Brief summary of linear programming and duality: Consider the linear program in standard form. (P ) min z = cx. x 0. (D) max yb. z = c B x B + c N x N

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1 Brief summary of linear programming and duality: Consider the linear program in standard form (P ) min z = cx s.t. Ax = b x 0 where A R m n, c R 1 n, x R n 1, b R m 1,and its dual (D) max yb s.t. ya c. By letting x = (x B, x N ), x B R m, x N R n m (a partition) we obtain the equivalent systems to (P) min s.t. z = c B x B + c N x N Bx B + Nx N = b x B, x N 0 Supposing that B is nonsingular, we obtain x B = B 1 b B 1 Nx N. By expressing the equation system w.r.t. x N we get min z = c B B 1 b + (c N c B B 1 N)x N (1) s.t. x B = B 1 b B 1 Nx N (2) x B, x N 0 (3) Then the simplex tableau is written as follows: BV z x B x N RHS z 1 0 c B B 1 N c N c B B 1 b x B 0 I B 1 N B 1 b We define the reduced costs according to (1), i.e. as c N := c B B 1 N c N. We generalize the definition to x as c := c B B 1 A c. Observe that c B = 0. Remember that any vertex of the feasible polyhedron can be identified by letting x N = 0 for some N. For x N = 0, we have a solution x = (B 1 b, 0), of value z = c B B 1 b. Such solution is feasible and optimal for the primal if and only if B 1 b 0 (4) c N = c N c B B 1 N 0 (5) 1

2 For a given B we can find: 1- Quantity of objective function:c B B 1 b 2- Quantity of basic variables: B 1 b 3- Reduced cost: c j := c j c B B 1 a j, for j N 4- Coefficients of x j in column j : B 1 a j, for j N Theorem If B is feasible and optimal for the primal, then y = c B B 1 is a feasible and optimal for the dual. Example1: Consider the following linear programming: By adding the slack variables: (P ) max z = 3x 1 + 5x 2 s.t. x 1 4 2x x 1 + 2x 2 18 x 1, x 2 0 (P ) max z = 3x 1 + 5x 2 s.t. x 1 + s 1 = 4 2x 2 + s 2 = 12 3x 1 + 2x 2 + s 3 = 18 x 1, x 2, s 1, s 2, s 3 0 The optimal simplex tableau is written as follows: BV z x 1 x 2 s 1 s 2 s 3 RHS z / s /3 1/3 2 x /2 0 6 x /3 1/3 2 s 1,x 2 and x 1 are basic variables; therefore,b and B 1 respectively are: /3-1/3 0 1/ /3 1/3 2

3 Sensitivity Analysis: Suppose you solve a linear program by hand ending up with an optimal table (or tableau to use the technical term). You know what an optimal tableau looks like: it has all non-negative values in Row 0 (which we will often refer to as the cost row), all non-negative right-hand-side values, and a basis (identity matrix) embedded. To determine the effect of a change in the data, I will try to determine how that change effected the final tableau, and try to reform the final tableau accordingly;therefore,sensitivity analysis allows us to determine how sensitive the optimal solution is to changes in data values. we analyzing changes in: 1-An Objective Function Coefficient (OFC) 2-A Right Hand Side (RHS) value of a constraint consider the example1 and its optimal tableau: 1-Suppose the cost for x 1 is changed to 4 in the original formulation, from its previous value 3.How this change affected the optimal tableau? a) c j := c B B 1 a j c j c 1 := c B B 1 a 1 c 1, c 1 := (0, 5, 4)B 1 (1, 0, 3) t 4 = 0 c 2 := c B B 1 a 2 c 2, c 2 := (0, 5, 4)B 1 (0, 2, 2) t 5 = 0 c 3 := c B B 1 a 3 c 3, c 3 := (0, 5, 4)B 1 (1, 0, 0) t 0 = 0 c 4 := c B B 1 a 4 c 4, c 4 := (0, 5, 4)B 1 (0, 1, 0) t 0 = 7/6 c 5 := c B B 1 a 5 c 5, c 5 := (0, 5, 4)B 1 (0, 0, 1) t 0 = 4/3 b) z := c B B 1 b z := (0, 5, 4)B 1 (4, 12, 18) t = 38 c) y = c B B 1 y = (0, 5, 4)B 1 = (0, 7/6, 4/3) 2-Suppose the cost for x 1 is changed to 3 + in the original formulation, from its previous value 3.How this change affected the optimal tableau? a) c j := c B B 1 a j c j c 4 := c B B 1 a 4 c 4 0 = c 4 := (0, 5, 3 + )B 1 (0, 1, 0) t 0 0 = 3/2 1/3 0 = 9/2 c 5 := c B B 1 a 5 c 5 0 = (0, 5, 3 + )B 1 (0, 0, 1) t 0 0 = 3 Therefore for any 3 9/2 the tableau remains optimal. Is there any effect on the optimality? Is there any effect on the feasible region? 3

4 3-Suppose the right hand side is changed to (3, 8, 20) t in the original formulation, from (4, 12, 18) t.how this change affected the optimal tableau? a) z := c B B 1 b z := (0, 5, 3)B 1 (3, 8, 20) t = 32 b) x B := (s 1, x 2, x 1 ) t = B 1 b = (s 1, x 2, x 1 ) t = B 1 (3, 8, 20) t = ( 1, 4, 4) t 4-Suppose the right hand side is changed to (4, 12 2, 18 4 ) t in the original formulation, from (4, 12, 18) t.how this change affected the optimal tableau? a) x B := (s 1, x 2, x 1 ) t = B 1 b 0 = (s 1, x 2, x 1 ) t = B 1 (4, 12 2, 18 4 ) t 0 = (2 1/3, 6, 2 2/3 ) t 0 = 3 Therefore for any 3 9/2 the tableau remains feasible. Is there any effect on the optimality? Is there any effect on the feasible region? Find: 1- z/ b :What is the shadow price for the third constraint? Interpret its value for management. 2- Dual problem of the example1 3- Relationship between optimal objective functions in primal and dual 4

5 Example2: A company has to determine the best number of three models of a product to produce in order to maximize profits. The models are, an economy model, a standard model, and a deluxe model. Constraints include production capacity limitations (time available in minutes) in each of three departments (cutting and dyeing, sewing, and inspection and packaging) as well as constraint that requires the production of at least 1000 economy models. The linear programming model is shown here: Maxz = 3x 1 + 5x x 3 s.t. 12x x 2 + 8x 3 18, 000 Cutting and dying 15x x x 3 18, 000 Sewing 3x 1 + 4x 2 + 2x 3 9, 000 Inspection and modeling x 1 1, 000 Economy model x 1, x 2, x 3 0 Solve the problem using cplex Solver. a) How many units of each model should be produced to maximize the total profit contribution? b) Which constraints are binding? c) Interpret slack and/or surplus in each constraint. d) Overtime rates in the sewing department are $ 12 per hour. Would you recommend that the company consider using overtime in that department? Explain. e) What is the shadow price for the fourth constraint? Interpret its value for management. f) Suppose that the profit contribution of the economy model is increased by $1. How do you expect the solution to change? What is the new value of the objective function (profit)? g) The profit contribution for the standard model is $5 per unit. How much would this profit contribution have to change to make it worthwhile to produce some units of standard model? 5

Worked Examples for Chapter 5

Worked Examples for Chapter 5 Example for Section 5.2 Construct the primal-dual table and the dual problem for the following linear programming model fitting our standard form. Maximize Z = 5 x 1 + 4 x