ECE 307 – Techniques for Engineering Decisions Networks and Flows George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved NETWORKS AND FLOWS A network is a system of lines or channels connecting different points Examples abound in nearly all aspects of life: electrical systems communication networks airline webs local area networks distribution systems © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved NETWORKS AND FLOWS The network structure is also common to many other systems that at first glance are not necessarily viewed as networks distribution system consisting of manufacturing plants, warehouses and retail outlets matching problems such as work to people, assignments to machines and computer dating © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved NETWORKS AND FLOWS river systems with pondage for electricity generation mail delivery networks project management of multiple tasks in a large undertaking such as construction or a space flight We consider a broad range of network and network flow problems © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved THE TRANSPORTATION PROBLEM The basic idea of the transportation problem is illustrated with the problem of distribution of a specified homogenous product from several sources to a number of localities at least cost We consider a system with m warehouses, n markets and links between them with the specified costs of transportation © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved ⎧ ⎪1 ⎪ ⎪ ⎪ ⎪2 ⎪ ⎪ ⎨ ⎪ ⎪i ⎪ ⎪ ⎪ ⎪ ⎪m ⎩ supply demand a1 b1 a2 transportation links with costs b2 ci j bj c i j = ∞ whenever am warehouse i cannot ship to market j bn © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved ⎫ ⎪⎪ ⎪ ⎪ ⎪ 2⎪ ⎪ ⎪ ⎬ ⎪ j⎪ ⎪ ⎪ ⎪ ⎪ n⎪ ⎪ ⎭ markets warehouses THE TRANSPORTATION PROBLEM THE TRANSPORTATION PROBLEM all the supply comes from the m warehouses; we associate the index i = 1, 2, … , m with a warehouse all the demand is at the n markets; we associate the index j = 1, 2, … , n with a market shipping costs c i j for each unit from the warehouse i to the market j © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved THE TRANSPORTATION PROBLEM The transportation problem is to determine the optimal shipping schedule that minimizes shipping costs for the set of m warehouses to the set of n markets : the quantities shipped from the warehouse i to each market j © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved LP FORMULATION OF THE TRANSPORTATION PROBLEM The decision variables are xij = quanity shipped from warehouse i to market j i = 1, 2, , m j = 1, 2, , n The objective function is m n cij xij ∑ ∑ i =1 j =1 © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved LP FORMULATION OF THE TRANSPORTATION PROBLEM The constraints are: n xij ∑ j =1 m xij ∑ i =1 ≤ i = 1, 2, , m ≥ bj j = 1, 2, , n i = 1, 2, , m xij ≥ j = 1, 2, , n © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 10 NONSTANDARD TRANSPORTATION PROBLEM For the overdemand case, we introduce the fictitious warehouse W m+1 to supply the shortage; ⎡ n ⎢ b j ⎢⎣ j =1 ∑ − m ∑ i =1 ⎤ ⎥ we set c m+1, j = ⎥⎦ for j = 1, 2, … , n and the problem is in standard form with i = 1, … , m + (augmented number of warehouses) © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 79 NONSTANDARD TRANSPORTATION PROBLEM Note that the variable x m+1, j is the shortage at market j and is the shortfall in the demand b j experienced by the market M j due to inadequate supplies For each market j , x m+1, j is a measure of the infeasibility of the problem © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 80 EXAMPLE: CANNING OPERATIONS SCHEDULING This problem is concerned with the schedule of plants A and B in the purchase of the raw supplies from growers grower availability (ton) price ( $ / ton ) Smith 200 10 Jones 300 Richard 400 © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 81 EXAMPLE: CANNING OPERATIONS SCHEDULING and shipping costs in $ / ton given by to from plant A B Smith 2.5 Jones 1.5 Richard © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 82 EXAMPLE: CANNING OPERATIONS SCHEDULING The plants’ labor costs and capacity limits are plant A B 450 550 25 20 capacity ( ton ) labor costs ( $ / ton ) © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 83 EXAMPLE: CANNING OPERATIONS SCHEDULING The selling price for canned goods is 50 $ / ton and the company can sell all it produces The problem is to determine the maximum profit schedule Note that this is an unbalanced problem since supply = 200 + 300 + 400 = 900 tons demand = 450 + 550 = 1000 tons > 900 tons Clearly, the decision variables are the amounts purchased from each grower and shipped to each plant © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 84 EXAMPLE: CANNING OPERATIONS SCHEDULING The objectives is formulated as max Z = ⎡ ⎤ ⎡ ⎤ ⎢ 50 − 25 − 10 − ⎥ x ⎢ 50 − 25 − − ⎥ x SA + ⎥ JA ⎢ ⎥ ⎢ 13 15 ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎢ 50 − 20 − 10 − 2.5 ⎥ x ⎥x + ⎢ 50 − 25 − − + ⎥ RA ⎢ ⎥ SB ⎢ 12 17.5 ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎥ x + ⎢ 50 − 20 − − ⎥ x + ⎢ 50 − 20 − − 1.5 ⎥ JB ⎢ ⎢ ⎥ RB 19.5 19 ⎣ ⎦ ⎣ ⎦ © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 85 EXAMPLE: CANNING OPERATIONS SCHEDULING The supply constraints are x SA + x SB ≤ 200 x JA + x JB ≤ 300 x RA + x RB ≤ 400 The demand constraints are x SA + x JA + x RA ≤ 450 x SB + x JA + x RB ≤ 550 © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 86 EXAMPLE: CANNING OPERATIONS SCHEDULING Clearly, all decision variables are nonnegative The unbalanced nature of the problem requires the introduction of a fictitious grower F with the supply 100 corresponding to the supply shortage; in this way the nonstandard problem becomes standard We set up the standard transportation problem © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 87 EXAMPLE: CANNING OPERATIONS SCHEDULING plant j A supply B grower i S 200 13 17.5 J 300 15 19.5 R 400 12 19 F 100 0 demand 450 550 © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 88 EXAMPLE: CANNING OPERATIONS SCHEDULING Please note that the objective is a maximization rather than a minimization We therefore recast the mechanics of the u-v scheme for the maximization problem As a homework exercise, show that the duality complementary slackness conditions allow us to change the u – v algorithm in the following way: © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 89 EXAMPLE: CANNING OPERATIONS SCHEDULING the selection of the nonbasic variable x i j to enter the basis is from those x i j where the corresponding c i j > ui + v j and we evaluate and focus on all c i j > so that x i j is a candidate to enter the basis we pick x pq c pq = max pq ∋ x pq is nonbasic {c } pq © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 90 EXAMPLE SOLUTION plant j A B supply 200 200 50 300 grower i S 13 J 250 R F demand 15 400 100 450 19.5 400 19 100 550 © 2006-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 91 ... i j © 200 6-2 009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 14 TRANSPORTATION PROBLEM EXAMPLE market j w/h i M1 W1 10 W3 M3 M4 W2 bj M2 7 6 4 © 200 6-2 009 George... © 200 6-2 009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 17 APPLICATION OF THE LEAST – COST RULE market j w/h i M1 M2 M3 M4 W1 W2 2 10 W3 bj 7 6 © 200 6-2 009 George... reduced tableau: c 24 is the lowest-valued c i j with i = 2, j = and we select x 24 as a basic variable © 200 6-2 009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved