted and are completely backordered. The proposed solution procedures are shown to generate global minimum replenishment schedules for both general increasing and decreasing demand patterns. An extensive empirical comparison using randomly generated linear and exponential demands revealed that the replenishment policy which starts with shortages in every cycle is the least cost policy and the replenishment policy which prohibits shortages in the last cycle exhibited the best service level effectiveness. An optimal procedure for the same problem with trended inventory subject to a single constraint on the minimum service level (maximum fraction of time the inventory system is out of stock during the planning horizon) is also proposed in this paper.     

An Inventory Model with Finite Horizon and Price Changes

An inventory model is developed for a finite horizon and price changes. The structure and form of the optimal policy is determined along with sensitivity analysis with respect to the length of the horizon. This is a prelude to considering an infinite horizon problem in which at some a priori known time the purchase price of the item will increase. In this paper appropriate ordering policies are determined with respect to known information about an ensuing price rise.     

Optimal and feedback strategies for managing multicohort populations

A Beverton and Holt type linear cohort dynamics model is integrated and combined with a nonlinear stock-recruitment relationship to obtain a discrete-time multicohort harvesting model. Assuming that each age class is individually controllable, it is shown, subject to certain assumptions, that the optimal harvesting strategy is to drive the population to the maximum sustainable yield solution in one time step. In most fisheries, this controllability assumption is not met and harvesting is agewise nonselective. In this case, it may be preferable to implement a harvesting policy based on suboptimal constant effort or stock level feedback strategies, rather than implement a more complicated optimal policy. This question is addressed through numerical studies on the management of an anchovy fishery.Dedicated to G. LeitmannThe author would like to thank M. Mangel, W. Reed, P. Sullivan, and G. Swartzman for commenting on a draft of this paper.     

Controlled jump processes

Finite and infinite planning horizon Markov decision problems are formulated for a class of jump processes with general state and action spaces and controls which are measurable functions on the time axis taking values in an appropriate metrizable vector space. For the finite horizon problem, the maximum expected reward is the unique solution, which exists, of a certain differential equation and is a strongly continuous function in the space of upper semi-continuous functions. A necessary and sufficient condition is provided for an admissible control to be optimal, and a sufficient condition is provided for the existence of a measurable optimal policy. For the infinite horizon problem, the maximum expected total reward is the fixed point of a certain operator on the space of upper semi-continuous functions. A stationary policy is optimal over all measurable policies in the transient and discounted cases as well as, with certain added conditions, in the positive and negative cases.     

A normative model of market response to sales force activities

This paper outlines a procedure to estimate optimal salesmen's call policies generating the highest long-run profits over an infinite planning horizon. This procedure is based on a simple model of market responses to sales calls, which accounts for the main characteristics of sales call responses such as carryover effects, minimum spacing, forgetting and decreasing responses after some no-activity period. In a sample application, the optimal call policy (under no salesforce time constraint) led to an estimate of the optimal sales force workload.     

Optimal advertising and pricing in a class of general new-product adoption models

In [21], Sethi et al. introduced a particular new-product adoption model. They determine optimal advertising and pricing policies of an associated deterministic infinite horizon discounted control problem. Their analysis is based on the fact that the corresponding Hamilton–Jacobi–Bellman (HJB) equation is an ordinary non-linear differential equation which has an analytical solution. In this paper, generalizations of their model are considered. We take arbitrary adoption and saturation effects into account, and solve finite and infinite horizon discounted variations of associated control problems. If the horizon is finite, the HJB-equation is a 1st order non-linear partial differential equation with specific boundary conditions. For a fairly general class of models we show that these partial differential equations have analytical solutions. Explicit formulas of the value function and the optimal policies are derived. The controlled Bass model with isoelastic demand is a special example of the class of controlled adoption models to be examined and will be analyzed in some detail.     

Groundwater management: Waiting for a drought*

In this article we present a stylized model for optimal management of an unconfined groundwater resource when the threat of drought exists. The drought is modeled as a stochastic event that hits at an uncertain date and two benchmark management policies are investigated: (a) A policy of optimal dynamic management ignoring the threat of drought; and (b) an economically optimal policy that accounts for the threat of a drought. We show that the optimal predrought steady‐state equilibrium stock size of groundwater under policy b is larger than that under policy (a) Furthermore, we show that an increase in the probability of a drought gives rise to two counteracting effects: One in the direction of a larger predrought steady‐state equilibrium stock size (a recovery effect) and one in the direction of a lower predrought steady‐state equilibrium stock (an extinction effect). We find that the recovery effect dominates the extinction effect. Recommendations for Resource Managers: We analyze two groundwater extraction policies that can be used when a threat of drought exists: (a) Dynamic optimal management ignoring the threat of drought; and (b) dynamic optimal management taking the threat of drought into account. We show that the predrought steady‐state equilibrium stock size of water should be larger under the policy (b) than under policy (a). This conclusion has three implications for resource managers:

• Current groundwater management should take future extraction possibilities into account.

• A resource manager ought to take the threat of drought into account in groundwater management.

• A buffer stock of water should be built‐up before the drought to be drawn upon during the event.

     

Cost Comparisons for Out-of-Phase Inventory Models

Inventory costs for a fixed time period have traditionally been determined by allocating total costs per cycle uniformly throughout that cycle as well as any partial cycles. This procedure for cost allocation has led to the solution of numerous inventory problems, most notable of which is the anticipated price-increase model. When comparing two out-of-phase inventory models, if costs are accounted for when they occur over a fixed planning horizon, inventory policies should be changed to reflect the impact of this different cost-allocation procedure. For the anticipated price-increase model, the ‘optimal’ order quantity as well as the implied savings in inventory costs will be different when cost models are developed based on these different cost-allocation methods. If the objective is to maximize over a fixed planning horizon the actual savings in inventory costs as they occur, the cost models presented here should be used.     

Optimal Quarantine Programmes for Controlling an Epidemic Spread

An optimal control problem is formulated with a simple epidemic model in which the control of the epidemic is effected by varying the scale of the quarantine program in a way which minimizes a discounted linear cost over an infinite horizon. An important function of the problem parameters is identified. It is shown that if this function has a value of less than or equal to one, then the optimal policy is not to quarantine at all. While if this functions assume a value in excess of one, then the optimal policy is not to quarantine at all if the initial fraction of infectives is sufficiently high; otherwise, it is optimal to have a full scale quarantine program. Slight modification in these policies are required for the finite horizon version of the problem.     

Stock repurchase with an adaptive reservation price: A study of the greedy policy

We consider the problem of stock repurchase over a finite time horizon. We assume that a firm has a reservation price for the stock, which is the highest price that the firm is willing to pay to repurchase its own stock. We characterize the optimal policy for the trader to maximize the total number of shares that they can buy over a fixed time horizon. In particular, we study a greedy policy, which involves in each period buying a quantity that drives stock price to the reservation price.     

A fair and time‐consistent sharing of the joint exploitation payoff of a fishery

We consider the problem of efficiently managing a fishery where pollution externalities are present. The open‐access bionomic model is analyzed in an ‐player differential game framework with two‐state variables, that is, the fish stock and the pollution stock. We characterize the noncooperative feedback‐Nash equilibrium and cooperative solution, and define an egalitarian sharing rule to allocate the joint welfare maximizing payoff over an infinite time horizon, and show that this rule is time consistent. Recommendations for Resource Managers

• ● Cooperation in management of a fishery where pollution externalities are present yields a higher payoff over time as compared to the noncooperative behavior.
• ● The dividend of cooperation can be allocated among the fisherpersons according to an egalitarian sharing rule.
• ● This allocation is time‐consistent, that is, no player will be tempted to deviate from cooperation as time goes by, and the initial agreement is sustainable.

     

Production-inventory control policy under warm/cold state-dependent fixed costs and stochastic demand: partial characterization and heuristics

The motivation for our study comes from some production and inventory systems in which ordering/producing quantities that exceed certain thresholds in a given period might eliminate some setup activities in the next period. Many examples of such systems have been discussed in prior research but the analysis has been limited to production settings under deterministic demand. In this paper, we consider a periodic-review production-inventory model under stochastic demand and incorporate the following fixed-cost structure into our analysis. When the order quantity in a given period exceeds a specified threshold value, the system is assumed to be in a “warm” state and no fixed cost is incurred in the next period regardless of the order quantity; otherwise the system state is considered “cold” and a positive fixed cost is required to place an order. Assuming that the unsatisfied demand is lost, we develop a dynamic programming formulation of the problem and utilize the concepts of quasi-K-convexity and non-K-decreasing to show some structural results on the optimal cost-to-go functions. This analysis enables us to derive a partial characterization of the optimal policy under the assumption that the demands follow a Pólya or uniform distribution. The optimal policy is defined over multiple decision regions for each system state. We develop heuristic policies that are aimed to address the partially characterized decisions, simplify the ordering policy, and save computational efforts in implementation. The numerical experiments conducted on a large set of test instances including uniform, normal and Poisson demand distributions show that a heuristic policy that is inspired by the optimal policy is able to find the optimal solution in almost all instances, and that a so-called generalized base-stock policy provides quite satisfactory results under reasonable computational efforts. We use our numerical examples to generate insights on the impact of problem parameters. Finally, we extend our analysis into the infinite horizon setting and show that the structure of the optimal policy remains similar.