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1.Introduction
In this modern world competition has increased a lot which ultimate has exerted a huge pressure on the organizations to come up each time with some kind of creativity in order to maintain their place in the market. Because of this tremendous pressure the life spans of almost all the things are getting shorter.

This shortage of the life span has further led to two problems which have relevance with the supply chain design in some sense. The first one is exploring several ways for the configuration of the new products life cycle. Whereas the second problem whereas the second problem is about the selection of the parts in the multi generation product. In this document we focused only on the first problem.

1.1 How to configure new products in supply chain
The supply chain network is the combination of both nodes and arcs where nodes represent functions and the arcs represent options. Suppose we have n functions then we have 2noptions which means 2 options per function.

Our main purpose here is to frame a decision support tool which can help the managers of the product to optimize the development of the process, in this process the design is fixed whereas the selection of the vendors for the raw material, selection of the assembly of those raw materials and the selection of the delivery options through which the product will be sent to the end customers is yet to be determined. For example: suppose we have 2 delivery options, one by road and the other by air both these options have different cost and the production time, if we are sending the product on a truck it will take 4-5 days and has a cost of 20$. On the other hand if we are sending it by the plane then it will take only 1 day but the cost is 50$.
The supply chain framework which we studied has consist of three specific costs namely:
Unit manufacturing Cost
Safety cost
Pipeline cost
And the purpose of the study is to minimize the sum of these cost while implementing a new supply chain.

2.Serial Line Formulation
This system consists of n stages connected in a serial line. The stage 1 would represent the raw material stage and the stage N is the finished good stage. Below is an example of a serial line having 4 stages.

In this example the stage 1 is having the raw material for house, stage 2 is about assembling those materials, stage 3 shows the distribution center from where it will be shipped to the final stage of customers.

2.1 Options
An option here means the selection of adequate part from the list of available choices depending upon their direct cost and the production lead time. It is basically a pair of {direct cost, production lead time}.
Production lead time is defined as the time taken by a stage to fulfill the demand of its immediate downstream stage. For example Suppose an order is placed to stage i by stage i+1 in specific time t and the stage i took t+3 time to fulfill that order this means the production time of stage i is 3. The production lead time includes the processing time and the waiting time and even includes the time take to transport the inventory if any.

The Direct cost includes both the cost of the material and the labor cost. For example considering stage 1 which is the raw material stage the direct cost is the combination of both the price paid to buy that product and the money given to the labors for unpacking the product.
The table shown below represents all the 4 stages shown in the diagram with their respective options. From this table it is quite clear that how the option scan be differentiated in terms of cost and time. Now here the main purpose is to design an algorithm which can help to select one option for each stage in such a way that the total cost of the supply chain should be decreased.

2.2 Periodic-review base-stock replenishment policy
This policy states that there is a common period in which all the stages will send their respective demands to their immediate upstream stages or fulfill the demands of their immediate downstream stages. So with this there will not be any ordering delay, which means the demand of the customer can be seen by all the stages at the same time.

2.3 External demand
External demand Itself represents that the demand which is outside the chain which means it only occurs at node N. The demand at node N in time (t) is represented asd?N t and the per time period average demand is ?N.2.3 Internal demand
Internal demand is the demand within the supply chain, which is the demand given by immediate downstream stages to its immediate upstream stages.

2.4 Guaranteed service times
This is the time given by the upstream stages by which they can fulfill the demands of either their downstream stages or the end customers. The guaranteed service time of stage N is SN.

2.5 Single-Stage Single-Option Model
This is the basic model and also the building block of the upcoming model in the document namely multi-stage multi- Option model. This model says that there is only one option per stage.

2.5.1Inventory model
Suppose Ii(t)represents the finished inventory at the stage i in an particular interval t.So taking into account the concept of base-stock policy and the service time the finished inventory can be expressed as:
Ii (t)=Bi-di(t-Si-1-Ti ,t-Si)2.5.2 Determination of base stock
In order to provide 100% services to the customer each stage should have the inventory level Ii(t)?0.So the base stock for stage i can be calculated as:
Bi?di(t-Si-1-Ti ,t-Si), For having the least base stock inventory we can have the bounded demand, now the base stock is:
Bi=Di(?),?=max{0,Si-1+Ti -Si}2.5.3 Safety stock model
With the help of the bounded base stock and the finished inventory level, we can calculate the expected safety stock as:
EIi=Bi-Edi(t-Si-1-Ti ,t-Si)
=Di(Si-1+Ti -Si)-(Si-1+Ti -Si)?i2.5.4 Pipeline Inventory
It is the inventory which is under process, so it can be calculated as
Wi(t)=di(t-Si-1-Ti,t-Si-1)Like safety stock, pipeline inventory is also calculated by bounded demand:EWi=Ti?i2.5.5 Safety stock cost calculation
For the calculation of safety stock cost of any stage we need the holding cost for that particular stage, suppose the holding cost rate for stage i is?then the total holding cost at stage i is ?Ci. Therefore, the safety stock cost is:?CiEIi2.5.6 Pipeline stock cost calculation
For the calculation of pipeline cost, we take into consideration the pipeline cost of its upstream stage and the average of the pipeline cost of that stage with the expected work in process.

(Ci-1+Ci/2)EWi2.6 Multi-Stage Multi-Option Serial Supply Chain Model
Multiple stage multiple option mainly builds based on single stage single option in order to model the pipeline stock levels and safety stock levels diagonally the serial supply chain. Due to keeping multiple options in supply chain it produces some additional complexity to the formulation. Their will be only one option will be selected in each single stage. In order to do this, we use 0-1 indicator variable yij for i = 1, 2, …, N and 1 ? j ? Oi. yij equals 1 if option j is selected for stage i and otherwise it equals 0 therefore yij = 1 implies.

Oij is selected. Stage i formulate is as follows

2.6.1Purpose of Multi-Stage Multi-Option Objective Function
In this they are mainly three relevant costs: manufacturing cost, safety stock cost, and pipeline stock cost. The only purpose of safety stock and pipeline stock costs is to hold the addition of multiple options at a stage. Depending upon the option chosen these three costs will be influenced. For better understanding, supply chain contains both lower direct cost and high safety stock cost in lower cost, goods are sold in long lead times but with a high safety stock cost. Selected options are taken by calculating Ci

for i = 1, 2, …, N where C0 = 0 with this we can control the supply chain’s cost.

Safety stock cost
Last stage in supply chain is safety stock during its processing activity has occurred. In stage i the product value of a unit of safety stock is equal to the cumulative cost of that stage
Expected safety stock cost at stage i

Here, ? represents the holding cost rate
Pipeline stock cost
By multiply the expected pipeline stock with the average cost of the product at the stage i, it is easy to control the cost of the pipeline stock for that stage.

Two equivalent cost calculations are as follows

Dynamic Programming Solution Procedure
Throw dynamic programming the problem of serial line case can be solved to optimality
2.6.2 State space determination
To solve network in a node-by-node process, define a state space that allows the algorithm which helps to solve and, in this process,, it will use only the information that is locally available at the node. Several key parameters are uniquely controlled by the options due to this single-option problem only requires one state variable where on other side multi-option serial supply chain problem use two state variables to modeled at that stage. Outbound service time at stage is represented by one state variable.

2.6.3 Forward recursive formulation
In starting Dynamic program is a forward recursion at stage 1 and proceeding to stage N. For each stage, the dynamic program estimates a functional equation represented by fi(C, S). fi(C, S) is defined as the minimum supply chain cost for node 1 to i given that stage i’s quotes a service time of S and cumulative cost is C.
The supply chain cost at stage i as a function of the service time quoted to stage i (SI), plus stage i’s service time (S), cumulative cost (C) and option (Oij) selected

In this stage to define the minimum supply chain cost for stages 1 through i given that stage i utilizes option Oij. Let fij(C, S) denote this option-specific optimal cost-to-go function as follows

3 Assembly Network Formulation
3.1 Network Representation
A supply chain network can be displayed as a assembly network in which one stage gets the inputs from many other stages (i.e. from other suppliers) but it can supply to only one of it’s downstream stage. In terms of network, an assembly network is nothing but a graph where each node can have multiple arriving arcs but only one leaving arc. Here, it’s considered as the nodes are arranged in topological manner. For every arc (i,j) belongs to A, i ; j. It means that It means that the finished node the Nth node.

Ni can be defined as the subset of the nodes {1, 2, … i} for each node i. Ni can be determined as using the following equation

Here, B(i) denotes the set of adjacent backward stages to stage i; B(i)={h:(h,i)?A}.
An example for the assembly network supply chain is as follows:

This figure shows the assembly network representation at the circuit board which has the two incoming components (controller and the motherboard) and a single outgoing component that is to the platform base. From the table of options here it has two sourcing options those are low direct cost, high production lead time and high direct cost and low production lead time for the supplier. Ni can be taken as {3} for i=3 and {1, 2, 4} for i=4.

3.2 Internal demand
The demand at each internal stage i is denoted as di(t) and it is given as:
di (t) = ?ijd j (t)
At each internal stage demand is constant and bounded. The average rate of demand for component I is given as:

?ij represents the demand of stage i required to produce one unit of stage j’s product.

3.3 Solution Procedure
3.3.1 Dynamic programming formulation
In assembly network formulation, cumulative cost and service time are the state variables.

3.3.2 Forward recursive formulation
Forward recursion is beginning at stage1 and proceeding till stage N. fi(C, S) is a functional equation and defined as minimum supply chain cost.
Supply chain cost gij is defined as the function of maximum service time quoted (SI) plus option selected (Oij) plus cumulative cost (C) and service time (s).

Option specific optimal cost function is given as below:
Cumulative cost at stage I is C and C – cij is the subgraph’s cumulative cost.

The maximum replenishment time at stage i,
4.Distribution Network Formulation
4.1Network Representation
Distributed network in supply chain can be the one which has many customers and only one supplier at each stage. In terms of network, distribution network is a graph where each node has only one entering arc and multiple leaving arcs (i.e. customers). All the nodes are arranged in topological manner That is, for every arc (i,j) ??A, i ; j. Here for the purpose of construction raw material node is represented as node 1. D(i) denotes the set of forward adjacent stages to stage i; D(i)={k:(i,k)?A}. Ni is used to explain the dynamic programming recursion.
An example for the distributed supply chain network is as follows:

The figure represents the distributed supply chain network system. The manufactured product serves both export and US distribution markets. For US distribution there are two types of customers (Class A and Class B). All these stages have two sourcing options containing premium and basic transportation vendors. Ni is {3} for i=3 and {2, 4 ,5} for i=2.

4.2 Demand assumption
4.2.1 External demand
External demand is the demand at the stage N and it is same as the demand in serial line at each stage. It is also assumed that each outer node is bounded.

4.2.2 Internal demand
Internal demand is the demand at each stage between stage 1 to stage N and each stage orders based on the base stock replenishment policy, the demand at each stage is given as:

For each component i the average demand rate is given as follows:

4.3 Service times
4.3.1 Internal service times
For each down stream stage guaranteed service time is represented as Sij at internal stage i for j, (i, j) ??A. Here it is assumed that for each end customer maximum amount of service time is given as the input.

4.4 Solution Procedure
4.4.1 Recursive formulation
In distinction to the other two networks, distribution network starts proceeding from the end nodes and works backward to the node which doesn’t have any incoming nodes. Fi(CI,SI) is a function used to define the minimum supply chain cost. CI denotes the entering cumulative cost and SI denotes the service time at each stage i.
Here the supply chain cost is a function of maximum service time quoted plus service time plus cumulative cost plus option selected at stage i.

Option specific optimal cost is defined as follows:

There are two conditions for the above equation. First one is for the internal stage i the service time s is positive and it must not exceed the sum of service time(SI) and option’s production time (Tij). Second one is the incoming cumulative cost (CI) have to be a feasible entering cumulative cost at stage i. That is, CI ??XIi.

Now the functional equation is represented as
below:
The maximum replenishment time at each internal stage i is given as:

5.Spanning Tree Network
5.1 Network Representation
In network they are N nodes and N-1 arcs with connecting each other in spanning tree. Spanning trees had two special cases one is distribution network and another is Assembly networks. In supply chain real world numerous are flexibility to capture in this spanning tree network.

As they are N nodes and N-1 arcs, for each node i is define Ni as the subset of nodes {1, 2, … i} that are connected to i on the sub-graph consisting of nodes {1, 2, … i}. Ni is used to describe the dynamic programming recursion. Ni is also determined by the following equation.

Here B(i) denote the set of backwards adjacent nodes and let D(i) denote the set of forward adjacent nodes.
This is the simple example of a typical supply chain

Ni is {1, 2, 8} for i=8 and {4, 6, 7, 8, 10} for i=10
Typical supply chain is the combination of two variants. These two variants share a common circuit board and are differentiated by their other assemblies in the network. Computer superstores and retail wholesalers combinedly share the standard product and specialty retailers can only uniquely sold the premium product.

5.2 Solution Procedure
By two-state dynamic program spanning tree networks are been solved. Nodes place a key role in spanning tree where two forms of the functional equation are depending on them in the network. As spanning tree contain two forms in one form fi(C, S), distinct as the minimum supply chain cost for the subgraph Ni given that stage i has a cumulative cost C and quotes a service time of S and coming to second form in the network is Fi(CI, SI), definite as the minimum supply chain cost for the subgraph Ni given that stage i’s arriving cumulative cost is CI and stage i is quoted a service time of SI. Correspondingly, these two forms are generalizations of the functional equation for the assembly networks and distribution networks.

Functional equation development
In supply chain, cost for the subgraph are rooted at stage i as a function of stage i’s incoming service time (SI), outgoing cumulative cost (C), outgoing service time (S), and option selected (Oij)
The supply chain cost at stage i and nodes in Ni that are upstream of stage i with the minimum cost of supply chain for the configuration upstream of stage i, as a function of the configuration’s cumulative cost and service time is represented as follows
6 Conclusion
7 References
http://web.mit.edu/~sgraves/www/MSOM%202000%20Graves%20and%20Willems%2005_19_2000.pdfhttps://www.researchgate.net/publication/227446187_Optimal_Investment_in_Product-Flexible_Manufacturing_Capacity.

https://link.springer.com/article/10.1023%2FA%3A1008142823872https://www.scribd.com/document/256445633/Willems-Th-1-25-99https://dspace.mit.edu/bitstream/handle/1721.1/9682/42587929-MIT.pdf?sequence=2