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Readings Readings Chapter 7 Integer Linear Programming BA 452 Lesson B.4 Binary Fixed Costs 1 Overview Overview BA 452 Lesson B.4 Binary Fixed Costs 2 Overview Fixed Costs of Production are those production costs that are present whenever production is positive. The simplest model of fixed costs in a linear program restricts decision variables to binary (0 or 1). Fixed Costs of Assignment are modeled like Fixed Costs of Production. The simplest model of fixed costs restricts decision variables (the fraction of work completed) to be binary. Location Covering Problems are a special kind of Linear Programming problem when outputs are fixed because the firm has only established customers, who require coverage by service centers. Worker Covering Problems are like Location Covering Problems, where customers require scheduling workers so that at each time period is covered by a prescribed minimal number of workers. Transportation Problems with New Origins are Transportation Problems extended so that new origins may be added, at a fixed cost. They choose the best plant locations and how much to ship. Transshipment Problems with New Nodes are Transshipment Problems extended so that new transshipment nodes may be added, at a fixed cost. They choose the best transshipment locations. BA 452 Lesson B.4 Binary Fixed Costs 3 Tool Summary Tool Summary Use binary variables to indicate whether an activity, such as a production run, is undertaken. Write a multiple-choice constraint: The sum of two or more binary variables equals 1, so any feasible solution choose one variable to equal 1. Write a mutually-exclusive constraint: The sum of two or more binary variables is at most 1, so any feasible solution chooses at most one variable to equal 1. All variables could equal 0. Write a conditional constraint: An inequality constraint so that one binary variable cannot equal unless certain other binary variables also equal 1. Write a corequisite constraint: An equality constraint of binary variables, so are either both 0 or both 1. BA 452 Lesson B.4 Binary Fixed Costs 4 Fixed Cost of Production Fixed Cost of Production BA 452 Lesson B.4 Binary Fixed Costs 5 Fixed Cost of Production Overview Fixed Costs of Production are those production costs that are present whenever production is positive. The simplest way to model fixed costs in a linear program is to restrict decision variables to be binary (0 or 1). For example, suppose the cost of producing quantity x is 5x. On the one hand, if x can take on any non-negative value (like x is the pounds of hamburger produced), then 5 is the constant unit cost of production. On the other hand, if x can take on only binary values (like x is the number of new books adopted), then cost is 0 if there is no production and 5 if there is positive production, so 5 is the fixed cost of production. BA 452 Lesson B.4 Binary Fixed Costs 6 Fixed Cost of Production Question: W. W. Norton & Company, the oldest and largest publishing company wholly owned by its employees, must decide which new textbooks to adopt next year. The books considered are described along with their expected three-year sales: Book Projected Sales Business Calculus 20 Finite Math 30 General Statistics 15 Mathematical Statistics 10 Business Statistics 25 Finance 18 Financial Accounting 25 Managerial Accounting 50 English Literature 20 German 30 BA 452 Lesson B.4 Binary Fixed Costs 7 Fixed Cost of Production Three individuals in the company can be assigned to these projects, all of whom have varying amounts of time available. John has 60 days, Susan has 52 days, and Monica has 43 days. The days required by each person to complete each project are showing in the following table. For example, if the business calculus book is published, it will require 30 days of John’s time and 40 days of Susan’s time. Book Projected Sales John Susan Monica Business Calculus 20 30 40 X Finite Math 30 16 24 X General Statistics 15 24 X 30 Mathematical Statistics 10 20 X 24 Business Statistics 25 10 X 16 Finance 18 X X 14 Financial Accounting 25 X 24 26 Managerial Accounting 50 X 28 30 English Literature 20 40 34 30 German 30 x 50 36 BA 452 Lesson B.4 Binary Fixed Costs 8 Fixed Cost of Production Norton will not publish more than two statistics books or more than one accounting book in a single year. In addition, one of the math books (business calculus or finite math) must be published, but not both. Which books should be published, and what are the projected sales? If Monica has 1 more day available, which books should be published, and what are the projected sales? Comment. BA 452 Lesson B.4 Binary Fixed Costs 9 Fixed Cost of Production i Book 1 Business Calculus 2 Finite Math 3 General Statistics 4 Mathematical Statistics 5 Business Statistics 6 Finance 7 Financial Accounting 8 Managerial Accounting 9 English Literature 10 German BA 452 Lesson B.4 Binary Fixed Costs 10 Fixed Cost of Production Max s.t. 20x1 30x1 40x1 x1 i Book Projected Sales John Susan Monica 1 Business Calculus 20 30 40 X 2 Finite Math 30 16 24 X 3 General Statistics 15 24 X 30 4 Mathematical Statistics 10 20 X 24 5 Business Statistics 25 10 X 16 6 Finance 18 X X 14 7 Financial Accounting 25 X 24 26 8 Managerial Accounting 50 X 28 30 9 English Literature 20 40 34 30 10 German 30 x 50 36 + 30x2 + 15x3 + 10x4 + 25x5 + + 16x2 24x2 + 24x3 + 20x4 + 10x5 30x3 x3 + + 24x4 x4 + + 16x5 x5 + + + 18x6 14x6 + + + 25x7 + 50x8 + 20x9 + + + 40x9 34x9 30x9 24x7 26x7 + + 28x8 30x8 x7 + x8 + + + x2 BA 452 Lesson B.4 Binary Fixed Costs 30x10 50x10 36x10 = 60 52 43 2 1 1 John Susan Monica No. of Stat Books Account Book Math Book 11 Fixed Cost of Production Max s.t. 20x1 30x1 40x1 + 30x2 + 15x3 + 10x4 + 25x5 + + 16x2 24x2 + 24x3 + 20x4 + 10x5 30x3 x3 x1 + + + 24x4 x4 + + 16x5 x5 + + 18x6 + 25x7 + 50x8 + 20x9 + 30x10 14x6 + + 24x7 26x7 + + 28x8 30x8 + + + 40x9 34x9 30x9 + + 50x10 36x10 x7 + x8 x2 BA 452 Lesson B.4 Binary Fixed Costs = 60 52 43 2 1 1 John Susan Monica No. of Stat Books Account Book Math Book 12 Fixed Cost of Production i Book 1 Business Calculus 2 Finite Math 3 General Statistics 4 Mathematical Statistics 5 Business Statistics 6 Finance 7 Financial Accounting 8 Managerial Accounting 9 English Literature 10 German Publish finite math, business statistics, and financial accounting. Projected sales are 80,000. BA 452 Lesson B.4 Binary Fixed Costs 13 Fixed Cost of Production i Book 1 Business Calculus 2 Finite Math 3 General Statistics 4 Mathematical Statistics 5 Business Statistics 6 Finance 7 Financial Accounting 8 Managerial Accounting 9 English Literature 10 German Publish finite math, business statistics, and financial accounting. Projected sales are 80,000. BA 452 Lesson B.4 Binary Fixed Costs 14 Fixed Cost of Production If Monica is available 1 more day (44 days total), optimal projected sales are now 98,000. A big (discrete) gain for a small increase in a constraint. BA 452 Lesson B.4 Binary Fixed Costs 15 Fixed Cost of Assignment Fixed Cost of Assignment BA 452 Lesson B.4 Binary Fixed Costs 16 Fixed Cost of Assignment Overview Fixed Costs of Assignment are modeled like Fixed Costs of Production. The simplest model of fixed costs restricts decision variables to be binary. For example, suppose the cost of worker i performing the fraction xij of job j is 5xij. On the one hand, if xij can take on any non-negative value between 0 and 1 (like xij is the fraction of the job performed), then 5 is the constant unit cost of assignment (like the time cost of working). On the other hand, if xij can take on only binary values, then cost is 0 if you are not assigned (xij = 0) and 5 if you are assigned the entire job (xij = 1), so 5 is the fixed cost of assignment (like the cost of commuting to a job across town). BA 452 Lesson B.4 Binary Fixed Costs 17 Fixed Cost of Assignment Question: Tina's Tailoring has five idle tailors and four custom garments to make. The estimated time (in hours) it would take each tailor to make each garment is shown in the next slide. (An 'X' in the table indicates an unacceptable tailor-garment assignment.) Garment Wedding gown Clown costume Admiral's uniform Bullfighter's outfit 1 19 11 12 X 2 23 14 8 20 Tailor 3 4 5 20 21 18 X 12 10 11 X 9 20 18 21 BA 452 Lesson B.4 Binary Fixed Costs 18 Fixed Cost of Assignment Formulate an integer program for determining the tailorgarment assignments that minimize the total estimated time spent making the four garments. No tailor is to be assigned more than one garment, and each garment is to be worked on by only one tailor. This particular problem can be formulated as either a binary program or as an integer program. Any feasible solution to the latter program is binary (0-1). BA 452 Lesson B.4 Binary Fixed Costs 19 Fixed Cost of Assignment Answer: Define the decision variables xij = 1 if garment i is assigned to tailor j = 0 otherwise. Find the number of decision variables = [(number of garments)x(number of tailors)] - (number of unacceptable assignments) = [4x5] - 3 = 17 Find the number of constraints. 1 for each garment and each tailor = 9. BA 452 Lesson B.4 Binary Fixed Costs 20 Fixed Cost of Assignment Define the objective function Minimize total time spent making garments: Min 19x11 + 23x12 + 20x13 + 21x14 + 18x15 + 11x21 + 14x22 + 12x24 + 10x25 + 12x31 + 8x32 + 11x33 + 9x35 + 20x42 + 20x43 + 18x44 + 21x45 BA 452 Lesson B.4 Binary Fixed Costs 21 Fixed Cost of Assignment Define the constraints of exactly one tailor per garment: 1) x11 + x12 + x13 + x14 + x15 = 1 2) x21 + x22 + x24 + x25 = 1 3) x31 + x32 + x33 + x35 = 1 4) x42 + x43 + x44 + x45 = 1 BA 452 Lesson B.4 Binary Fixed Costs 22 Fixed Cost of Assignment Define the constraints of no more than one garment per tailor: 5) x11 + x21 + x31 < 1 6) x12 + x22 + x32 + x42 < 1 7) x13 + x33 + x43 < 1 8) x14 + x24 + x44 < 1 9) x15 + x25 + x35 + x45 < 1 BA 452 Lesson B.4 Binary Fixed Costs 23 Fixed Cost of Assignment Minimum time: 55 hours Optimal assignments: Garment Wedding gown Clown costume Admiral's uniform Bullfighter's outfit BA 452 Lesson B.4 Binary Fixed Costs 1 19 11 12 X Tailor 2 3 4 5 23 20 21 18 14 X 12 10 8 11 X 9 20 20 18 21 24 Location Covering Location Covering BA 452 Lesson B.4 Binary Fixed Costs 25 Location Covering Overview Location Covering Problems are a special kind of Linear Programming problem when outputs are fixed because the firm has only established customers. Commitments to established customers require building service centers so that each customer area is covered by a prescribed minimal number of service centers. The objective is to minimize the cost of building service centers. If each center is equally costly, the objective reduces to minimizing the number of service centers. The simplest way to model the all-or-nothing decision to build or not build a serve center is to restrict the fraction of the center built to be a binary (0 or 1) decision variable. BA 452 Lesson B.4 Binary Fixed Costs 26 Location Covering Question: UPS is drawing up new zones for the location of drop boxes for customers. The city has been divided into the four zones shown below. You have targeted six possible locations for drop boxes (numbered 1 through 6). The list of which drop boxes could be reached easily from each zone is listed below. Zone Can be Served by Drop Box Locations: Downtown Financial 1, 2, 5, 6 Downtown Legal 2, 4, 5 Retail South 1, 2, 4, 6 Retail North 3, 4, 5 BA 452 Lesson B.4 Binary Fixed Costs 27 Location Covering Formulate and solve a model to provide the fewest dropbox locations yet make sure that each zone is covered by at least two boxes. BA 452 Lesson B.4 Binary Fixed Costs 28 Location Covering Answer: Min x1 + x2 + x3 + x4 + x5 + x6 s.t. x1 + x2 + x5 + x6 > 2 x2 + x4 + x5 > 2 x1 + x2 + x4 + x6 > 2 x3 + x4 + x5 > 2 BA 452 Lesson B.4 Binary Fixed Costs 29 Worker Covering Worker Covering BA 452 Lesson B.4 Binary Fixed Costs 30 Worker Covering Overview Worker Covering Problems are like Location Covering Problems. Outputs are fixed because the firm has only established customers. Commitments to established customers require scheduling workers so that at each time period customer needs are covered by a prescribed minimal number of workers. The objective is to minimize the cost of scheduling workers. If each worker is equally costly, the objective reduces to minimizing the number of workers scheduled. The simplest way to model the all-ornothing decision to schedule a worker is to restrict the number of workers scheduled at each time period to be an integer decision variable. BA 452 Lesson B.4 Binary Fixed Costs 31 Worker Covering Question: Amazon.com is open 24 hours a day. The number of phone operators need in each four hour period of a day is listed below. Period Operators Needed 10 p.m. to 2 a.m. 8 2 a.m. to 6 a.m. 4 6 a.m. to 10 a.m. 7 10 a.m. to 2 p.m. 12 2 p.m. to 6 p.m. 10 6 p.m. to 10 p.m. 15 BA 452 Lesson B.4 Binary Fixed Costs 32 Worker Covering Suppose operators work for eight consecutive hours. Formulate and solve the company’s problem of determining how many operators should be scheduled to begin working in each period in order to minimize the number of cashiers needed? (Hint: Workers can work from 6 p.m. to 2 a.m.) BA 452 Lesson B.4 Binary Fixed Costs 33 Worker Covering Answer: Define the decision variables TNP = the number of operators who begin working at 10 p.m. TWA = the number of operators who begin working at 2 a.m. SXA = the number of operators who begin working at 6 a.m. TNA = the number of operators who begin working at 10 a.m. TWP = the number of operators who begin working at 2 p.m. SXP = the number of operators who begin working at 6 p.m. Min TNP + TWA + SXA + TNA + TWP + SXP s.t.TNP + TWA > 4 TWA + SXA > 7 SXA + TNA > 12 TNA + TWP > 10 TWP + SXP > 15 SXP + TNP > 8, all variables > 0 BA 452 Lesson B.4 Binary Fixed Costs 34 Worker Covering BA 452 Lesson B.4 Binary Fixed Costs 35 Transportation with New Origins Transportation with New Origins BA 452 Lesson B.4 Binary Fixed Costs 36 Transportation with New Origins Transportation Problems with New Origins are Transportation Problems extended so that new origins may be added, at a fixed cost. Also called Distribution System Design Problems, they choose the best plant locations and determine how much to ship from each plant. The simplest way to model those fixed costs in a linear program is with binary (0 or 1) variables. For example, consider a Transportation Problem extended by allowing the possibility of developing a new origin. Suppose the fixed development cost is 5 and the new origin’s supply capacity is 7. That is a linear program with a supply of 7Y at the new origin and added cost term of 5Y, where Y = 0 indicates the new origin is not developed and Y = 1 indicates the new origin is developed. BA 452 Lesson B.4 Binary Fixed Costs 37 Transportation with New Origins Question: Linksys operates a plant to produce its wireless routers in St. Louis with an annual capacity of 30,000 units. Product is shipped to regional distribution centers in Boston, Atlanta, and Houston. Because of an anticipated increase in demand, Linksys plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas City. The estimated annual fixed cost and annual capacity for the four proposed plants are as follows: Proposed Plant Annual Fixed Cost Annual Capacity Detroit $175,000 10,000 Toledo $300,000 20,000 Denver $375,000 30,000 Kansas City $500,000 40,000 BA 452 Lesson B.4 Binary Fixed Costs 38 Transportation with New Origins The company’s long-range planning group forecasts of the anticipated annual demand at the distribution centers are as follows: Distribution Center Annual Demand Boston 30,000 Atlanta 20,000 Houston 20,000 BA 452 Lesson B.4 Binary Fixed Costs 39 Transportation with New Origins The shipping cost per unit from each plant to each distribution center is as follows: Plant\Distribution Boston Atlanta Houston Detroit $5 $2 $3 Toledo $4 $3 $4 Denver $9 $7 $5 Kansas City $10 $4 $2 St. Louis $8 $4 $3 Formulate and solve the problem of minimizing the cost of meeting all demands. BA 452 Lesson B.4 Binary Fixed Costs 40 Transportation with New Origins Answer: Define binary variables for plant construction, Y1 = 1 if a plan is constructed in Detroit; 0, if not Y2 = 1 if a plan is constructed in Toledo; 0, if not Y3 = 1 if a plan is constructed in Denver; 0, if not Y4 = 1 if a plan is constructed in Kansas City; 0, if not Define shipment variables just as in transportation problems, Xij = the units shipped (in thousands) from plant i (i = 1, 2, 3, 4, 5) to distribution center j (j = 1, 2, 3) each year. BA 452 Lesson B.4 Binary Fixed Costs 41 Transportation with New Origins The objective is minimize total cost. From cost data Plant\Distribution Boston Atlanta Houston Detroit $5 $2 $3 Toledo $4 $3 $4 Denver $9 $7 $5 Kansas City $10 $4 $2 St. Louis $8 $4 $3 shipping costs (in thousands of dollars) are 5X11 + 2X12 + 3X13+ 4X21 + 3X22 + 4X23 + 9X31 + 7X32 + 5X33 + 10X41 + 4X42 + 2X43 + 8X51 + 4X52 + 3X53 BA 452 Lesson B.4 Binary Fixed Costs 42 Transportation with New Origins From cost data Proposed Plant Annual Fixed Cost Annual Capacity Detroit $175,000 10,000 Toledo $300,000 20,000 Denver $375,000 30,000 Kansas City $500,000 40,000 plant construction costs (in thousands of dollars) are 175Y1 + 300Y2 + 375Y3 + 500Y4 Hence, the objective to minimize total costs is Min 5X11 + 2X12 + 3X13+ 4X21 + 3X22 + 4X23 + 9X31 + 7X32 + 5X33 + 10X41 + 4X42 + 2X43 + 8X51 + 4X52 + 3X53 + 175Y1 + 300Y2 + 375Y3 + 500Y4 BA 452 Lesson B.4 Binary Fixed Costs 43 Transportation with New Origins From capacity data Proposed Plant Annual Fixed Cost Annual Capacity Detroit $175,000 10,000 Toledo $300,000 20,000 Denver $375,000 30,000 Kansas City $500,000 40,000 Detroit capacity constraint is X11 + X12 + X13 < 10Y1 Toledo capacity constraint is X21 + X22 + X23 < 20Y2 Denver capacity constraint is X31 + X32 + X33 < 30Y3 Kansas City capacity constraint is X41 + X42 + X43 < 40Y4 And St. Louis capacity constraint is X51 + X52 + X53 < 30 BA 452 Lesson B.4 Binary Fixed Costs 44 Transportation with New Origins From demand data Distribution Center Annual Demand Boston 30,000 Atlanta 20,000 Houston 20,000 Boston demand constraint is X11 + X21 + X31 + X41 + X51 = 30 Atlanta demand constraint is X12 + X22 + X32 + X42 + X52 = 20 Houston demand constraint is X13 + X23 + X33 + X43 + X53 = 20 Non-negativity constraints complete the linear programming formulation. BA 452 Lesson B.4 Binary Fixed Costs 45 Transportation with New Origins From demand data Distribution Center Annual Demand Boston 30,000 Atlanta 20,000 Houston 20,000 Boston demand constraint is X11 + X21 + X31 + X41 + X51 = 30 Atlanta demand constraint is X12 + X22 + X32 + X42 + X52 = 20 Houston demand constraint is X13 + X23 + X33 + X43 + X53 = 20 Non-negativity constraints complete the linear programming formulation. BA 452 Lesson B.4 Binary Fixed Costs 46 Transportation with New Origins The Management Scientist solves this mixed integer linear program of 4 binary variables Yj and 15 continuous variables Xij, and 8 constraints. BA 452 Lesson B.4 Binary Fixed Costs 47 Transportation with New Origins BA 452 Lesson B.4 Binary Fixed Costs 48 Transportation with New Origins The Management Scientist solves this mixed integer linear program of 4 binary variables Yj and 15 continuous variables Xij, and 8 constraints. BA 452 Lesson B.4 Binary Fixed Costs 49 Transportation with New Origins All variables at the optimum are zero except: X42 = 20, X43 = 20, X51 = 30, and Y4 = 1. So, the Kansas City plant should be built; 20,000 units should be shipped from Kansas City to Atlanta; 20,000 units should be shipped from Kansas City to Houston; and 30,000 units should be shipped from St. Louis to Boston. BA 452 Lesson B.4 Binary Fixed Costs 50 Transshipment with New Nodes Transshipment with New Nodes BA 452 Lesson B.4 Binary Fixed Costs 51 Transshipment with New Nodes Overview Transshipment Problems with New Transshipment Nodes are Transshipment Problems extended so that new transshipment nodes may be added, at a fixed cost. They choose the best transshipment locations and determine how much to ship through each location. The simplest way to model those fixed costs in a linear program is with binary (0 or 1) variables. BA 452 Lesson B.4 Binary Fixed Costs 52 Transshipment with New Nodes Question: Zeron Industries supplies three firms (Zrox, Hewes, Rockrite) with customized shelving for its offices. Zeron orders shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc. Currently, weekly demands by the users are 50 for Zrox, 60 for Hewes, and 40 for Rockrite. Both Arnold and Supershelf can supply up to 75 units to its customers. Zeron currently ships from its Northside facilities, but it can develop Southside facilities for a fixed cost of 8. Unit costs from the manufacturers to the suppliers are: Zeron N Zeron S Arnold 5 8 Supershelf 7 4 The costs to install the shelving at the various locations are: Zrox Hewes Rockrite Zeron N 1 5 8 Zeron S 3 4 4 BA 452 Lesson B.4 Binary Fixed Costs 53 Transshipment with New Nodes 75 ARNOLD Arnold 5 Zeron N 8 75 4 50 Hewes HEWES 60 RockRite 40 1 5 8 3 7 Super Shelf Zrox Zeron WASH BURN S 4 4 BA 452 Lesson B.4 Binary Fixed Costs 54 Transshipment with New Nodes Define decision variables: xij = amount shipped from manufacturer i to supplier j xjk = amount shipped from supplier j to customer k where i = 1 (Arnold), 2 (Supershelf) j = 3 (Zeron N), 4 (Zeron S) k = 5 (Zrox), 6 (Hewes), 7 (Rockrite) y = 1 if Zeron S is developed, y = 0 if Zeron S is not developed BA 452 Lesson B.4 Binary Fixed Costs 55 Transshipment with New Nodes Define objective function: Minimize total shipping costs plus setup costs. Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37 + 3x45 + 4x46 + 4x47 + 8y Constrain amount out of Arnold: x13 + x14 < 75 Constrain amount out of Supershelf: x23 + x24 < 75 Constrain amount through Zeron N: x13 + x23 - x35 - x36 - x37 = 0 Constrain amount through Zeron S: x14 + x24 - x45 - x46 - x47 = 0 Constrain amount into Zrox: x35 + x45 = 50 Constrain amount into Hewes: x36 + x46 = 60 Constrain amount into Rockrite: x37 + x47 = 40 Setup indicator for Zeron S: x45 + x46 + x47 < 150y (The first 4 constraints imply x45 + x46 + x47 < 150, so the setup indicator constraint “x45 + x46 + x47 < 150y” means, at an optimum, y = 1 if any material x45 or x46 or x47 transships through Zeron S, and y = 0 if no material x45 or x46 or x47 transships through Zeron S.) BA 452 Lesson B.4 Binary Fixed Costs 56 BA 452 Quantitative Analysis End of Lesson B.4 BA 452 Lesson B.4 Binary Fixed Costs 57