CaseStudyForm  

Title  The American Steel Transshipment Problem 
DateSubmitted  15 Feb 2008 
CaseStudyType  TeachingCaseStudy 
OperationsResearchTopics  LinearProgramming, IntegerProgramming, NetworkOptimisation, TransshipmentProblem 
ApplicationAreas  Logistics 
ProblemDescription 
American Steel, an Ohiobased steel manufacturing company, produces steel at its two steel mills located at Youngstown and Pittsburgh. The company distributes finished steel to its retail customers through the distribution network of regional and field warehouses shown below:
The network represents shipment of finished steel from American Steel's two steel mills located at Youngstown (node 1) and Pittsburgh (node 2) to their field warehouses at Albany, Houston, Tempe, and Gary (nodes 6, 7, 8 and 9) through three regional warehouses located at Cincinnati, Kansas City, and Chicago (nodes 3, 4 and 5). Also, some field warehouses can be directly supplied from the steel mills. Table 1 presents the minimum and maximum flow amounts of steel that may be shipped between different cities along with the cost per 1000 ton/month of shipping the steel. For example, the shipment from Youngstown to Kansas City is contracted out to a railroad company with a minimal shipping clause of 1000 tons/month. However, the railroad cannot ship more than 5000 tons/month due the shortage of rail cars. Table 1 Arc Costs and Limits  From node  To node  Cost  Minimum  Maximum   Youngstown  Albany  500    1000   Youngstown  Cincinnati  350    3000   Youngstown  Kansas City  450  1000  5000   Youngstown  Chicago  375    5000   Pittsburgh  Cincinnati  350    2000   Pittsburgh  Kansas City  450  2000  3000   Pittsburgh  Chicago  400    4000   Pittsburgh  Gary  450    2000   Cincinnati  Albany  350  1000  5000   Cincinnati  Houston  550    6000   Kansas City  Houston  375    4000   Kansas City  Tempe  650    4000   Chicago  Tempe  600    2000   Chicago  Gary  120    4000  The current monthly demand at American Steel's four field warehouses is shown in Table 2.  Field Warehouses  Monthly Demand   Albany, N.Y.  3000   Houston  7000   Tempe  4000   Gary  6000  The Youngstown and Pittsburgh mills can produce up to 10,000 tons and 15,000 tons of steel per month, respectively. The management wants to know the least cost monthly shipment plan. 
ProblemFormulation 
The American Steel Problem can be solved as a transshipment problem. The supply at the supply nodes is the maximum production at the steel mills, i.e., 10,000 and 15,000 for Youngstown and Pittsburgh respectively. The demand at demand nodes in given by the demand at the field warehouses and the other nodes are transshipment nodes. The costs and bounds on flow through the network are also given. The most compact formulation for this problem is a network formulation (see The Transshipment Problem for details).

ComputationalModel 
We can use the AMPL model file transshipment.mod (see The Transshipment Problem in AMPL for details) to solve the American Steel Transshipment Problem. We need a data file to describe the nodes, arcs, costs and bounds. The node set is simply a list of the node names:
set NODES := Youngstown Pittsburgh Cincinnati 'Kansas City' Chicago Albany Houston Tempe Gary ;
Note that The arc set is 2dimensional and can be defined in a number of different ways: # List of arcs set ARCS := (Youngstown, Albany), (Youngstown, Cincinnati), ... , (Chicago, Gary) ; # Table of arcs set ARCS: Cincinnati 'Kansas City' Chicago Albany Houston Tempe Gary := Youngstown + + + +    Pittsburgh + + +    + . . . # Array of arcs set ARCS := (Youngstown, *) Cincinnati 'Kansas City' Chicago Albany (Pittsburgh, *) Cincinnati 'Kansas City' Chicago Gary . . . (Chicago, *) Tempe Gary ; Since the node set is small and the arc set is "dense", i.e., many of the node pairs are represented in the arc set, we choose a table to define ARCS: set ARCS: Cincinnati 'Kansas City' Chicago Albany Houston Tempe Gary := Youngstown + + + +    Pittsburgh + + +    + Cincinnati    + +   'Kansas City'     + +  Chicago      + + ;
The param NetDemand := Youngstown 10000 Pittsburgh 15000 Albany 3000 Houston 7000 Tempe 4000 Gary 6000 ;
We can use lists, tables or arrays to define the parameters for the American Steel Transhippment Problem,
but in this case we will use a list and define multiple parameters at once. This allows us to "cutandpaste" the parameters from the problem description. Note the use of
param: Cost Lower Upper:= Youngstown Albany 500 . 1000 Youngstown Cincinnati 350 . 3000 Youngstown 'Kansas City' 450 1000 5000 Youngstown Chicago 375 . 5000 . . . Chicago Gary 120 . 4000 ; Note that the cost is in $/1000 ton, so we must divide the cost by 1000 in our script file before solving: reset; model transshipment.mod; data steel.dat; let {(m, n) in ARCS} Cost[m, n] := Cost[m, n] / 1000; option solver cplex; solve; display Flow;
We can define the PuLP/Dippy model using functions in transshipment_funcy.py

Results 
Using transshipment.mod , and the data and script files defined in Computational Model we can solve the American Steel Transshipment Problem to get the optimal Flow variables:
If the total supply is greater than the total demand, the transshipment problem will solve, but flow may be left in the network (in this case at the Pittsburgh node). In If total supply is less than demand (hence the problem is infeasible) we can add a dummy supply node (see with arcs to all the demand nodes. The optimal solution will show the "best" nodes to send the extra supply to.

Conclusions 
In order to minimise the monthly shipment costs, American Steel should follow the shipment plan shown in Table 3.
Table 3 Optimal Shipment Plan  From/To  Cincinnati  Kansas City  Chicago  Albany  Houston  Tempe  Gary   Youngstown  3000  3000  3000  1000      Pittsburgh  2000  3000  3000     2000   Cincinnati     2000  3000     Kansas City      4000  2000    Chicago       2000  4000  As with many network problems, it can be illuminating to display the solution graphically as shown in Figure 1. Figure 1 Optimal Shipment Plan

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