Allocative Collaborative Profit in Supply Chains

Introduction and motivation Game theory Model description Conclusion

Allocative Collaborative Profit in Supply Chains Anna Tatarczak Maria Curie-Skłodowska University in Lublin, Poland

October 4, 2017

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Allocative Collaborative Profit in Supply Chains


Cooperation in supply chains Setting up a cooperation to improve the own performance is an effective way to improve logistic operations. Types of problems: transportation planning traveling salesman vehicle routing joint distribution inventory

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Joint Replenishment Problem (JRP) is defined as the coordination of multiple items that are jointly ordered and shipped by using a single truck, has grown in importance over the past decade and is of interest to researchers. Scope of the paper This paper examines the subject of cooperation in a JRP for one-supplier multi-retailer systems under the same cost structure. JRP is a model of the supply chain formed by one supplier and n retailers to deal with optimizing the shipment of goods through shared warehouses.

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Motivation Although the benefits from collaboration are appealing, the key question is how to distribute the collaborative savings/profits among the partners. The question is of great importance because a right allocation assures stability and fairness in a cooperation, and, hence cooperative game theory would provide interesting solution approaches to handle such problems. Hence, in this research, we used game theory solution concepts to divide the cost among the members of the coalition. Furthermore, by integrating cooperative game theory with the JRP we propose a new method of cost allocation. The method is based on different distribution keys e.g. volume, capacity.

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A cooperative game is a pair G = (N, ν), where N = {1, 2, ..., n}. N is called the grand coalition.The characteristic function ν assign to every nonempty coalition S ⊆ N a value ν(S), with ν(∅) = 0. When coalition S cooperates, the total cost C (S) and the resulting cost saving is given by X ν(S) = C ({i}) − C (S), for all S ⊆ N. i∈S

Some important properties of cooperative games are listed below in order to derive insights concerning the application of solution concepts.

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Some important properties of cooperative games are listed below in order to derive insights concerning the application of solution concepts. Definition A coalition S is profitable if and only if ν(S) ≥ 0.

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Some important properties of cooperative games are listed below in order to derive insights concerning the application of solution concepts. Definition A coalition S is profitable if and only if ν(S) ≥ 0. Definition The game G = (N, ν) is superadditive if the value ν satisfies the following equation ν(S ∪T ) ≥ ν(S)+ν(T ) for all coalitions S, T ⊂ N Tatarczak

and S ∩T = ∅.



Definition The core of coalitional game G = (N, ν) is defined as n X o X Core(N, ν) = x : φi (ν) = ν(N) ∧ φi (ν) ≥ ν(S), for all S ⊆ N . i∈N

i∈S

The core is the most attractive solution concept in cooperative game theory. It is a set of valid, efficient allocations that cannot be improved upon by a coalition. For a collaborative game, the core may be empty, that is, a balanced budget with stable cost allocations may not be achievable.

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Properties of cost allocations The major problem in cooperative game theory is how to allocate the costs of the grand coalition among the players. The allocation vector satisfies a variety of properties, some of them are listed below. Coalitional game G = (N, ν) is P X efficient if i∈N φi (ν) = ν(N), this property ensures that the total value of the grand coalition is distributed among the players,

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Properties of cost allocations The major problem in cooperative game theory is how to allocate the costs of the grand coalition among the players. The allocation vector satisfies a variety of properties, some of them are listed below. Coalitional game G = (N, ν) is P X efficient if i∈N φi (ν) = ν(N), this property ensures that the total value of the grand coalition is distributed among the players, X individual rationality if ∀i∈N φi (ν) ≥ ν({i}), it guarantees that each player should at least get what the player would get individually,

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Properties of cost allocations The major problem in cooperative game theory is how to allocate the costs of the grand coalition among the players. The allocation vector satisfies a variety of properties, some of them are listed below. Coalitional game G = (N, ν) is P X efficient if i∈N φi (ν) = ν(N), this property ensures that the total value of the grand coalition is distributed among the players, X individual rationality if ∀i∈N φi (ν) ≥ ν({i}), it guarantees that each player should at least get what the player would get individually, X collective rationality if ∀i∈S φi (ν) ≥ ν(S), ∅ ⊂ S ⊂ N. If this property is not satisfied, then the players have an incentive to drop out of the grand coalition in order to gain a higher payoff allocation. The payoffs that secures efficiency and individual rationality are called the imputation. No partner would accept an allocation that has no imputation, therefore most of the solution concepts are a set of imputations. Tatarczak



Let us consider the example of a group of retailers who decide to cooperate by ordering jointly via a single large order. The mathematical model is developed based on the following assumptions the group of retailers can only make orders for full truckload delivery, the demand of each retailer is deterministic, and there is no shortage, the transportation cost is not relevant to the transportation quantity. In the optimal replenishment strategy under full truckload (FTL) shipments (Vi · Qi = CAPi ), the carriers have a similar cost structure, and FS =

X Di · Vi X = Fi , Capi i∈S

Qi Vi Di

∅ ⊂ S ⊆ N, i ∈ {1, 2, ..., n}.

i∈S

order size of i retailer volume of i retailer’ product demand of retailer i

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Capi Fi

vehicle capacity ordering frequency of retailer i



C (S) The total average cost of the coalition ∅ ⊂ S ⊆ N is the sum of the ordering cost, holding cost and transportation cost, as follows P X X Hi + βi Disti C (S) = A · Fi + Pi∈S i∈S Fi i∈S i∈S {z } | | {z } | {z } ordering cost

=

Disti βi Hi Fi A

holding cost

transportation cost

CO(S) + CH(S) + CT (S),

travel distance cost per kilometer holding cost ordering frequency of retailer i fixed ordering cost

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CO(S) CH(S) CT (S) C (S)

i ∈ {1, 2, ..., n}.

ordering cost of coalition S holding cost of coalition S travel cost of coalition S total cost of coalition S



Lemma The cost savings function C (S) is a profitable supperadditive function. Proof. The costP saving function is profitable if for every S ⊆ N we have C (S) ≤ i∈S C (i). C (S)

≥ = =

P

=

A

≤ =

P A i∈S Fi + P i∈S C (i).

i∈S

Fi +

P P H Pi∈S i + βi Disti Pi∈S FiHi Pi∈S i∈S βi Disti i∈S Fi +

C (S) + C (T ) = P P Hi Hi P P P P A( i∈S Fi + i∈T Fi ) + Pi∈S + Pi∈T + i∈S βi Disti + i∈T βi Disti P i∈S Fi P i∈T Fi Hi + i∈T Hi P P P P P A( i∈S Fi + i∈T Fi ) + Pi∈S + i∈S βi Disti + i∈T βi Disti F + F i∈S i i∈T i P P P i∈S∪T Hi P A i∈S∪T Fi + + i∈S∪T βi Disti i∈S∪T Fi C (S ∪ T ). Tatarczak



Corollary P Since C (S) ≤ i∈S C (i) then ν(S) ≥ 0. This implies that every coalition S is profitable.

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Corollary P Since C (S) ≤ i∈S C (i) then ν(S) ≥ 0. This implies that every coalition S is profitable. The total holding and transportation cost is reduced for each retailer i when they order jointly. Ordering Holding Transportation solution The OHT-solution φi (ν), that assigns to each retailer i in coalition S, ∅ ⊂ S ⊆ N, their cost savings allocation is defined by the formula φi (ν) = βi (1 − γ)Disti +

Hi Disti

Hi Hi , −P Fi j∈S Fj

holding cost travel distance

Fi βi

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i, j ∈ {1, 2, ..., n}, γ ∈ (0, 1).

ordering frequency of retailer i cost per kilometer



Theorem The OHT-solution φi (ν) lies in the core, φi (ν) ∈ Core(N, ν). Proof. For every φi (ν) = βi (1 − γ)Disti +

Hi Fi

−

P Hi

j∈N

Fj

and i ∈ S we have

Efficiency X

φi (ν)

=

P

=

P

i∈N (βi (1

− γ)Disti +

Hi Fi

−

P Hi

j∈S

Fj

)

i∈N

= =

i∈N (βi (1

P

− γ)Disti +

(1 − γ) i∈N βi Disti + ν(N), for all i ∈ N.

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Hi

Hi FN ) P Hi i∈N Hi − i∈N Fi FN

Fi P

−




Hi Fi

−

P Hi

j∈N

Fj

and i ∈ S we have

Individual rationality for all i ∈ N, FN ≥ Fi , 0 < γ < 1, we have φi (ν) = βi (1 − γ)Disti +

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Hi Hi − ≥ 0 = ν(i), Fi FN




Hi Fi

−

P Hi

j∈N

Fj

and i ∈ S we have

Collective rationality Hi P Hi ) i∈S (βi (1 − γ)Disti + Fi − j∈S Fj Hi Hi i∈S (βi (1 − γ)Disti + Fi − FN ) P P P Hi (1 − γ) i∈S βi Disti + i∈S HFii − Pi∈S FN P P Hi (1 − γ) i∈S βi Disti + i∈S HFii − i∈S = ν(S). FS

P

i∈S

= = ≥

φi (ν) =

P

P

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Summary There exists a lot of empirical evidence to prove the importance of cooperation among supply chain members. To successfully cooperations, the costs of the cooperations must be fairly divided between the coalition members. This paper examines a set of cost allocation methods which depends on the situation whereby there are multiple decision makers who obtain benefits by full truckload shipments cooperation. Motivated by this problem, we introduce new allocation rules that arise from the practical situation (OHT - solution). The results indicated that the methods proposed can lead to a fair cost allocation, since a method such as OHT-solution may establish an adequate incentive for collaboration. Further, the OHT-solution is a suitable allocation since it is designed especially for this kind of cooperation. Tatarczak



Further research may take a number of different directions. First of all, it is worth to discuss the applicability of the proposed approach to dynamic distribution systems. Second direction is to consider multiple suppliers. Another possible direction is to extend the model by considering multiple objectives, such as the supply chain risk. Finally, an assumption of the approach discussed in the paper is that all the trips are full-truck-load. The long-term goal is to elaborate the paper to consider less-than-truck load scenarios. However, this approach may increase the model’s computational complexity.

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Further research may take a number of different directions. First of all, it is worth to discuss the applicability of the proposed approach to dynamic distribution systems. Second direction is to consider multiple suppliers. Another possible direction is to extend the model by considering multiple objectives, such as the supply chain risk. Finally, an assumption of the approach discussed in the paper is that all the trips are full-truck-load. The long-term goal is to elaborate the paper to consider less-than-truck load scenarios. However, this approach may increase the model’s computational complexity. Thank you for your attention Anna Tatarczak [email protected]

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