UECM2043, UECM2093 Operations Research Tutorial 2 1. Consider the following LP: Maximize = 21 + 32 subject to + 32 ≤ 6 31 + 22 ≤ 6 1 , 2 ≥ 0 1
a) Express the problem in equation form. b) Determine all the basic solutions of the problem, and classify them as feasible and infeasible. c) Use direct substitution in the objective function to determine the optimum basic feasible solution. d) Verify graphically that the solution obtained in (c) is the optimum LP solution. Hence, conclude that the optimum solution can be determined algebraically by considering the basic feasible solutions only. e) Show how the infeasible basic solutions are represented on the graphical solution space 2. Consider the following LP with two variables: Max = 81 + 42 Subject to 21 + 2 ≤ 4 1 + 22 ≤ 5 1 , 2 ≥ 0 Consider the graphical solution of the problem. Identify the path of the simplex method and the basic and nonbasic variables that define this path. 3. Consider the following set of constraints: 1 + 22 + 23 + 44 ≤ 40 21 − 2 + 3 + 24 ≤ 8 41 − 22 + 3 − 4 ≤ 10 1 , 2 , 3 , 4 ≥ 0 Solve the problem for each of the following objective functions. (a) Maximize = 21 + 2 − 33 + 54 . (b) Minimize = 51 − 42 + 63 − 84 . 4. The following tableau represents a specific simplex iteration. All variables are nonnegative. The tableau is not optimal for either a maximization or a minimization problem. Thus, when a nonbasic variable enters the solution it can either increase or 1
decrease z or leave it unchanged, depending on the parameters of the entering nonbasic variable. Basic z x8 x3 x1
x1 0 0 0 1
x2 -5 3 1 -1
x3 0 0 1 0
x4 4 -2 3 0
x5 -1 -3 1 6
x6 -10 -1 0 -4
x7 0 5 3 0
x8 0 1 0 0
Solution 620 12 6 0
a) Categorize the variables as basic and nonbasic and provide the current values of all the variables. b) Assuming that the problem is of the maximization type, identify the nonbasic variables that have the potential to improve the value of z. If each such variable enters the basic solution, determine the associated leaving variable, if any, and the associated change in z. Do not use the Gauss-Jordan row operations. c) Repeat part (b) assuming that the problem is of the minimization type. d) Which nonbasic variable(s) will not cause a change in the value of Z when selected to enter the solution? 5. The Gutchi Company manufactures purses, shaving bags, and backpacks. The construction includes leather and synthetics, leather being the scarce raw material. The production process requires two types of skilled labor: sewing and finishing. The following table gives the availability of the resources, their usage by the three products, and the profits per unit.
Resource Leather (ft2) Sewing (hr) Finishing (hr) Selling price ($)
Resource requirement per unit Purse Bag Backpack 2 1 3 2 1 2 1 0.5 1 24 22 45
Daily availability 42 ft2 40 hr 45 hr
Formulate the problem as a linear program and find the optimum solution by using the simplex method. 6. Use the Big M method to solve the following LP: Maximize = 21 + 32 − 53 Subject to 1 + 2 + 3 = 7 21 − 52 + 3 ≥ 10 1 , 2 , 3 ≥ 0
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7. Use the Big M method to solve the following LP: Minimize = 41 − 82 + 33 Subject to 1 + 2 + 3 = 7 21 − 52 + 3 ≥ 10 1 , 2 , 3 ≥ 0
8. Consider the following LP: Maximize = 31 + 22 subject to 41 − 2 ≤ 4 41 + 32 ≤ 6 41 + 2 ≤ 4 1 , 2 ≥ 0 (a) Show that the associated simplex iterations are temporarily degenerate. (b) Verify the result by solving the problem graphically. 9. For the following LP, identify three alternative optimal basic solutions, and then write a general expression for all the nonbasic alternative optima comprising these three basic solutions. Maximize = 1 + 22 + 33 subject to 1 + 22 + 33 ≤ 10 1 + 2 ≤5 1 ≤1 1 , 2 , 3 ≥ 0 10. Consider the LP: Maximize = 201 + 102 +
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subject to 31 − 32 + 53 ≤ 50 1 + 3 ≤ 10 1 − 2 + 43 ≤ 20 1 , 2 , 3 ≥ 0 (a) By inspecting the constraints, determine the direction (1 , 2 , 3 ) in which the solution space is unbounded. (b) Without further computations, what can you conclude regarding the optimum objective value?
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11. Consider the LP model Maximize = 31 + 22 + 33 subject to 21 + 2 + 3 ≤ 2 31 + 43 + 23 ≥ 8 1 , 2 , 3 ≥ 0 Show that the optimal solution includes an artificial basic variable at zero level. Does the problem have a feasible optimal solution? 12. The NW Shopping Mall conducts special events to attract potential patrons. Among the events that seem to aUract teenagers, the young/middle-aged group, and senior citizens, the two most popular are band concerts and art shows. Their costs per presentation are $1500 and $3000, respectively. The total (strict) annual budget allocated to the two events is $15,000. The mall manager estimates the attendance as follows: Number attending per presentation Teenagers Young/middle age Seniors 200 100 0 0 400 250
Event Band concert Art show
The manager has set minimum goals of 1000, 1200, and 800 for the attendance of teenagers, the young/middle-aged group, and seniors, respectively. Formulate the problem as a goal programming model. Suppose that the goal of attracting young/middle-aged people is twice as important as for either of the other two categories (teens and seniors). Find the associated solution, and check if all the goals have been met. 13. Camyo Manufacturing produces four parts that require the use of a lathe and a drill press. The two machines operate 10 hours a day. The following table provides the time in minutes required by each part:
Part 1 2 3 4
Production time in min Lathe Drill press 5 3 6 2 4 6 7 4
It is desired to balance the two machines by limiting the difference between their total operation times to at most 30 minutes. The market demand for each part is at least 10 units. Additionally, the number of units of part 1 may not exceed that of part 2. Formulate the problem as a goal programming model. Suppose that the market demand goal is twice as important as that of balancing the two machines, and that no overtime is allowed. Solve the problem, and determine if the goals are met. 4
