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operations research an introduction

Questions and Answers of Operations Research An Introduction

Consider the linear program max 3 z1 + z2 s.t. -2z1 + z2 … 2 z1 + z2 … 6 z1 … 4 z1, z2 Ú 0(a) Solve the problem graphically.(b) Add slacks z3, z4, and z5 to place the model in standard
Do Exercise 5-9 for the LP max 2 z1 + 5z2 s.t. 3z1 + 2z2 … 18 z1 … 5 z2 … 3 z1, z2 Ú 0
Consider the linear program max 10y1 + y2 s.t. 3y1 + 2y2 Ú 6 2y1 + 4y2 … 8 y1, y2 Ú 0(a) Solve the problem graphically. Be sure to identify all constraints, show contours of the objective,
Consider the standard-form linear program min x2 + x4 + x5 s.t. -2x1 + x2 + 2x4 = 7 x4 + x5 = 5 x2 + x3 - x4 = 3 x1,c, x5 Ú 0(a) Compute the basic solution corresponding to x1, x3, x4 basic, and
Do Exercise 5-12 on standard-form linear program max 5x1 - 10x2 s.t. 1x1 - 1x2 + 2x3 + 4x5 = 2 1x1 + 1x2 + 2x4 + x5 = 8 x1,c, x5 Ú 0 starting with x3 and x4 basic.
Do Exercise 5-12 on standard-form linear program min 2 x1 + 4x2 + 6x3 + 10x4 + 7x5 s.t. x1 + x4 = 6 x2 + x3 - x4 + 2x5 = 9 x1,c, x5 Ú 0 starting with x1 and x2 basic.
The following plot shows several feasible points in a linear program and contours of its objective function.Determine whether each of the following sequences of solutions could have been one followed
Construct the simplex dictionary form 5.28 corresponding to each of the following.(a) The model and basis shown in Exercise 5-7(b) The model and basis shown in Exercise 5-8
Rudimentary simplex Algorithm 5A is being applied to optimize a linear program with objective function min 3w1 + 11w2 - 8w3 Determine whether each of the following simplex directions for w4 leads to
Consider the linear program max 4y1 + 5y2 s.t. -y1 + y2 … 4 y1 - y2 … 10 y1, y2 Ú 0(a) Show graphically that the model is un bounded.(b) Add slacks y3 and y4 to place the model in standard
Do Exercise 5-18 for the LP min -10y1 + y2 s.t. -5y1 + 3y2 … 15 3y1 - 5y2 … 8 y1, y2 Ú 0
Setup each of the following to begin Phase I of two-phase simplex Algorithm 5B. Also indicate the basic variables of the initial Phase I solution.(a) max 2 w1 + w2 + 9w3 s.t. w1 + w2 … 18-2w1 + w3
Setup each of the models in Exercise 5-20 to begin a big-M solution using rudimentary simplex Algorithm 5A. Also indicate the basic variables of the initial solution.
Consider the linear program max 9 y1 + y2 s.t. -2y1 + y2 Ú 2 y2 … 1 y1, y2 Ú 0(a) Show graphically that the model is infeasible.(b) Add slacks and artificials y3,c, y5 to setup the model for
Do Exercise 5-22 for the linear program min 2 y1 + 8y2 s.t. y1 + y2 … 5 y2 Ú 6 y1, y2 Ú 0
Assuming that step size l 7 0 at every step, compute a finite bound on the number of iterations of Algorithm 5A for each of the following standard-form linear programs.(a) The model in Exercise
Rudimentary simplex Algorithm 5A is being applied to a standard-form linear program with variables x1,c, x5. Determine whether each of the following basic solutions is degenerate for the given basic
Return to Exercise 5-4 and consider adding additional constraint y2 … 4 to the original LP.(a) Repeat parts (a)-(c) of Exercise 5-4 with the extra constraint, and additional slack y5 included in
Do Exercise 5-26 on the LP of Exercise 5-5 with additional constraint y1 … 6 and focusing degeneracy parts on extreme point 1y1, y22 =16, 02.
Consider the linear program max x 1 + x2 s.t. x1 + x2 … 9-2x1 + x2 … 0 x1 - 2x2 … 0 x1, x2 Ú 0(a) Solve the problem graphically.(b) Add slacks x3,c, x5 to place the model in standard form.(c)
Do Exercise 5-28 for the LP max x 1 s.t. 6x1 + 3x2 … 18 12x1 - 3x2 … 0 x1, x2 Ú 0
Return to the LP of Exercise 5-7.(a) Compute the basis matrix inverse corresponding to the basic variables indicated.(b) Compute the corresponding pricing vector of 5.45 .(c) Without generating the
Do Exercise 5-30 for the LP of Exercise 5-8.
Consider applying revised simplex Algorithm 5C to the tabulated standard-form LP min c =x1 x2 x3 x4 x5 5 4 3 2 16 b A = 2 0 1 0 6 8 0 1 1 2 3 12 starting with x1 and x2 basic.(a) Determine the
Solve each of the following standard form linear programs by revised simplex Algorithm 5C, showing the basic inverse, the pricing vector, and update matrix E used at each iteration. Start from the
Suppose lower- and upper-bounded simplex Algorithm 5D is being applied to a problem with objective function max 3x1 - 4x2 + x3 - 4x4 + 10x5 3 main constraints, and bounds 0 … xj … 5 j = 1,c, 5
Consider the linear program min 5z1 + 6z2 s.t. z1 + z2 Ú 3 3z1 + 2z2 Ú 8 0 … z1 … 6 0 … z2 … 5(a) Solve the problem graphically.(b) Add slacks z3 and z4 to place the model in standard form
Do Exercise 5-35 on the LP max 6 z1 + 8z2 s.t. z1 + 3z2 … 10 z1 + z2 … 5 0 … z1 … 4 0 … z2 … 3 Start with original variable z1 nonbasic lowerbounded and z2 upper-bounded.
Solve each of the following standard form linear programs by lower- and upper-bounded simplex Algorithm 5D, showing the basic inverse, the pricing vector, and update matrix E used at each
As a result of a recent decision to stop production of toy guns that look too real, the Super Slayer Toy Company is planning to focus its production on two futuristic models: beta zappers and freeze
Eli Orchid can manufacture its newest pharmacutical product in any of three processes.One costs $14,000 per batch, requires 3 tons of one major ingredient and 1 ton of the other, and yields 2 tons of
Professor Proof is trying to arrange for the implementation in a computer program of his latest operations research algorithm. He can contract with any mix of three sources for help:unlimited hours
The NCAA is making plans for distributing tickets to the upcoming regional basketball championships. The up to 10,000 available seats will be divided between the media, the competing universities,
For each of the following constraint coefficient changes, determine whether the change would tighten or relax the feasible set, whether any implied change in the optimal value would be an increase or
Determine whether adding each of the following constraints to a mathematical program would tighten or relax the feasible set and whether any implied change in the optimal value would be an increase
For each of the following objective coefficient changes, determine whether any implied change in the optimal value would be an increase or a decrease and whether the rate of any such optimal value
Return to Super Slayer Exercise 6-1.(a) Assign dual variables to each main constraint of the formulation in part (a), and define their meanings and units of measurement.(b) Show and justify the
Do Exercise 6-8 for the problem of Exercise 6-2 using Table 6.7.
Do Exercise 6-8 for the problem of Exercise 6-3 using Table 6.8.
Do Exercise 6-8 for the problem of Exercise 6-4 using Table 6.9.
State the dual of each of the following LPs.(a) min 17x1 + 29x2 + x4 s.t. 2x1 + 3x2 + 2x3 + 3x4 … 40 4x1 + 4x2 + x4 Ú 10 3x3 - x4 = 0 x1,c, x4 Ú 0(b) min 44x1 - 3x2 + 15x3 + 56x4 s.t. x1 + x2 +
State (primal and dual) complementary slackness conditions for each LP in Exercise 6-12.
Each of the following LP has a finite optimal solution. State the corresponding dual, solve both primal and dual graphically, and verify that optimal objective function values are equal.(a) max 14x1
Compute the dual solution corresponding to each of the following basic sets in the standard-form LP max 6x1 + 1x2 + 21x3 - 54x4 - 8x5 s.t. 2x1 + 5x3 + 7x5 = 70+ 3x2 + 3x3 - 9x4 + 1x5 = 1 x1,c, x5 Ú
Each of the following is a linear program with no optimal solution. State the corresponding dual, solve both primal and dual graphically, and verify that whenever primal or dual is unbounded, the
For each of the following LPs and solution vectors, demonstrate that the given solution is feasible, and compute the bound it provides on the optimal objective function value of the corresponding
For each of the following, verify that the given formulation is the dual of the referenced primal in Exercise 6-17, demonstrate that the given solution is dual feasible, and compute the bound it
Demonstrate for each linear program in Exercise 6-17 that the dual of its dual is the primal.
Razorback Tailgate (RT) makes tents for football game parking lot cookouts at its two different facilities, with two processes that may be employed in each facility. All facilities and processes
As spring approaches, the campus grounds staff is preparing to buy 500 truckloads of new soil to add around buildings and in gulleys where winter has worn away the surface. Three sources are
Return to Exercise 6-1. Answer each of the following as well as possible from the results in Table 6.6.(a) Is the optimal solution sensitive to the exact value of the trimming hours available?At what
Return to Eli Orchid Exercise 6-2. Answer each of the following as well as possible from the results in Table 6.7.(a) What is the marginal cost of production(per ton of output)?(b) How much would it
Return to Exercise 6-3. Answer each of the following as well as possible from the results in Table 6.8.(a) What is the marginal cost per professional-equivalent hour of programming associated with
Return to NCAA ticket Exercise 6-4.Answer each of the following as well as possible from the results in Table 6.9.(a) What is the marginal cost to the NCAA of each seat guaranteed the media?(b)
Paper can be made from new wood pulp, from recycled office paper, or from recycled newsprint. New pulp costs $100 per ton, recycled office paper, $50 per ton, and recycled newsprint,$20 per ton. One
Silva and Sons Ltd. (SSL)2 is the largest coconut processor in Sri Lanka. SSL buys coconuts at 300 rupees per thousand to produce two grades (fancy and granule) of desiccated (dehydrated)coconut for
Tube Steel Incorporated (TSI) is optimizing production at its 4 hot mills. TSI makes 8 types of tubular products which are either solid or hollow and come in 4 diameters. The following two tables
Consider the primal linear program max 13z2 - 8z3 s.t. - 3z1 + z3 … 19 4z1 + 2z2 + 7z3 = 10 6z1 + 8z3 Ú 0 z1, z3 Ú 0(a) Formulate the corresponding dual in terms of variables v1, v2, v3.(b)
Return to the primal LPs of Exercise 6-17 and corresponding duals of Exercise 6-18.(a) State and justify all Karush-Kuhn-Tucker conditions for each pair of models.(b) Compute solution values for
Consider the standard-form linear program max 5x1 - 10x2 s.t. 1x1 - 1x2 + 2x3 + 4x5 = 2 1x1 + 1x2 + 2x4 + x5 = 8 x1, x2, x3, x4, x5 Ú 0(a) Taking x1 and x2 as basic, identify all elements of the
Return to the standard-form LP of Exercise 6-31, and do parts (a)–(d), this time using basis(x3, x4). Then(e) State all KKT conditions for your primal solution of part (b) and dual solution of part
Consider the linear program min 2x1 + 3x2 s.t. -2x1 + 3x2 Ú 6 3x1 + 2x2 Ú 12 x1, x2 Ú 0(a) Establish that subtracting nonnegative surplus variables x3 and x4 leads to the equivalent
Consider the linear program min 2x1 + 3x2 + 4x3 s.t. x1 + 2x2 + x3 Ú 3 2x1 - x2 + 3x3 Ú 4 x1, x2, x3 Ú 0(a) Use nonnegative surplus variables x4 and x5 to place the model in standard form.(b)
Consider the standard form linear program min 3x1 + 4x2 + 6x3 + 7x4 + x5 s.t. 2x1 - x2 + x3 + 6x4 - 5x5 - x6 = 6 x1 + x2 + 2x3 + x4 + 2x5 - x7 = 3 x1,c, x7 Ú 0(a) State the dual of this model using
Return to the standard-form LP of Exercise 6-33(a).(a) Solve the model by Dual Simplex Algorithm 6A starting from the all surplus basis (x3, x4). At each step, identify the basis matrix B, its
Return to the standard-form LP of Exercise 6-34(a).(a) Solve the model by Primal-Dual Simplex Algorithm 6B starting from dual solution v = 10, 02. At each major step, state the restricted primal, the
Consider the following linear program:max 3z1 + z2 s.t. -2z1 + z2 … 2 z1 + z2 … 6 z1 … 4 z1, z2 Ú 0 After converting to standard form, solution of the model via Rudimentary Simplex Algorithm
Consider the linear program max 2w1 + 3w2 s.t. 4w1 + 3w2 … 12 w2 … 2 w1, w2 Ú 0(a) Solve the problem graphically.(b) Determine the direction w of most rapid improvement in the objective
Do Exercise 7-1 for the LP min 9w1 + 1w2 s.t. 3w1 + 6w2 Ú 12 6w1 + 3w2 Ú 12 w1, w2 Ú 0 and point w102 = 13, 12.
The following plot shows several feasible points in a linear program and contours of its objective function.Determine whether each of the following sequences of solutions could have been one followed
Determine whether each of the following is an interior point solution to the standard-form LP constraints 4x1 + 1x3 = 13 5x1 + 5x2 = 15 x1, x2, x3 Ú 0(a) x = 13, 0, 12(b) x = 12, 1, 52(c) x = 11, 2,
Write all conditions that a feasible directionw must satisfy at any interior point solution to each of the following standard-form systems of LP constraints.(a) 2w1 + 3w2 - 3w3 = 5 4w1 - 1w2 + 1w3 =
Table 7.4 shows several constraint matrices A (or At) of standard-form LPs and the corresponding projection matrices P (or Pt). Use these results to compute the feasible direction for the specifed
Consider the standard-form LP min 14z1 + 3z2 + 5z3 s.t. 2z1 - z3 = 1 z1 + z2 = 1 z1, z2, z3 Ú 0(a) Determine the direction of most rapid objective function improvement at any solution z.(b) Compute
Do Exercise 7-7 for max 5z1 - 2z2 + 3z3 s.t. z1 + z3 = 4 2z2 = 12 z1, z2, z3 Ú 0
An interior point search using scaling has reached current solution x172 = 12, 5, 1, 92.Compute the affine scaled y that would correspond to each of the following x’s.(a) (1, 1, 1, 1)(b) (2, 1, 4,
Do Exercise 7-9 taking the listed vectors as scaled y’s and computing the corresponding x.
Consider the standard-form LP min 2x1 + 3x2 + 5x3 s.t. 2x1 + 5x2 + 3x3 = 12 x1, x2, x3 Ú 0 with current interior point solution x132 = 12, 1, 12.(a) Sketch the feasible space in a diagram like
Do Exercise 7-11 using the LP max 6x1 + 1x2 + 2x3 s.t. x1 + x2 + 5x3 = 18 x1, x2, x3 Ú 0 and current solution x132 = 16, 7, 12.
Return to the LP of Exercise 7-11.(a) Compute in both x and y space the next move direction that would be pursued by affine scaling Algorithm 7A.(b) Verify that the x of part (a) is both improving
Do Exercise 7-13 on the LP of Exercise 7-12.
Consider the standard-form LP min 10x1 + 1x2 s.t. x1 - x2 + 2x3 = 3 x2 - x3 = 2 x1, x2, x3 Ú 0(a) Show that x102 = 14, 3, 12 is an appropriate point to start affine scaling Algorithm 7A.(b) Derive
Do Exercise 7-15 using the LP max 6x1 + 8x2 + 10x3 s.t. 9x1 - 2x2 + 4x3 = 6 2x2 - 2x3 = -1 x1, x2, x3 Ú 0 and initial solution x102 = (29, 1, 32).
Suppose that Affine Scaling Algorithm 7A has reached current solution x1112 = 13, 1, 92.Determine whether each of the following next move directions would cause the algorithm to stop and why. If not,
Consider solving the following standard-form primal LP by Affine Scaling Algorithm 7A starting from solution x = 12, 3, 2, 3, 1>32.(a) Verify that the given solution is an appropriate point at
Consider the standard-form LP max 13w1 - 2w2 + w3 s.t. 3w1 + 6w2 + 4w3 = 12 w1, w2, w3 Ú 0(a) Sketch the feasible space in a plot like Figure 7.6, identify an optimal extreme point, and show w112 =
Do Exercise 7-19 for the LP min 2w1 + 5w2 - w3 s.t. w1 + 6w2 + 2w3 = 18 w1, w2, w3 Ú 0 and points w112 = 18, 1, 22, w122 = 10.01, 0.02, 8.9352.
Determine whether each of the following could be the first four barrier multiplier values m used in the outer loop of a barrier search Algorithm 7B.(a) 100,80,64,51.2(b) 100,200,100,800(c)
Assume that barrer Algorithm 7B has computed an appropriate move direction x for its standard-form LP and a maximum feasible step size lmax. For each of the original objective functions below,
Consider the LP min 4x1 - x2 + 2x3 s.t. 4x1 - 3x2 + 2x3 = 13 3x2 - x3 = 1 x1, x2, x3 Ú 0(a) Show that x102 = 13, 1, 22 is an appropriate point to start barrier Algorithm 7B.(b) Form the
Do Exercise 7-23 for the LP max -x1 + 3x2 + 8x3 s.t. x1 + 2x2 + x3 = 14 2x1 + x2 = 11 x1, x2, x3 Ú 0 at x102 = 15, 1, 72
Return to the LP and x102 of Exercise 7-15.(a) Show that x102 = 14, 3, 12 is an appropriate point to start barrier Algorithm 7B.(b) Compute the move direction x that would be pursued from x102 by
Do Exercise 7-25 for the LP and x102 of Exercise 7-16.
Return to the LP of Exercise 7-18 and consider solving by Primal-Dual Interior Algorithm 7C.(a) Derive the corresponding dual formulation over variables v and place it in standard form by adding
Do Exercise 7-27 on the LPwith starting solutions x = 12, 1, 5, 3, 12 and v = 12, 52 x1 x2 X3 X4 x5 min 14 30 11 9 10 RHS s.t. 2 -1 0 3 -2 10 1 5 2 -1 1 15
Return to the standard form LP instance of Exercise 7-18.(a) Explain why this LP model is an instance of problem (LP) defined as min a nj = 1 cj xj s.t. a nj = 1 aij xj = bi i = 1,c, m xj Ú 0, j =
Repeat Exercise 7–29 for the LP instance of(7.29).
Consider the 2-variable LP instance in Figure 7.9 and equation (7.29).(a) State the instance in the (LP) standard form format above, fixing a = 1>4.(b) Construct the basic solution corresponding to
The sketch that follows shows the District 88 river system in the southwestern United States.All water arises at mountain reservoir R1. The estimated flow is 294 million acre-feet. At least 24
A semiconductor manufacturer has three different types of silicon wafers in stock to manufacture its three varieties of computer chips. Some wafer types cannot be used for some chips, but there are

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