Network Modeling Location Problems Summary Module 8 – From One-Echelon to Multiple-Echelon Dr Yong Wu 7108IBA Supply Chain Modeling Network Modeling Location Problems Summary Echelon What is an echelon? The concept of echelon is a fundamental concept in multi-stage systems: The echelon of stage i comprises stage i itself and all downstream stages. Network Modeling Location Problems Summary Network Optimization Modeling Network optimization models have a natural graphical network representation. It is a core discipline in logistics and supply chain management. Many specialised solution techniques have been developed specifically for network models. Network Modeling Location Problems Summary A Network and Its Nodes and Arcs Network Modeling Location Problems Summary Nodes Nodes represent facilities in the network. In our examples discussed so far, a node can be a hotel, a power plant, a city, etc. Decisions around the nodes generally include whether a node (facility) should open or not. Attributes associated with nodes generally include: Fixed cost, such as the set up cost Variable cost, such as variable cost per unit production Capacity, such as how many units could be produced per day/week/month/year Network Modeling Location Problems Summary Arcs Arcs connect nodes They represent activities or processes, such as a transportation activity which moves product from node A to node B. Attributes associated with arcs generally include: Cost per unit of flow, such as $0.05 per product per kilometer Lower and upper flow limits, such as it needs to be at least 100 units (lower) before it’s viable, and the capacity you can handle is at most 10,000 units (upper). An arc can be directional, i.e., you might only be able to move from one node to the other, but cannot reverse the flow. Network Modeling Location Problems Summary Network Optimization Model Helps Identify service improvements Manage capital Reduce costs / increase profit Balance service, capital and costs Enhance shareholder value Network Modeling Location Problems Summary Network Optimization Example Network Modeling Location Problems Summary Network Optimization Outcome Network Modeling Location Problems Summary Graphical Representation Network Modeling Location Problems Summary Outflows and Inflows An arrow pointed out of a node is called an outflow. An arc pointed into a node is called an inflow. General networks can have both inflows and outflows for any given node. Typically in Excel, network models have one changing cell per arc. For each node in the network, there is a flow balance constraint. Network Modeling Location Problems Summary In Reality The objective of many real-world network models is to ship goods from one set of locations to another set of locations at minimum cost, subject to various constraints. There are many variations of these models. The general logistics problem is similar to the transportation problem except for two possible differences. First, arc capacities are often imposed on some or all of the arcs. These become simple upper bound constraints in the model. Second and more significantly, inflows and outflows can be associated with any node. Nodes are generally categorized as origins, destinations, and transshipment (transfer) points. Network Modeling Location Problems Summary Origin, Transfer Point, and Destination Network Modeling Location Problems Summary Net Inflow and Net Outflow The best way to think of these categories is in terms of net inflow and net outflow. The net inflow for any node is defined as total inflow minus total outflow for that node. The net outflow is the negative of this, i.e., total outflow minus total inflow. Therefore, we can see that: An origin node should have a (positive) net outflow, which should be capped by its supply capacity. Similarly, for a destination node, we need an inflow which should equal to or exceed the demand. For transshipment points, the net inflow (or the net outflow) should be zero, i.e., the sum of inflows should equal to the sum of outflows. Network Modeling Location Problems Summary Continuous Location Models Objective: Find a single location that minimizes the total transportation cost from this location to all destination points. Assumptions All points within Euclidean space; Center location can be anywhere on the space; Transportation cost is the distance multiplied by volume (demand). Network Modeling Location Problems Summary Problem Characteristics Physically this is a relatively simple problem—It may be a problem to find the center of gravity. Mathematically this is a rather challenging problem as it is a nonlinear optimization problem. Let’s assume we have n customers located at different places. The coordinates of these customers and their demand information are known. We want to work out one location (e.g., a warehouse) which minimizes the total shipment-distance (e.g., ton-kilometer) traveled. Network Modeling Location Problems Summary The Formulation If we use di to represent the distance between the location and customer i, and Di for the demand of customer i. Then we want to minimize : min X i diDi Using (xi,yi) to represent the location of customer i, and xw,yw to represent the location of the warehouse (the decision variables), we have: min X i p(xw −xi)2 + (yw −yi)2Di Network Modeling Location Problems Summary Warehouse Location Optimization Network Modeling Location Problems Summary Problems with the Continuous Location Model The distances might not be Euclidean distances. Not every location is feasible. Some locations might be more desirable than others. Therefore, most facility location problem selects optimal locations from a candidate list. Network Modeling Location Problems Summary Network Facility Location Problem Objective: Find the location(s) from a list of candidates that minimizes the total cost of delivering from this point to n destinations over a given network. minX i X j cijxij +X i Fioi Subject to: X j xij 6 Si ∀i ∈ S X i xij > Dj ∀j ∈ D xij 6 Moi ∀i,j X i oi > Nmin X i oi 6 Nmax Network Modeling Location Problems Summary Notation i : Distribution centers j : Customers Si : Available supply at DC i Dj : Demand by customer j cij : Cost to serve customer j from DC i Fi : Fixed cost for opening DC i Nmin : Minimum number of DCs to open Nmax : Maximum number of DCs to open M : The big M, a sufficiently large number xij : Flow on arc from DC i to customer j oi : Binary variable for DC i, = 1 is open Network Modeling Location Problems Summary Selecting the Right Locations Network Modeling Location Problems Summary Trade-Offs for Multiple Locations Supply chain management is about finding balance between: Costs Inventory costs Facility costs Transportation costs Operating costs Asset costs Service Customer satisfaction Sales growth Increase market share Response times Level of service Network Modeling Location Problems Summary Cost Impacts Network Modeling Location Problems Summary Level of Service Impacts Tactical inventory replenishment policies Cycle service level Item fill rate Cost of stock out event Cost of item short These are not incorporated in the strategic network design! Distance to customer can be used as a proxy for level of service. Network Modeling Location Problems Summary Distance to Customers How to measure? Maximum distance can be set so no customer is more than certain kilometers away from a DC Percentage of all customers need to be within certain kilometers of a DC Average distance for customers to DC must be less than certain kilometers Weighted average distance for customer to DC must be less than certain kilometers How to model? Report distance results from each run Add constraints to enforce Change the objective function Network Modeling Location Problems Summary Building Scenarios We can build scenarios to check the performance of each scenario. For example, we can choose to set up one, two, three, four, or five DCs. Increasing number of DCs reduces distance to customers and we can assume it increases level of service. However, increasing number of DCs doesn’t mean the cost will increase in the same direction. The fixed costs increase with each new DC opened. The outbound transportation decreases with each new DC opened. Network Modeling Location Problems Summary Selecting the Right Locations Network Modeling Location Problems Summary Costs versus Average Demand Distance Network Modeling Location Problems Summary Level of Service as Constraints Level of service requirements can be added into the model as constraints. Usually these constraints can drive the solution. We can also estimate the cost of meeting a specific level of service. Note the level of service can be specified in many different ways with different metrics. Network Modeling Location Problems Summary Take Away Points Network optimization can be a challenging task. Proper understanding of the concepts related to network optimization may help a lot. Modeling in Excel and observe the changes can lead to some insightful observations. Optimizing a network does not explicitly consider the level of service, but we can build these requirements into the model.