CS代写 MBA 8419 - Decision Making Technology

Over. Def. App.
Distribution and Network Models
Master of Business Administration
MBA 8419 – Decision Making Technology

1 Distribution and Network Models

Over. Def. App. Overview of the presentation
Definitions Network
General Optimization Model
Applications
Multi-period planning
General principles
Production planning basic case
Production planning with general deliveries
Logistics and transportation Transportation problem
Distribution and Network Models

Over. Def. App. Net. Flo. Gen. Opt. Mod.
Definitions Network
Defined using a graph, which is a structure defined as a set of nodes for which some pairs of the nodes are connected via arcs.
Arc (i, j) : where i = initial node and j = terminal node
(i,j) ⇒ emerges (leaves) node i and is incident to (arrives) at node j
Arc : defines a specific relationship between two nodes Examples
route linking intersection i to j assignment of employee i to task j renting a vehicle i to a client j
Distribution and Network Models

Over. Def. App.
Net. Flo. Gen. Opt. Mod.
Definitions Network
FIGURE – Visualization of social network analysis
4 Distribution and Network Models

Over. Def. App. Net. Flo. Gen. Opt. Mod.
Definitions Flow
In a network, flow refers to the units (e.g., goods, materials, people, etc.) that move on the arcs following their specific direc- tion.
Associated with each arc (i,j)
xij = the number of units (i.e., quantity of flow) that move
along the arc (i, j)
cij = unit cost for the flow moving along (i,j) Bounds on the quantity of flow associated with xij
uij = maximum quantity of flow that can move along arc (i,j) lij = minimum quantity of flow that can move along arc (i,j)
lij ≤ xij ≤ uij
Distribution and Network Models

Over. Def. App. Net. Flo. Gen. Opt. Mod.
Definitions Flow
Flow (cont’d) :
Associated with each node i
b(i) = demand value associated with i
There are three possible cases with respect to the demand values :
b(i) > 0 ⇒ node i defines an origin for the flow (i.e., flow enters the network at this node).
b(i) < 0 ⇒ node i defines a destination for the flow (i.e., flow leaves the network at this node). b(i) = 0 ⇒ node i is a transhipment node (i.e., flow simply transits at this node and remains within the network). If the values b(i), for all nodes i, are integer, then solution to the network flow problem will also be integer (i.e., without the need to impose the integrality requirements). 6 Distribution and Network Models Over. Def. App. Net. Flo. Gen. Opt. Mod. Definitions General Optimization Model Decision variables xij = number of units of flow that transit on arc (i,j) Objective Function Total cost incureed to distribute the flow through the network 􏰂 cijxij for all arcs in the network (i,j) Subject to Flow conservation constraints at all nodes For each node i → total flow on the arcs leaving i - total flow on the arcs arriving at i = b(i) Bounds on the flow transiting through each arc Foreacharc(i,j)→lij ≤xij ≤uij Distribution and Network Models Over. Def. App. Net. Flo. Gen. Opt. Mod. Definitions General Optimization Model General Example : (a) Costs, bounds, b(i) (b) Network FIGURE – Network flow problem 8 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Multi-period planning General Principles Single period planning Consists of decisional problems that occur for a single moment in a time horizon and that only considers the ressources available (supply) and state of the market (demand) for that particular mo- ment. Multi-period planning Consists of decisional problems that occur over multiple moments in a time horizon and that explicitly take into account the dynamic by which available ressources and market conditions can evolve (i.e., change) through time. 9 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Multi-period planning General Principles (cont’d) FIGURE – Single period planning process 10 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Multi-period planning General Principles (cont’d) FIGURE – Multiple periods planning process 11 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Multi-period planning Production planning basic case : Pastissimo inc. Pastissimo is an Italian company that specializes in the production of high- quality pasta for a variety of clients. The company has recently received an important order from one of its client, Hyper-Halli. Following this order, for the next 6 months, Pastissimo will deliver (in 1kg bag units) the spaghetti that is sold by Hyper-Halli as its own brand. Therefore, at the end of each month, Pas- tissimo will deliver 4 tons of spaghetti to Hyper-Halli, which has agreed to pay 5.28$ per bag for these deliveries. The production of spaghetti requires the use of wheat. To ensure that enough wheat will be available, Pastissimo has nego- tiated a contract with a local producer. The details of the contract are provided in the following table : Month Price in $ tà Minimum in tà Maximum in tà ààà 4 6 2 9à5 3 4 3 ààà 5 à 4 98à 2 3 5 à2à 4 à 6 à25 5 6 FIGURE – Contract with wheat producer 12 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Multi-period planning Production planning basic case : Pastissimo inc. (cont’d) To store either the wheat that is bought from the producer or the spaghetti that is produced, Pastissimo has both a silo, where wheat can be stockpiled, and a warehouse, where the final products can be kept. At the beginning of month 1, the silo already has 2 tons of wheat and the company would like to keep the same amount at the end of the 6th month. The silo can store up to 3 tons of wheat and the monthly storing cost is 20$/t. As for the store, its capacity is 1 ton of spaghetti and the monthly storing cost is 25$/t. To ensure that Pastissimo delivers the required amounts of spaghetti to Hyper-Halli for the next 6 months, the manager planned the production capacity and costs as follows : Month Production Capacity in tà Production Costs in $ tà 6 6à 4 4 6à 5 4 à5 6 3 65 FIGURE – Production capacity and costs 13 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Multi-period planning Production planning basic case : Pastissimo inc. (cont’d) Question Pastissimo is interested in planning its operations to perform the order to Hyper-Halli for the next 6 months. Therefore, The company is looking to determine the following : Supply of wheat Inventory (wheat and spaghetti) Production levels Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Multi-period planning Production planning basic case : Pastissimo inc. (cont’d) Network flow model Identify the Beginning and End of each month Bi, i = 1,...,6 Ei, i = 1,...,6 Supply of wheat : • → Bi Production : Bi → Ei Storing wheat : Bi−1 → Bi Storing spaghettis : Ei−1 → Ei Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Multi-period planning Arcs i, j Costs cij Bound lij Bound uij •,B ààà 4 6 •,B2 9à5 3 4 •,B3 ààà 5 à •,B4 98à 2 3 •,B5 à2à 4 à •,B6 à25 5 6 B,E 6à à 6 B2,E2 5à à 5 B3,E3 5à à 4 B4,E4 6à à 4 B5,E5 à5 à 4 B6,E6 65 à 3 B,B2 2à à 3 B2,B3 2à à 3 B3,B4 2à à 3 B4,B5 2à à 3 B5,B6 2à à 3 E,E2 25 à E2,E3 25 à E3,E4 25 à E4,E5 25 à E5,E6 25 à b B b B2 b B3 b B4 b B5 b B6 2 à à à à -2 b E b E2 b E3 b E4 b E5 b E6 -4 -4 -4 -4 -4 -4 (a) Costs, bounds, b(i) FIGURE – Network flow for Pastissimo (b) Network 16 Distribution and Network Models Applications Multi-period planning Over. Def. App. M.-P. Pla. Log. Tra. Decision variables X•Bi = number of t. of wheat that are bought at the beginning of month i, where i = 1,2,...,6 XBi Bi +1 = number of t. of wheat that is stored in the silo from the beginning of month i to the beginning of month i + 1, where i = 1,2,...,5 XBi Ei = number of t. of spaghetti produced during the month i , where i = 1,2,...,6 X +1 = number of t. of spaghetti that are stored in the warehouse from the end of month i to the end of month i + 1, where i = 1, 2, . . . , 5 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Multi-period planning Objective function min1000X•B1 +975X•B2 +...+20(XB1B2 +XB2B3 +...+XB5B6)+ 160XB1E1 +150XB2E2 +...+25(XE1E2 +XE2E3 +...+XE5E6) Subject to Flow conservation constraints at all nodes For example NodeB4 →XB4B5 +XB4E4 −X•B4 −XB3B4 =0 NodeE2 →XE2E3 −XB2E2 −XE1E2 =−4 Bounds on the arcs For example 4≤X•D1 ≤6 0≤XD5F5 ≤4 0≤XF2F3 ≤1 etc. Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Multi-period planning Production planning with general deliveries Assuming the Pastissimo renegotiates its distribution contract with Hyper-Halli, which now accepts advanced deliveries, or, late deliveries. Specifically, the new contract allows the following delivery options : Late deliveries by one month can be accepted by Hyper-Halli, provided that Pastissimo pays a fee of 35$/t. for all spaghetti that is delivered late. Advanced deliveries by one or two moths can be accepted by Hyper-Halli, provided that Pastissimo pays a fee of either 14$/t. or 17$/t. for all spaghetti that is delivered in advance by one and two months, respectively. In addition, each time a bag of spaghetti is delivered to Hyper-Halli, Pastissimo pays a cost of 0.05$/kg. in transportation fees. Question : How can the previous network flow model be adjusted to represent this new situation ? 19 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Multi-period planning Arcs i, j Costs cij Bound lij E,D 5à à E,D2 64 à E,D3 6à à E2,D 85 à E2,D2 5à à E2,D3 64 à E2,D4 6à à E3,D2 85 à E3,D3 5à à E3,D4 64 à E3,D5 6à à E4,D3 85 à E4,D4 5à à E4,D5 64 à E4,D6 6à à E5,D4 85 à E5,D5 5à à E5,D6 64 à E6,D5 85 à E6,D6 5à à Bound uij à b B b B2 b B3 b B4 2 à à à à -2 b E b E2 b E3 b E4 b E5 b E6 àààààà b D b D2 b D3 b D4 b D5 b D6 -4 -4 -4 -4 -4 -4 (a) Costs, bounds, b(i) FIGURE – Network flow for Pastissimo with generalized deliveries (b) Network 20 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Logistics and transportation General Context Transportation is at the heart of logistics operations and one of the major drivers of economic activities. People and goods need to be efficiently moved throughout the world in order for societies and economies to function and thrive. Transportation Problem Base case : Adopting the point of view of either the shipper or the receiver Specific detailed routes are not considered Service from origin-destination and the overall cost is important Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Logistics and transportation Transportation Problem (cont’d) Consider the problem of a company who needs to supply its ware- houses with finished products that are then distributed to clients. The products are produced at a series of plants and, at the end of each month, they are transported towards the different ware- houses of the company. For the next month, the company needs to perform the following operations : Chicago Kansas City Houston 120 u. 80 u. 80 u. Warehouses Atlanta Los Angeles 150 u. 60 u. 70 u. 22 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Logistics and transportation Transportation Problem (cont’d) The cost of shipping between cities is usually a function of, given a set of possible services, min distances between the cities × a tarif per unit. Assuming that the following unit costs (i.e., $/unit) apply : Chicago Kansas City Houston Atlanta Los Angeles 865 15 12 10 3 10 9 Question : How can this problem be formulated as a network flow problem ? 23 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Logistics and transportation FIGURE – Illustration of the transportation problem 24 Distribution and Network Models Applications Logistics and transportation Over. Def. App. M.-P. Pla. Log. Tra. Defining the network 2 types of nodes (Plants and Warehouses) : Pj,wherej =1→Chicago,j =2→KansasCity,j =1→ Houston Wi,wherei =1→NewYork,i =2→Atlanta,i =3→Los Angeles Arcs represent the transportation of units Plants → Warehouses Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Logistics and transportation (a) Transportation problem network FIGURE – Network flow - Transportation problem minz = 8xPW + 6xPW2 + 5xPW3 + 5xP2W + 2xP2W2 + àxP2W3 + 3xP3W + àxP3W2 + 9xP3W3 xPW + xPW2 + xPW3 = 2à xP2W + xP2W2 + xP2W3 = 8à xP3W + xP3W2 + xP3W3 = 8à xPW xP2W xP3W = 5à xPW2 xP2W2 xP3W2 = 6à xPW3 xP2W3 xP3W3 = àà (b) Optimization model 26 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Logistics and transportation Assignment problem The assignment problem is a special case of the transportation problem. Definition The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment. It is required to perform all tasks by assigning exactly one agent to each task and exactly one task to each agent in such a way that the total cost of the assignment is minimized. 27 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Logistics and transportation Assignment problem (cont’d) Assuming that in the previous context, the company was looking to assign a single production plant to a single warehouse to perform the necessary supply activities by simply considering the unit transportation costs. minz = 8xPW + 6xPW2 + 5xPW3 + 5xP2W + 2xP2W2 + àxP2W3 + 3xP3W + àxP3W2 + 9xP3W3 xPW + xPW2 + xPW3 = xP2W + xP2W2 + xP2W3 = xP3W + xP3W2 + xP3W3 = xP W xP 2W xP 3W = xP W 2 xP 2W 2 xP 3W 2 = xP W 3 xP 2W 3 xP 3W 3 = (a) Assignment problem FIGURE – Network flow - Assignment problem (b) Optimization model 28 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Logistics and transportation Assignment problem (cont’d) Intercity truck transportation Distances (km) : Loads 1234567 1 Scranton 2 Honesdale 3 Y NY Dover Paterson 229 229 139 176 212 212 114 155 111 111 32 54 Newton 146 116 125 153 123 91 108 81 25 4Edison 62 62 69 68 5 Princeton 6 Warwick 7 Newark Question : 92 92 84 95 116 116 62 69 54 54 43 26 88 89 111 44 101 76 How should the company proceed to solve this transportation problem ? 29 Distribution and Network Models Over. Def. App. M.-P. Pla. Log. Tra. Applications Logistics and transportation FIGURE – Network - Intercity truck transportation problem 30 Distribution and Network Models 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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