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