Multiobjective long-term planning of biopharmaceutical manufacturing facilities

Biopharmaceutical companies with large portfolios of clinical and commercial products typically need to allocate production across several multiproduct facilities, including third party contract manufacturers. This poses several capacity planning challenges which are further complicated by the need to satisfy different stakeholders often with conflicting objectives. This work addresses the question of how a biopharmaceutical manufacturer can make better long-term capacity planning decisions given multiple strategic criteria such as cost, risk, customer service level, and capacity utilization targets. A long-term planning model that allows for multiple facilities and accounts for multiple objectives via goal programming is developed. An industrial case study based on a large scale biopharmaceutical manufacturer is used to illustrate the functionality of the model. A single objective model is used to identify how best to use existing capacity so as to maximize profits for different demand scenarios. Mitigating risk due to unforeseen circumstances by including a dual facility constraint is shown to be a reasonable strategy at base case demand levels but unacceptable if demands are 150% higher than expected. The capacity analysis identifies where existing capacity fails to meet demands given the constraints. A multiobjective model is used to demonstrate how key performance measures change given different decision making policies where different weights are assigned to cost, customer service level, and utilization targets. The analysis demonstrates that a high profit can still be achieved while meeting key targets more closely. The sensitivity of the optimal solution to different limits on the targets is illustrated.     

Divide and conquer: when and how should competitors share?

When two individuals are unwilling to fight over a valuable resource, then they may obtain it with equal probability, or they may choose to divide the resource in some way. Although both strategies have been observed in nature, modelers have so far implicitly assumed that their long-term payoffs are the same. First we show that increasing returns to size in the value of a resource favor random allocation over sharing, whereas diminishing returns favor the reverse. Next we extend our approach to understand the conditions under which sharing will evolve when contestants vary in their resource-holding potential. We show that although closely matched individuals are more likely to share, it is by no means a prerequisite when contestants have limited information about one another’s abilities. Collectively, our models support recent observations of physical sharing as a solution to conflict resolution, and elucidate the conditions under which sharing will arise.     

Tolerance of understory plants subject to herbivory by roe deer

Classical foraging theory states that animals feeding in a patchy environment can maximise their long term prey capture rates by quitting food patches when they have depleted prey to a certain threshold level. Theory suggests that social foragers may be better able to do this if all individuals in a group have access to the prey capture information of all other group members. This will allow all foragers to make a more accurate estimation of the patch quality over time and hence enable them to quit patches closer to the optimal prey threshold level. We develop a model to examine the foraging efficiency of three strategies that could be used by a cohesive foraging group to initiate quitting a patch, where foragers do not use such information, and compare these with a fourth strategy in which foragers use public information of all prey capture events made by the group. We carried out simulations in six different prey environments, in which we varied the mean number of prey per patch and the variance of prey number between patches. Groups sharing public information were able to consistently quit patches close to the optimal prey threshold level, and obtained constant prey capture rates, in groups of all sizes. In contrast all groups not sharing public information quit patches progressively earlier than the optimal prey threshold value, and experienced decreasing prey capture rates, as group size increased. This is more apparent as the variance in prey number between patches increases. Thus in a patchy environment, where uncertainty is high, although public information use does not increase the foraging efficiency of groups over that of a lone forager, it certainly offers benefits over groups which do not, and particularly where group size is large.     

Designing equitable antiretroviral allocation strategies in resource-constrained countries

Background

Recently, a global commitment has been made to expand access to antiretrovirals (ARVs) in the developing world. However, in many resource-constrained countries the number of individuals infected with HIV in need of treatment will far exceed the supply of ARVs, and only a limited number of health-care facilities (HCFs) will be available for ARV distribution. Deciding how to allocate the limited supply of ARVs among HCFs will be extremely difficult. Resource allocation decisions can be made on the basis of many epidemiological, ethical, or preferential treatment priority criteria.

Methods and Findings

Here we use operations research techniques, and we show how to determine the optimal strategy for allocating ARVs among HCFs in order to satisfy the equitable criterion that each individual infected with HIV has an equal chance of receiving ARVs. We present a novel spatial mathematical model that includes heterogeneity in treatment accessibility. We show how to use our theoretical framework, in conjunction with an equity objective function, to determine an optimal equitable allocation strategy (OEAS) for ARVs in resource-constrained regions. Our equity objective function enables us to apply the egalitarian principle of equity with respect to access to health care. We use data from the detailed ARV rollout plan designed by the government of South Africa to determine an OEAS for the province of KwaZulu–Natal. We determine the OEAS for KwaZulu–Natal, and we then compare this OEAS with two other ARV allocation strategies: (i) allocating ARVs only to Durban (the largest urban city in KwaZulu–Natal province) and (ii) allocating ARVs equally to all available HCFs. In addition, we compare the OEAS to the current allocation plan of the South African government (which is based upon allocating ARVs to 17 HCFs). We show that our OEAS significantly improves equity in treatment accessibility in comparison with these three ARV allocation strategies. We also quantify how the size of the catchment region surrounding each HCF, and the number of HCFs utilized for ARV distribution, alters the OEAS and the probability of achieving equity in treatment accessibility. We calculate that in order to achieve the greatest degree of treatment equity for individuals with HIV in KwaZulu–Natal, the ARVs should be allocated to 54 HCFs and each HCF should serve a catchment region of 40 to 60 km.

Conclusion

Our OEAS would substantially improve equality in treatment accessibility in comparison with other allocation strategies. Furthermore, our OEAS is extremely different from the currently planned strategy. We suggest that our novel methodology be used to design optimal ARV allocation strategies for resource-constrained countries.     

Workers, sexuals, or both? Optimal allocation of resources to reproduction and growth in annual insect colonies

Understanding decisions about the allocation of resources into colony growth and reproduction in social insects is one of the challenging issues in sociobiology. In their seminal paper, Macevicz and Oster predicted that, for most annual insect colonies, a bang–bang strategy should be favoured by selection, i.e. a strategy characterised by an “ergonomic phase” with exponential colony growth followed by a “reproductive phase” with all resources invested into the production of sexuals. Yet, there is empirical evidence for the simultaneous investment into the production of workers and sexuals in annual colonies (graded control). We, therefore, re-analyse and extend the original model of Macevicz and Oster. Using basic calculus, we can show that sufficiently strong negative correlation between colony size and worker efficiency or increasing mortality of workers with increasing colony size will favour the evolution of graded allocation strategies. By similar reasoning, graded control is predicted for other factors limiting colony productivity (for example, if queens’ egg laying capacity is limited).     

Reengineering the Order Fulfillment Process in Supply Chain Networks

An order fulfillment process (OFP) starts with receiving orders from customers and ends with delivering the finished goods. The order fulfillment process is complex because it is composed of several activities, executed by different functional entities, and heavily interdependent among the tasks, resources, and agents involved in the process. A supply chain network (SCN) is a network of autonomous or semi-autonomous business entities involved, through upstream and downstream links, in the different processes and activities that produce goods or services to customers. As manufacturing practice shifts toward the outsourcing paradigm, the OFP is more likely to be executed throughout SCNs. It becomes imperative to integrate the OFP into SCNs to improve the OFP. Generalizing from the variety and complexity of SCNs, this study identifies several main types of SCN structures and addresses OFP issues based on them. The objective of reengineering the OFP is to achieve agility in the process in terms of efficiency, flexibility, robustness, and adaptability. We propose a multiagent information system (MAIS) approach for reengineering the OFP in SCNs. The MAIS models the OFP in SCNs and evaluates OFP performance by applying the proposed strategies. A multiagent simulation platform, called Swarm, is enhanced and applied for modeling the MAIS; and experiments are conducted to simulate the OFP and then evaluate the potential improvement strategies to identify useful strategies for improving the OFP. The strategies we evaluated include (1) coordinating demand management policies, (2) information sharing strategies, (3) synchronizing material and capacity availability, (4) dynamic resource allocation, and (5) the combination of various strategies. The results shed light on identifying the main effects of various strategies on OFP performance. The insights from utilizing various strategies in different SCNs help reengineer the OFP in SCNs.     

Sustainable sourcing of strategic raw materials by integrating recycled materials

In this paper we investigate a manufacturer’s sustainable sourcing strategy that includes recycled materials. To produce a short life-cycle electronic good, strategic raw materials can be bought from virgin material suppliers in advance of the season and via emergency shipments, as well as from a recycler. Hence, we take into account virgin and recycled materials from different sources simultaneously. Recycling makes it possible to integrate raw materials out of steadily increasing waste streams back into production processes. Considering stochastic prices for recycled materials, stochastic supply quantities from the recycler and stochastic demand as well as their potential dependencies, we develop a single-period inventory model to derive the order quantities for virgin and recycled raw materials to determine the related costs and to evaluate the effectiveness of the sourcing strategy. We provide managerial insights into the benefits of such a green sourcing approach with recycling and compare this strategy to standard sourcing without recycling. We conduct a full factorial design and a detailed numerical sensitivity analysis on the key input parameters to evaluate the cost savings potential. Furthermore, we consider the effects of correlations between the stochastic parameters. Green sourcing is especially beneficial in terms of cost savings for high demand variability, high prices of virgin raw material and low expected recycling prices as well as for increasing standard deviation of the recycling price. Besides these advantages it also contributes to environmental sustainability as, compared to sourcing without recycling, it reduces the total quantity ordered and, hence, emissions are reduced.     

Aligning demand and supply flexibility in custom product co-design

Flexibility of supply and demand is essential for successful implementation of a mass customization strategy that delivers sustained competitive advantage. Supply flexibility, i.e., a choice of alternative products designed to perform the same basic function, is made possible by the range of capabilities available in flexible and agile manufacturing systems and in supply chains. Demand flexibility is derived from the degree to which a customer is willing to compromise on product features or performance levels in order to meet budgetary (reflected in price) or schedule (reflected in delivery) constraints. Flexibility of both supply and demand can have significant strategic and financial value if they are properly aligned. However, customers are mostly unaware of mapping of demand flexibility on to supply flexibility and its impact on production cost and time. Recent advances in information technology have made it possible to co-design a product that involves customer on one end and the manufacturer on the other. This creates an aura and an opportunity where a middle ground between the supply and demand flexibility can be explored and a “deal” can be struck where both parties settle for a product that is beneficial to both through a negotiated settlement. In this paper, we develop a framework for such negotiations. The customer requirements are treated as a range of negotiable options instead of a set of fixed inputs. Demand and supply for customization is then matched by aligning the flexibility of manufacturing systems with customers’ requirement options. Based on this framework, a negotiation scheme is developed to assist customers and manufacturers in exploring and utilizing demand and supply flexibility information in co-design. The negotiation scheme is formulated using goal programming. Finally, an interactive problem-solving procedure is developed and implemented with an illustrative example.     

Component Partitioning Under Demand and Capacity Uncertainty in Printed Circuit Board Assembly

This paper considers the problem of configuring a printed circuit board (PCB) assembly line experiencing uncertainty in demand and capacity. The PCB assembly process involves a single line of automatic placement machines, a variety of board types, and a number of component types. The line is set up only once, at the beginning of a production cycle, to eliminate setups between board types. Using this strategy, the line therefore can assemble all different types of PCBs without feeder changes. The problem then becomes to partition component types to the different machines in the hope of processing all boards quickly with a good workload balance. In this paper, the board demands and machine breakdowns are random but follow some probability distribution, which can be predicted from past observations of the system. We formulate this problem as a stochastic mixed-integer programming formulation with the objective of minimizing the expected makespan for assembling all PCBs during a production cycle. The results obtained indicate significant improvement over the existing methods. We hope that this research will provide more PCB assembly facilities with models and techniques to hedge against variable forecasts and capacity plans     

Completeness in structural genomics

Structural genomics has the goal of obtaining useful, three-dimensional models of all proteins by a combination of experimental structure determination and comparative model building. We evaluate different strategies for optimizing information return on effort. The strategy that maximizes structural coverage requires about seven times fewer structure determinations compared with the strategy in which targets are selected at random. With a choice of reasonable model quality and the goal of 90% coverage, we extrapolate the estimate of the total effort of structural genomics. It would take approximately 16,000 carefully selected structure determinations to construct useful atomic models for the vast majority of all proteins. In practice, unless there is global coordination of target selection, the total effort will likely increase by a factor of three. The task can be accomplished within a decade provided that selection of targets is highly coordinated and significant funding is available.     

Global Documentation of Fish Introductions: the Growing Crisis and Recommendations for Action

Fish provides 15% of the total animal protein in human diets. It is also the primary source of livelihood for 35 million people (30 M in Asia and 2.6 M in Africa). The increase in global population and demand for fish protein cannot be met by capture fisheries alone. Governments are turning towards aquaculture as the source of fish protein. However, it has also led to the introduction and establishment of non-native species in local ecosystems through their escapement from aquaculture facilities to the wild. In freshwater ecosystems with relatively high endemism, this has become a significant problem. Documenting the international movement of fish is one way of providing a general view of the magnitude of these movements and the existing and potential threat faced by ecosystems due to species invasiveness. Information, however, is limited and scattered in different journals and agency/project reports. Several agencies, both local and international, have databases that provide information on invasive species (terrestrial and aquatic, local, regional or international in scope). The critical challenge is for consolidation, common access through data sharing and development of risk assessment and management tools. This is proposed through the use of Internet technology, sharing of databases or having a gateway or portal to which all introduced and invasive fish species related databases link. The fusion of these information sources will allow access to updated and reliable information. The experience of the WorldFish Center in documenting these phenomena through developing the FishBase information system and global partnerships is presented with recommendations for harmonizing approaches. An erratum to this article is available at .     

A two-stage stochastic programming approach for new tape-out allocation decisions for demand fulfillment planning in semiconductor manufacturing

Demand fulfillment and capacity utilization directly affects customer satisfaction, market growth, and the profitability of the company in the semiconductor industry. These characteristics boost the significance of allocating various customer demands to a number of wafer fabrication facilities (fabs) with different capacity configurations. Before volume production, the introduction of new semiconductor product, namely new tape-out (NTO), requires extremely sophisticated and lengthy qualification with high-cost masks and pilot runs in the qualified fabs. Thus, the NTO allocation will affect future product mix of the qualified fabs, and the flexibility to fulfill the volume demands of the allocated NTOs in the corresponding fabs. This research aims to construct a two-stage stochastic programming (2-SSP) demand fulfillment model. The first stage considers NTO allocation decisions to a number of qualified fabs before the corresponding demand volume is realized. The second stage allocates the capacity to fulfill demand requirements based on the results of four options of capacity reconfiguration: (1) qualifying a product to more than one fab (share); (2) physically transferring a set of masks for a product from one fab to another, where a requalification is required (transfer); (3) moving tools from under-loaded fabs to over-utilized fabs (backup); and (4) utilizing different technologies to capacity inside a fab to support the technology with insufficient capacities (exchange). Both the share and transfer options require long lead time for qualification, whereas the backup and exchange options can be accomplished within a planned timeframe. A numerical study based on real settings is conducted to estimate the validity of the proposed 2-SSP model via values of stochastic solution (VSS) and expected values of perfect information (EVPI). The results showed the benefits of adopting 2-SSP models, especially in an environment with high-demand fluctuation. Furthermore, the proposed 2-SSP can provide near-optimal solutions similar to those of deterministic models with perfect information.     

Improved dairy cattle mating plans at herd level using genomic information

From 2012 to 2018, 223 180 Montbéliarde females were genotyped in France and the number of newly genotyped females increased at a rate of about 33% each year. With female genotyping information, farmers have access to the genomic estimated breeding values of the females in their herd and to their carrier status for genetic defects or major genes segregating in the breed. This information, combined with genomic coancestry, can be used when planning matings in order to maximize the expected on-farm profit of future female offspring. We compared different mating allocation approaches for their capacity to maximize the expected genetic gain while limiting expected progeny inbreeding and the probability to conceive an offspring homozygous for a lethal recessive allele. Three mate allocation strategies (random mating (RAND), sequential mating (gSEQ€) and linear programing mating (gLP€)) were compared on 160 actual Montbéliarde herds using male and female genomic information. Then, we assessed the benefit of using female genomic information by comparing matings planned using only female pedigree information with the equivalent strategy using genomic information. We measured the benefit of adding genomic expected inbreeding and risk of conception of an offspring homozygous for a lethal recessive allele to Net merit in mating plans. The influence of three constraints was tested: by relaxing the constraint on availability of a particular semen type (sexed or conventional) for bulls, by adding an upper limit of 8.5% coancestry between mate pairs or by using a more stringent maximum use of a bull in a herd (5% vs 10%). The use of genomic information instead of pedigree information improved the mate allocation method in terms of progeny expected genetic merit, genetic diversity and risk to conceive an offspring homozygous for a lethal recessive allele. Optimizing mate allocation using linear programming and constraining coancestry to a maximum of 8.5% per mate pair reduced the average coancestry with a small impact on expected Net Merit. In summary, for male and female selection pathways, using genomic information is more efficient than using pedigree information to maximize genetic gain while constraining the expected inbreeding of the progeny and the risk to conceive an offspring homozygous for a lethal recessive allele. This study also underlines the key role of semen type (sexed vs conventional) and the associated constraints on the mate allocation algorithm to maximize genetic gain while maintaining genetic diversity and limiting the risk to conceive an offspring homozygous for a lethal recessive allele.     

Flee or fight uncertainty: plant strategies in relation to anticipated damage

In order to cope with damage, plants have evolved a number of strategies. We incorporate two of those strategies, compensatory regrowth and escaping damage in time, into a mathematical model in an attempt to outline under what circumstances one or the other of these phenotypic traits will evolve. Escaping damage in time is accomplished by flowering and setting seeds at a point of time when the risk of damage is low, whereas a compensatory capacity is made possible by activating a proportion of meristems that are left dormant. Our analysis suggests that damage that is predictable in time will favour phenotypes that flower late in the season and that have a good compensatory capacity. As damage becomes less predictable in time, a strategy that implies flowering as early as possible in the season and with no compensatory capacity at all, becomes advantageous.     

How Herbivore Browsing Strategy Affects Whole-Plant Photosynthetic Capacity

If a browse damage index indicates that a tree has been 50% browsed by herbivores, does this mean half the leaves are entirely eaten or are all the leaves half eaten? Were the affected leaves old or young? Large or small? In sunshine or shade? Understanding what effect browsing will have on the photosynthetic capacity and the plant’s survival ability clearly requires a greater understanding of browsing strategy across the canopy than can be given by a single index value. We developed stochastic models of leaf production, growth and consumption using data from kamahi (Weinmannia racemosa) trees in New Zealand which have been browsed by possums (Trichosurus vulpecula), to ascertain which of six feasible browsing strategies possums are most likely to be employing. We compared the area distribution of real fallen leaves to model output in order to select the best model, and used the model to predict the age distribution of leaves on the tree and thus infer its photosynthetic capability. The most likely browsing strategy that possums employ on kamahi trees is a preference for virgin (i.e. previously unbrowsed) leaves, consistent with the idea that browsing increases the production of chemical plant defences. More generally, our results show that herbivore browsing strategy can significantly change the whole-plant photosynthetic capability of any plant and hence its ability to survive, and therefore, herbivore damage indices should be used in conjunction with more detailed information about herbivore browsing strategy.     

Why Are Product Prices in Online Markets Not Converging?

Why are product prices in online markets dispersed in spite of very small search costs? To address this question, we construct a unique dataset from a Japanese price comparison site, which records price quotes offered by e-retailers as well as customers’ clicks on products, which occur when they proceed to purchase the product. The novelty of our approach is that we seek to extract useful information on the source of price dispersion from the shape of price distributions rather than focusing merely on the standard deviation or the coefficient of variation of prices, as previous studies have done. We find that the distribution of prices retailers quote for a particular product at a particular point in time (divided by the lowest price) follows an exponential distribution, showing the presence of substantial price dispersion. For example, 20 percent of all retailers quote prices that are more than 50 percent higher than the lowest price. Next, comparing the probability that customers click on a retailer with a particular rank and the probability that retailers post prices at a particular rank, we show that both decline exponentially with price rank and that the exponents associated with the probabilities are quite close. This suggests that the reason why some retailers set prices at a level substantially higher than the lowest price is that they know that some customers will choose them even at that high price. Based on these findings, we hypothesize that price dispersion in online markets stems from heterogeneity in customers’ preferences over retailers; that is, customers choose a set of candidate retailers based on their preferences, which are heterogeneous across customers, and then pick a particular retailer among the candidates based on the price ranking.     

Growth, Allocation and Water Relations of Shade-grownQuercus rubraL. Saplings Exposed to a Late-season Canopy Gap

For understory saplings to exploit canopy gaps successfully,carbon gain must increase in the gap environment. We predictedthat total biomass of shade-grown red oak saplings would increaseafter exposure to a late-season canopy gap, and that increasedwater and nutrient demand within the canopy gap would drivechanges in the allocation of this carbon. Shade-grown red oaksaplings acclimated to gaps by increasing biomass during theseason of gap formation and increasing the potential for carbongain in the following summer. Within-season carbon gain didnot result from greater production of leaf area, so it mostlikely arose from higher photosynthetic rates of existing shade-developedfoliage, which may be linked to accumulation of leaf nitrogen.During the season of gap formation, shade-gap plants increasedallocation to storage of total non-structural carbohydrates(TNC), and to root growth. The increase in TNC storage suggeststhat shade-developed saplings exposed to gaps were also primedfor fast growth and carbon gain in the following summer. Theincrease in root growth suggests that higher nutrient and waterdemand drove allocation shifts to enhance the capacity for nutrientand water uptake in the gap. Plant hydraulic conductivity (K_a)of shade-grown plants was limited upon exposure to the gap,possibly because of embolism formation resulting from the abruptincrease in water demand. Greater water potential gradientscompensated for limitations to K_a, allowing saplings to maintainhigh transpiration rates, suggesting that actual water uptakeof shade-gap plants was unaffected by gap exposure. Acclimation; canopy gaps; carbon allocation; hydraulic conductivity; nitrogen allocation; non-structural carbohydrates;Quercus rubra L.; red oak; stomatal conductance; transpiration; water-relations     

How to predict community responses to perturbations in the face of imperfect knowledge and network complexity

Recent attempts to predict the response of large food webs to perturbations have revealed that in larger systems increasingly precise information on the elements of the system is required. Thus, the effort needed for good predictions grows quickly with the system''s complexity. Here, we show that not all elements need to be measured equally well, suggesting that a more efficient allocation of effort is possible. We develop an iterative technique for determining an efficient measurement strategy. In model food webs, we find that it is most important to precisely measure the mortality and predation rates of long-lived, generalist, top predators. Prioritizing the study of such species will make it easier to understand the response of complex food webs to perturbations.     

Economic allocation: Examples and derived decision tree

Goal, Scope and Background  In the recently published (Dutch) Handbook on LCA, economic allocation is advised as baseline method for most allocation situations in a detailed LCA. Although the Handbook on LCA aimed to provide a ‘cookbook’ with operational guidelines for conducting each step of an LCA, this was not completely achieved for the allocation step. The guidelines for allocation largely remained at the level of principles. This restricted elaboration of economic allocation may hamper application in practice. Therefore, this paper elaborates some examples applying economic allocation. Method  Two concepts are of particular importance when applying economic allocation: functional flow and multi-functional process. The definitions of these concepts are presented and discussed. The basic principle of economic allocation is that having determined the various functional flows of a multi-functional process, all other flows need to be allocated to these functional flows according to their shares in the total proceeds. Proceeds are based on prices and these are not always easy to determine for a process. A summary of possible solutions for different problems when determining prices is given. Results and Discussion  The examples presented focus on co-production and various recycling situations. All examples are hypothetical in order to avoid discussions on the data. The examples show that the prices of the functional flows determine the allocation results. It is of importance to have correct information on the relative prices of the functional flows at stake, especially whether they are negative or positive. Learning from these examples, we establish a decision tree for economic allocation. The decision tree is meant for identifying and handling multi-functionality situations starting from a defined (product) system. This decision tree is with minor adaptations also applicable to other allocation methods and has a more general value than for the economic allocation method only. Conclusions and perspective  The examples have helped us to establish a decision tree for handling the multi-functionality problem by economic allocation. The examples can be broadened to other materials and allocation situations. We would encourage others to provide other examples and experiences as we expect that these will help to further improve and refine the guidelines and decision tree for economic allocation in future.     

An Optimal Strategy for Energy Allocation in a Multiple Resource Environment

We consider a model for a population in discrete time with nonoverlapping generations that has fecundity proportional to the amounts of two essential resources obtained up to a saturating level. A population’s strategy defines how individuals divide their total available energy between efforts to obtain the two resources. We assume that the total amount of each resource obtained is a positive, increasing, concave down function of the total energy exerted toward the resource. By considering two competing subpopulations that have different energy allocation strategies, we characterize the stability of all possible equilibria and find a unique optimal strategy where a fixed subpopulation resists invasion by a small competing subpopulation using any other strategy. Except when one of the resources is readily obtained above the saturation level, this optimal strategy is to divide effort equally between the resources. We illustrate the behavior of the model, directly showing the effects of an invading subpopulation with pairwise invasibility plots.