set CANDIDATE; set ZONE; param revenue{candidate, ZONE} >= 0; var Assign{CANDIDATE, ZONE} binary;

Transcription

1 Assignment Problem Data Zone j = Candidate i = c ij = annual sales volume in $million if candidate i is assigned to zone j 1

2 6! = ! = ! = A size-24 assignment problem would require 1.1 billion centuries to enumerate at 1 billion assignments per second. The greedy solution is pessimal, with value $91 million. The optimal solution is (1,4), (2,5), (3,6), (4,1), (5,3), (6,2), with value $124 million. The solution on page 13 has a value of $121 million. 1-1

3 Assignment Problem Formulation Let c ij be as above. Let x ij = { 1 if candidate i is assigned to zone j 0 otherwise. n n Maximize c ij x ij i=1 j=1 subject to n x ij = 1 for j = 1,...,n i=1 n x ij = 1 for i = 1,...,n j=1 x ij = 0 or 1 for i = 1,...,n, j = 1,...,n 2

4 Assignment Problem Formulation (AMPL) set CANDIDATE; set ZONE; param revenue{candidate, ZONE} >= 0; var Assign{CANDIDATE, ZONE} binary; maximize Total_revenue: sum{i in CANDIDATE, j in ZONE} revenue[i, j] * Assign[i, j]; subject to Zone_limit{i in CANDIDATE}: sum{j in ZONE} Assign[i, j] = 1; subject to Cand_limit{j in ZONE}: sum{i in CANDIDATE} Assign[i, j] = 1; set CANDIDATE := Ann Bob Carol Debbie Ed Frank; set ZONE := NE NC NW SE SC SW; param revenue: NE NC NW SE SC SW := Ann Bob Carol Debbie Ed Frank ; 3

5 Components of an Optimization Model A set or sets of objects, and possibly derived sets of relations between objects. Attributes of the elements of the sets, some parameters and some decision variables. All attributes have units associated with them. Each attribute is stored as a vector, indexed by the elements of the associated set. An objective function to be maximized or minimized by selecting values for the decision variables. Constraints on the relationships among values of the decision variables. Values of the parameters. 4

6 A Product Mix Problem A bookcase requires three hours of work, one unit of metal, and four units of wood, and it brings in a net prot of $19. A desk requires two hours of work, one unit of metal, and three units of wood, and it brings in a net prot of $13. A chair requires one hour of work, one unit of metal, and three units of wood, and it brings in a net prot of $12. A bedframe requires two hours of work, one unit of metal, and three units of wood, and it brings in a net prot of $17. Only 225 hours of labor, 117 units of metal, and 420 units of wood are available per day. 5

7 Mathematical Program For parameters b i (and possibly other parameters hidden in the denitions of f and g i ) and decision variables x j : Min (or Max) f(x1,x2,...,x n ) subject to g i (x1,x2,...,x n ) = b i i= 1,2,...,m 6

8 Linearity Assumptions Divisibility Allowable values of a decision variable x j lie in some interval of real numbers l j x j u j (where l j = and u j = are allowed). Proportionality A function f(x1,x2,...,x n ) has the property that f(cx1,cx2,...,cx n ) = cf(x1,x2,...,x n ) for any constant c. Additivity A function f(x1,x2,...,x n ) has the property that f(x1 + y1,x2 + y2,...,x n + y n ) = f(x1,x2,...,x n ) + f(y1,y2,...,y n ) for any values (x1,x2,...,x n ) and (y1,y2,...,y n ) of the decision variables. A function function f that satises additivity and proportionality is called linear. (Note that divisibility is a property of the decision variables, not the function.) A linear function can always be written in the form f(x1,x2,...,x n ) = c1x1 + c2x2 + + c n x n. where c1,c2,...,c n are constants. A mathematical program is called a linear program if the objective function f and all constraint functions g i are linear and all decision variables are divisible. 7

9 Linear Program For parameters c j, a ij, and b i, and decision variables x j : Min (or Max) subject to n c j x j n a ij x j j=1 j=1 = b i x j 0 j = 1,2,...,n i = 1, 2,...m 8

10 Blending Problem Blend silicon and nitrogen to produce two types of fertilizer. Fertilizer 1 must be at least 40% nitrogen and sells for $70/lb. Fertilizer 2 must be at least 70% silicon and sells for $40/lb. Can purchase up to 80 lb of nitrogen at $15/lb and up to 100 lb of silicon at $10/lb. Assuming all product can be sold, maximize prot. 9

11 Diet Problem Four foods (brownies, chocolate ice cream, cola, and cheesecake). Brownies cost $.50 each, provide 400 Calories, 3 oz of chocolate, 2 oz of sugar, and 2 ounces of fat. Ice cream costs $.20 per scoop, and a scoop provides 200 Calories, 2 oz of chocolate, 2 oz of sugar, and 4 ounces of fat. Cola costs $.30 per glass, and a glass provides 150 Calories and 4 oz of sugar. Cheesecake costs $.80 per slice, and provides 500 Calories, 4 ounces of sugar, and 5 ounces of fat. My diet requires 1000 Calories, 6 oz of chocolate, 10 oz of sugar, and 8 oz of fat. Find a diet that meets my nutritional needs at the lowest cost. 10

12 Transportation Problem Power plants 1, 2, and 3 supply 35 million kwh, 50 million kwh, and 40 million kwh, respectively. Cities 1, 2, 3, and 4 require 45, 20, 30, and 30 million kwh, respectively. Cost of delivery ($ per million kwh) depends on origin and destination. City Plant

13 Multiperiod Production Planning A dynamic model takes account of changing conditions over time. A static model applies to a single period or moment. Demand in coming quarters: 40 boats in rst, 60 in second, 75 in third, 25 in fourth. Demand must be met on time. Starting inventory of 10 boats. In any quarter, boats can be produced to sell or produced to inventory. Inventory carrying charge is $20 per boat per period. Production on regular time is $400 per boat, and on overtime is $450 per boat. Regular-time capacity is 40 boats per period, but unlimited overtime is available. How many small sailboats in each quarter to meet demand at minimum cost? 12

14 Reporting Solutions to Applied Problems 1. A brief \executive summary" reporting your recommended course of action and any implications, in the language of the original problem, answering all questions raised in the problem statement. 2. The mathematical model, including The symbol dictionary, including units associated with all symbols; The mathematical formulation, including objective and constraints, with explanations. Any tables showing parameter values. 3. The AMPL model (with judicious use of names and comments). 4. AMPL reports to support your recommendations. 13