Managerial Decision Making Chapter 5 Network Modelling

A Transshipment Problem: Cars must be shipped from ports in Newark and Jacksonville to other locations at minimum cost. The costs, supplies, and demands are shown on the network. Newark 1 Boston 2 Columbus 3 Atlanta 5 Richmond 4 J'ville 7 Mobile 6 $30 $40 $50 $35 $40 $30 $35 $25 $50 $45 $50 –200 –

Defining the Decision Variables For each arc in a network flow model we define a decision variable as: X ij = the amount being shipped (or flowing) from node i to node j For example… X 12 = the # of cars shipped from node 1 (Newark) to node 2 (Boston) X 56 = the # of cars shipped from node 5 (Atlanta) to node 6 (Mobile) Note: The number of arcs determines the number of variables!

Defining the Objective Function Minimize total shipping costs. MIN : 30X X X X X X X X X X X 76

Total Supply = 500 cars Total Demand = 480 cars If Total Supply > Total Demand then for every node, Inflow - Outflow >= Supply or Demand For example at Node 3, Inflow = X 23 + X 53 Outflow = X 35 => X 23 + X 53 – X 35 ≥ 60

Constraint for node 1: –X 12 – X 14 >= – 200 (Note: there is no inflow for node 1!) This is equivalent to: +X 12 + X 14 <= 200

Defining the Constraints Flow constraints –X 12 – X 14 >= –200} node 1 +X 12 – X 23 >= +100} node 2 +X 23 + X 53 – X 35 >= +60} node 3 + X 14 + X 54 + X 74 >= +80} node 4 + X 35 + X 65 + X 75 – X 53 – X 54 – X 56 >= +170} node 5 + X 56 + X 76 – X 65 >= +70} node 6 –X 74 – X 75 – X 76 >= –300} node 7 Nonnegativity conditions X ij >= 0 for all ij

Constraints for Network Flow Problems: The Balance-of-Flow Rules For Minimum Cost Network Apply This Balance-of-Flow: Flow Problems Where:Rule At Each Node: Total Supply > Total DemandInflow-Outflow >= Supply or Demand Total Supply < Total DemandInflow-Outflow <=Supply or Demand Total Supply = Total DemandInflow-Outflow = Supply or Demand

LP Model for Network Model  The optimal solution of a Network Model can be found by solving an associated Linear Programming model:  x ij = # of units of flow from node i to node j (for some pairs i-j)  Minimize the total cost:∑ c ij x ij  Balance flow in each node i:  If Total Supply > Total Demand:Inflow – Outflow ≥ b i  If Total Supply < Total Demand:Inflow – Outflow ≤ b i  If Total Supply = Total Demand:Inflow – Outflow = b i  Limit each variable x ij :l ij ≤ x ij ≤ u ij 9

LP Model for Network Model  Remember:  For each arc there always is one decision variable  For each node there always is one flow balancing constraint  Special structure of the network flow constraints:  Each variable appears in two constraints only, once with coefficient equal to 1, and once with coefficient equal to -1.  If Demand and Supply are integers, then the optimal solution is always integer 10

Optimal Solution to the LP Newark 1 Boston 2 Columbus 3 Atlanta 5 Richmond 4 J'ville 7 Mobile 6 $30 $40 $50 $40 $50 $

LP Model in Excel See file Fig5-2.xls and Fig5-2generic.xlsFig5-2.xlsFig5-2generic.xls 12

The Shortest Path Problem  Many decision problems boil down to determining one shortest (or cheapest) route or path through a network.  Shortest way to drive a car from a city to another city  Fastest way for an ambulance to get to an accident  Easiest way to walk across the mountains  The Shortest Path Problem is a special case of the Network Flow Problem:  There is one source node with 1 unit of supply (b i = –1)  There is one destination node with 1 unit of demand (b i = 1)  All other nodes do not have any supply or demand (b i = 0) 13

The American Car Association B'ham Atlanta G'ville Va Bch Charl. L'burg K'ville A'ville G'boro Raliegh Chatt hrs 3.0 hrs 1.7 hrs 2.5 hrs 1.7 hrs 2.8 hrs 2.0 hrs 1.5 hrs 2.0 hrs 5.0 hrs 3.0 hrs 4.7 hrs 1.5 hrs 2.3 hrs 1.1 hrs 2.0 hrs 2.7 hrs 3.3 hrs +1 What is the fastest way to get from B’ham to Virginia Beach? 14 +0

B'ham Atlanta G'ville Va Bch Charl. L'burg K'ville A'ville G'boro Raliegh Chatt pts 4 pts 3 pts 5 pts 7 pts 8 pts 2 pts 9 pts 4 pts 9 pts 3 pts 4 pts 5 pts What is the most scenic way to get from B’ham to Va Bch?

Solving the Problem There are two possible objectives for this problem  Finding the quickest route (minimizing travel time)  Finding the most scenic route (maximizing the scenic rating points) See file Fig5-7.xlsFig5-7.xls

The Equipment Replacement Problem  The problem of determining when to replace equipment is another common business problem.  It can also be modeled as a shortest path problem…

The Compu-Train Company  Compu-Train provides hands-on software training.  Computers must be replaced at least every two years.  A lease contract is being considered: Initially the equipment is bought at a cost of $62,000  Prices increase 6% per year  60% trade-in for 1 year old equipment  15% trade-in for 2 year old equipment

 Cost of trading after 1 year = 1.06*62, *62,000 = $28,520  Cost of trading after 2 years = *62, *62,000 = $60,

Network for Contract $28,520 $60,363 $30,231 $63,985 $32,045 $67,824 $33,968 Cost of trading after 1 year: 1.06*$62, *$62,000 = $28,520 Cost of trading after 2 years: *$62, *$62,000 = $60,363 etc, etc….

 Cost of going from state 2 to state 3, (Cost of trading after 1 year then trading after 1 year) = * – 0.6*1.06 * = $30,231  Cost of going from state 2 to state 4,) Cost of trading after 1 year then trading after 2 years )= * – 0.15*.106 * = $63,

Compu-Train – model x ij = 1 if replacement associated with arc i-j is chosen = 0 otherwise Min28,520x ,363x ,231x ,985x ,045x ,824x ,968x 45 (total cost) s.t.– x 12 – x 13 = –1(1) x 12 – x 23 – x 24 = 0(2) x 13 + x 23 – x 34 – x 35 = 0(3) x 24 + x 34 – x 45 = 0(4) x 35 + x 45 = 1(5) All x ij >= 0  See file Fig5-12.xls – one model for both contracts.Fig5-12.xls 22

Worker Assignment Problem  Three workers, Ann, Bob, and Cindy, should be assigned jobs such that the total work time is minimized.  Three jobs are available and the work time (in hours) for each worker on each job is given in the table below. Jobs Worker123 Ann Bob Cindy  Each worker should be assigned exactly one job. 23

Assignment Model 24 X ij = 1 if worker i is assigned to job j, =0 otherwisei=A..C, j=1..3 min21X A1 + 50X A2 + 40X A3 + 35X B1 + 30X B2 + 22X B3 + 55X C1 + 20X C2 + 25X C3 (total work time) X A1 + X A2 + X A3 = 1(Ann) X B1 + X B2 + X B3 = 1 (Bob) X C1 + X C2 + X C3 = 1(Cindy) X A1 + X B1 + X C1 = 1(job 1) X A2 + X B2 + X C2 = 1(job 2) X A3 + X B3 + X C3 = 1(job 3) All X ij >= 0 What if another worker was available? What if job 3 required 2 workers?

Some network flow problems don’t have trans-shipment nodes; only supply and demand nodes These problems are implemented more effectively using the technique described in Chapter 3.

Transportation Problem: Tropicsun  Transportation problem is a special case of the network flow X ij = # of bushels shipped from node i to node j, i=1..3, j=4..6 min21X X X X X X X X X 36 (total bushel-miles) X 14 + X 15 + X 16 = 275,000 (1) X 24 + X 25 + X 26 = 400,000 (2) X 34 + X 35 + X 36 = 300,000 (3) X 14 + X 24 + X 34 <= 200,000 (4) X 15 + X 25 + X 35 <= 600,000 (5) X 16 + X 26 + X 36 <= 225,000 (6) All X ij >= 0  Note: can state the constraints as inflow-outflow<=0 26

LP Model for Network Model  The optimal solution of a Network Model can be found by solving an associated Linear Programming model:  x ij = # of units of flow from node i to node j (for some pairs i-j)  Minimize the total cost:∑ c ij x ij  Balance flow in each node i:  If Total Supply > Total Demand:Inflow – Outflow ≥ b i  If Total Supply < Total Demand:Inflow – Outflow ≤ b i  If Total Supply = Total Demand:Inflow – Outflow = b i  Limit each variable x ij :l ij ≤ x ij ≤ u ij 27

Relationship of Network Flow Problems  Some problems that do not look like network flow problems may have a network flow problem structure.  Example: equipment replacement problem Generalization Special Case Integer solutions guaranteed if all b i, l ij, u ij are integer 28

Generalized Network Flow Problems  In some problems, a gain or loss occurs in flows.  Example of gain  Interest or dividends on investments  Examples of loss  Oil or gas shipped through a leaky pipeline  Imperfections in raw materials entering a production process  Spoilage of food items during transit  Theft during transit  Generalized Network Flow Problem can be used:  Arc i-j has yield a ij (gain if >1, loss if <1)  Outflow from i is x ij  Inflow to j is a ij x ij ij c ij a ij [l ij, u ij ] 29

Coal Bank Hollow Recycling Problem  Coal Bank Hollow Recycling specializes in collecting and recycling paper. Collected used paper goes through recycling process 1 or 2, and then it is used to produce paper pulp for newsprint, packaging paper, and print stock. Supply, demand, costs, and yields (changes of mass) are shown below.  Taking the Raw Material trough Process 1 and 2:  To Process 1To Process 2  MaterialCost YieldCost YieldSupply  Newspaper$1390%$1285%70 tons  Mixed Paper$1180%$1385%50 tons  White Office Paper$995%$1090%30 tons  Cardboard$1375%$1485%40 tons  Using the paper material from process 1 and 2 to produce final products  Newsprint Packaging PaperPrint Stock  Pulp SourceCostYieldCostYieldCostYield  From Process 1$595%$690%$890%  From Process 2$690%$895%$795%  Demand60 tons40 tons50 tons 30

Network for Recycling Problem Newspaper 1 Mixed paper 2 3 Cardboard 4 Recycling Process Newsprint pulp 7 Packing paper pulp 8 Print stock pulp White office paper Recycling Process 2 $13 $12 $11 $13 $9 $10 $14 $13 90% 80% 95% 75% 85% 90% 85% $5 $6 $8 $6 $7 $8 95% 90% 95% +0 31

Model part 1  X ij = # of tons of paper material flowing out of node i on arc i-j (note: flow into node j = yield*X ij )  Minimize total cost  Min13X X X X X X X X X X X X X X 69  Raw materials constraints  – X 15 – X 16 >= –70 (node 1)  – X 25 – X 26 >= –50 (node 2)  – X 35 – X 36 >= –30 (node 3)  – X 45 – X 46 >= –40 (node 4) 32

Model part 2  Recycling processes constraints  0.9X X X X 45 -X 57 - X 58 -X 59 >= 0 (node 5)  0.85X X X X 46 -X 67 -X 68 -X 69 >= 0 (node 6)  Paper pulp  +0.95X X 67 >= 60 (node 7)  +0.90X X 67 >= 40 (node 8)  +0.90X X 67 >= 50 (node 9)  Nonnegativity  All X ij >= 0  See file Fig5-17.xlsFig5-17.xls 33

Important Modeling Point  In generalized network flow problems, gains and/or losses associated with flows across each arc effectively increase and/or decrease the available supply.  This can make it difficult to tell if the total supply is adequate to meet the total demand. When in doubt, do the following:  Assume sufficient supply (inflow – outflow >= b i ) and try to solve the problem in Excel.  If infeasible, assume insufficient supply (inflow – outflow <= b i ) and solve the problem again.  Unlike in Network Flow Problems, in Generalized Network Flow Problems integer solutions are not guaranteed because of fractional gains or losses. 34

End of Chapter 5  Extra material in the chapter:  Maximal Flow Problem – find the maximum number of units that could be pushed through the network  Minimum Spanning Tree – find the cheapest tree spanning over the set of nodes  Other well-known Network Models (see wikipedia.org)  Traveling Salesman Problem – finding the shortest route visiting all houses in a neighborhood. 35