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## Operations Research Modeling Toolset

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**Operations Research Modeling Toolset**Queueing Theory Markov Chains PERT/ CPM Network Programming Dynamic Programming Simulation Markov Decision Processes Inventory Theory Linear Programming Stochastic Programming Forecasting Integer Programming Decision Analysis Nonlinear Programming Game Theory**Network Problems**• Linear programming has a wide variety of applications • Network problems • Special types of linear programs • Particular structure involving networks • Ultimately, a network problem can be represented as a linear programming model • However the resulting A matrix is very sparse, and involves only zeroes and ones • This structure of the A matrix led to the development of specialized algorithms to solve network problems**Types of Network Problems**• Shortest Path Special case: Project Management with PERT/CPM • Minimum Spanning Tree • Maximum Flow/Minimum Cut • Minimum Cost Flow Special case: Transportation and Assignment Problems • Set Covering/Partitioning • Traveling Salesperson • Facility Location and many more**The Transportation Problem**• The problem of finding the minimum-cost distribution of a given commodity from a group of supply centers (sources) i=1,…,mto a group of receiving centers (destinations) j=1,…,n • Each source has a certain supply (si) • Each destination has a certain demand (dj) • The cost of shipping from a source to a destination is directly proportional to the number of units shipped**Simple Network Representation**Sources Destinations 1 Supply s1 1 Demand d1 Supply s2 2 2 Demand d2 … … xij n Demand dn Supply sm m Costs cij**Example: P&T Co.**• Produces canned peas at three canneries Bellingham, WA, Eugene, OR, and Albert Lea, MN • Ships by truck to four warehouses Sacramento, CA, Salt Lake City, UT, Rapid City, SD, and Albuquerque, NM • Estimates of shipping costs, production capacities and demands for the upcoming season is given • The management needs to make a plan on the least costly shipments to meet demand**Example: P&T Co. Map**1 2 3 3 2 1 4**Example: P&T Co. Data**Shipping cost per truckload**Example: P&T Co.**• Network representation**Example: P&T Co.**• Linear programming formulation Let xij denote… Minimize subject to**Feasible Solutions**• A transportation problem will have feasible solutions if and only if • How to deal with cases when the equation doesn’t hold?**Integer Solutions Property: Unimodularity**• Unimodularity relates to the properties of the A matrix(determinants of the submatrices, beyond scope) • Transportation problems are unimodular, so we get the integers solutions property: For transportation problems, when every si and dj have an integer value, every BFS is integer valued. • Most network problems also have this property.**Transportation Simplex Method**• Since any transportation problem can be formulated as an LP, we can use the simplex method to find an optimal solution • Because of the special structure of a transportation LP, the iterations of the simplex method have a very special form • The transportation simplex method is nothing but the original simplex method, but it streamlines the iterations given this special form**Transportation Simplex Method**Initialization (Find initial CPF solution) Is the current CPF solution optimal? Yes Stop No Move to a better adjacent CPF solution**Prototype Problem**• Holiday shipments of iPods to distribution centers • Production at 3 facilities, • A, supply 200k • B, supply 350k • C, supply 150k • Distribute to 4 centers, • N, demand 100k • S, demand 140k • E, demand 300k • W, demand 250k • Total demand vs. total supply**Finding an Initial BFS**• The transportation simplex starts with an initial basic feasible solution (as does regular simplex) • There are alternative ways to find an initial BFS, most common are • The Northwest corner rule • Vogel’s method • Russell’s method (beyond scope)**The Northwest Corner Rule**• Begin by selecting x11, let x11 = min{ s1, d1 } • Thereafter, if xij was the last basic variable selected, • Select xi(j+1) if source i has any supply left • Otherwise, select x(i+1)j**The Northwest Corner Rule**Z = 10770**Vogel’s Method**• For each row and column, calculate its difference: = (Second smallest cij in row/col) - (Smallest cij in row/col) • For the row/col with the largest difference, select entry with minimum cij as basic • Eliminate any row/col with no supply/demand left from further steps • Repeat until BFS found**Vogel’s Method (3): update supply, demand and differences**90**Vogel’s Method (5): update supply, demand and differences**100 90**Vogel’s Method (7): update supply, demand and differences**100 50 90**Vogel’s Method (8): select xAS as basic variable**140 100 50 90**Vogel’s Method (9): update supply, demand and differences**140 100 50 90**Vogel’s Method (10): select xAW as basic variable**140 60 100 50 90**Vogel’s Method (11): update supply, demand and differences**140 60 100 50 90**Vogel’s Method (12): select xBW and xBE as basic variables**140 60 210 140 100 50 90 Z = 10330**Optimality Test**• In the regular simplex method, we needed to check the row-0 coefficients of each nonbasic variable to check optimality and we have an optimal solution if all are 0 • There is an efficient way to find these row-0 coefficients for a given BFS to a transportation problem: • Given the basic variables, calculate values of dual variables • ui associated with each source • vj associated with each destination using cij – ui – vj = 0 for xij basic, or ui + vj = cij (let ui = 0 for row i with the largest number of basic variables) • Row-0 coefficients can be found from c’ij=cij-ui-vj for xij nonbasic**Optimality Test (1)**60 140 210 140 100 50 90**Optimality Test (2)**• Calculate ui, vj using cij – ui – vj = 0 for xij basic (let ui = 0 for row i with the largest number of basic variables) 60 140 210 140 100 50 90**Optimality Test (3)**• Calculate c’ij=cij-ui-vj for xij nonbasic 60 140 140 210 50 100 90**Optimal Solution**Sources Destinations N Demand = 100 A 140 Supply = 200 60 S Demand = 140 210 Supply = 350 B 140 E Demand = 300 (shortage of 90) 100 Supply = 150 C 50 W Demand = 250 Cost Z = 10330**An Iteration**• Find the entering basic variable • Select the variable with the largest negative c’ij • Find the leaving basic variable • Determine the chain reaction that would result from increasing the value of the entering variable from zero • The leaving variable will be the first variable to reach zero because of this chain reaction**Initial Solution Obtained by the Northwest Corner Rule**100 100 40 300 10 150 90**100**100 - + 40 300 10 150 + - 90 ? Iteration 1**End of Iteration 1**100 100 40 210 100 150 90**Optimality Test**100 100 40 210 100 150 90**Iteration 2**+ - 100 100 + - 210 100 40 ? 150 90**End of Iteration 2**140 60 40 210 100 150 90**Optimality Test**140 60 40 210 100 150 90 Z = 10330**Optimal Solution**Sources Destinations N Demand = 100 60 A Supply = 200 140 S Demand = 140 40 Supply = 350 B 210 E Demand = 300 (shortage of 90) 100 Supply = 150 C 150 W Demand = 250 Cost Z = 10330**The Assignment Problem**• The problem of finding the minimum-costly assignment of a set of tasks (i=1,…,m) to a set of agents (j=1,…,n) • Each task should be performed by one agent • Each agent should perform one task • A cost cij associated with each assignment • We should have m=n (if not…?) • A special type of linear programming problem, and • A special type of transportation problem,with si=dj= ?