Development of a mathematical model for 'Hayward' kiwifruit softening in the supply chain : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Food Technology at Massey University, New Zealand

Fruit loss is a major concern to the kiwifruit industry as it incurs a high cost to monitor and remove over soft or rotten fruit to meet export standards. Kiwifruit is exposed to various temperature scenarios due to different packhouse cooling practices, and temperature control is difficult to maintain throughout the supply chain. Fruit pallet temperatures are wirelessly monitored in the supply chain. This time temperature data provides valuable rich information which could be used to predict kiwifruit quality. In the laboratory, green ‘Hayward’ kiwifruit were exposed to industry coolchain scenarios to investigate their influence on fruit firmness in subsequent storage. Cooling rate and storage temperature were identified to affect fruit firmness and chilling injury development significantly, where accelerated softening and increased chilling injury development was observed in late storage (> 100 d) when fruit were cooled directly to 0 °C. However, when fast cooled fruit were stored at 2 °C instead of 0 °C, low incidence of chilling injury was observed. The influence of cooling rate and storage temperature on kiwifruit quality suggests that industry should focus on the management practices adopted by packhouses in order to maintain acceptable quality after long term storage. A proportion of the firmness data results were used to develop a mechanistic style mathematical model of kiwifruit softening. Kiwifruit softening was mathematically described based on the correlation with starch degradation, breakdown of cell wall structure, and a description of the incidence of chilling injury development during storage. The model inputs consist of solely commonly collected at-harvest attributes: firmness, dry matter and soluble solids content and time-temperature data. Applying at-harvest attributes as model inputs enabled a capability to predict different softening curves as influenced by fruit maturity, and grower line differences. The developed model demonstrated promising softening prediction with mean absolute errors (MAE) between 0.8 to 2.1 N when fruit were exposed to fluctuating temperatures and cooling profiles. A logistic model was used to estimate the proportion of chilling injured fruit. Based on the given time temperature information, the logistic model was able to predict the proportion of chilling injured fruit reasonably well (R2 = 0.735). This chilling injury prediction was subsequently used to adjust the softening prediction during the late storage period (>100 d). Model validation was performed using the remaining data, identifying a lack of fit in both the rapid (MAE of 20.8 N) and gradual (MAE of 8.0 N) softening phase. The lack of fit in the rapid softening phase is proposed to be explained by the presence of an initial lag phase in softening which the developed model is unable to predict. The magnitude of firmness associated with starch content and cell wall integrity heavily influenced the lack of fit in the gradual softening phase. Fixing the initial amount of firmness associated to cell wall integrity to be constant for all maturities and grower lines improved the softening prediction. Overall, this thesis contributes to the challenge of predictively modelling kiwifruit quality in the supply chain. However, there are still many opportunities for improvement including introducing the influence of: variation within the same batch; fruit maturity on chilling injury development; ethylene in the environment; pre-harvest management practices and extending the model to have more focus on high temperature conditions such as those experienced in the marketplace. Conducting studies on: the effect of curing on kiwifruit; using non-destructive techniques to provide information to help define model parameters for prediction; effect of high temperature exposure on kiwifruit softening are possible opportunities that may contribute to enable better prediction of kiwifruit quality in the supply chain in the future

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In a cost sharing problem on a weighted undirected graph, all other nodes want to connect to the source node for some service. Each edge has a cost denoted by a weight and all the connected nodes should share the total cost for the connectivity. The goal of the existing solutions (e.g. folk solution and cycle-complete solution) is to design cost sharing rules with nice properties, e.g. budget balance and cost monotonicity. However, they did not consider the cases that each non-source node has a budget which is the maximum it can pay for its cost share and may cut its adjacent edges to reduce its cost share. In this paper, we design two cost sharing mechanisms taking into account the nodes' budgets and incentivizing all nodes to report all their adjacent edges so that we can minimize the total cost for the connectivity.Comment: arXiv admin note: substantial text overlap with arXiv:2201.0597

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We consider a cost sharing problem on a weighted directed acyclic graph (DAG) with a source node to which all the other nodes want to connect. The cost (weight) of each edge is private information reported by multiple contractors, and among them, only one contractor is selected as the builder. All the nodes except for the source need to share the total cost of the used edges. However, they may block others' connections to the source by strategically cutting their outgoing edges to reduce their cost share, which may increase the total cost of connectivity. To minimize the total cost of connectivity, we design a cost sharing mechanism to incentivize each node to offer all its outgoing edges and each contractor to report all the edges' weights truthfully, and show the properties of the proposed mechanism. In addition, our mechanism outperforms the two benchmark mechanisms

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