Late homework will be corrected but will receive no credit.
p.146 (section
4.4) #4.1, 4.2, 4.6, 4.11, 4.12
p.154 (section 4.6) #4.16, 4.17, 4.19
No additional written work. - Computer solution for Project due 11/20
These exercises should help to focus your study on the most recent material - they will not be collected.
p.131 #3.23,
# 3.25 (3.25 should say "in exercise 3.14") - but only 3.14g -- you have already solved this part of 3.14 and
have initial and final tableaus - no simplex calculation is required for
exercise 3.25(3.14g) .
p. 136 #3.28,
p.44#1.4:[Use the "penalty method" ("Big M" method) solution - available in Exercise
4.1 solution - no further simplex work is required for this problem]
What happens to the total shipping cost if the demand at Retail outlet 2 increases to 130 units?
What if it increases to 160 units (Can we give an exact amount for the change?)?
If we insist on shipping 20 units from center A to retail outlet 1, what will this do to the total cost?
[Notice the difference from last week's assignment, which asked -"Can we predict the effects?" - These questions ask you to actually do the prediction - but only on the value of the objective, not the actual solution- if it is possible].
1. p.110 # 3.13d - Show standard form (done for previous HW) & solve with SIMPLEX - hand in initial & final tableaus,and write optimal solution and value[remember it's a "maximize"]
2. p.124 #3.20 (You need the setup and initial and final tableaus – use SIMPLEX to calculate).
a.) Use the inital and final tableaus to identify [no calculations are needed for these] B , D, C(B)^T, C(D)^T, B^(-1), B^(-1)*D, C(B)^T*B^(-1)*D, B^(-1)*b , C(B)^T*B^(-1)*b
b.) Use these calculations (as in activity 17) to answer the questions in 3.20 [Maple is likely to be helpful for algebraic work - solution of systems etc.]. Actually carry out the calculations - check your results using the sensitivity analysis printed by SIMPLEX..3. p. 115 #3.14 b (you have already solved 3.14b, using Maple)
a. Solve with SIMPLEX. [show first and last tableaus of phase 1, first and last tableaus of phase 2] – Give the solution
b. Use the tableaus (no further calculation) to identify B1, B2, B1^(-1), B2^(-1), B , D, C(B)^T, C(D)^T, B^(-1), B^(-1) * D, C(B)^T * B^(-1) *D, B^(-1) * b , C(B)^T * B^(-1) * b and use these to calculate B^(-1) (= B2^(-1) * B1^(-1).
c. Use these to answer the following. Note that the sensitivity analysis printed by SIMPLEX will not be correct because of the rewriting of the constraints based on phase I, but the calculations based on B and B^(-1)(as in activity 17, and as in ex 2 above will be correct - but the calculations based on the overall B and B^(-1) will work.
Do not re-solve the problem If a question cannot be answered without re-solving the problem, then stating that fact is your answer.i. how does the optimal solution change if the coefficient on x3 in the (original - maximize) objective function is changed from 3 to 6?
ii. How low would the coefficient on x1 have to go (original - maximize - objective) to drive x1 out of the basis?
iii. What happens to the optimal solution if the right-hand side of the first constraint changes from 6 to 12?
iv. What happens if the coefficient of x3 in the first constraint is changed from 2 to 5 ?
v. What happens if a new constraint is added, requiring that 2x1 – x3 ≤ 13?4. p.40 #1.4 [You set this up for HW 2 - posted solutions show a setup]
Here's how we can make the sensitivity analysis in simplex work even when we need artificial variables. To work with the simplex algortithm (and SIMPLEX program) you will need to add artificial variables X15 – X19 to constraints 1, 4 – 7. Add the artificial variables to the objective function with very large coefficients (500 should do - if the simplex algorithm does not drive them out of the basis, try again with larger coefficients) - This is often called the "Big M" method of dealing with artificial variables. Notice that you have to leave the 450 (of the objective function) out of the calculations and stick it back on in the answers.a.) Solve the problem with SIMPLEX
b.) answer the following questions, using the sensitivity analysis from simplex. Do not re-solve the problem – if the answer would require re-solving, say so]i. What happens to the optimal shipping plan and cost if the number of units available at Center A is only 125 (rather than 150)?
ii. How high would the (unit) cost of shipping from Center B to Outlet 2 have to go to cause a change in the basis [and drive B-2 out of the basis]? [As usual, assume no other changes take place at the same time]
iii. If the unit cost of shipping from Center C to outlet drops to $56 [be careful - the coefficient in the object isn't simply the shipping cost], how will this affect the optimal shipping plan and cost?
iv If no more than 50 units can be stored at B (adding a constraint X5 + X6 + X7 + X8 ≥ 100) what happens to the optimal shipping plan and cost?
p.110 #3.13 d For each step (each tableau) identify the Basis matrix B (identify basis variables - in order), the coefficients vector C(subB), the "altered" vector C(subB)(Trans)B(inv)A – C(Trans) and the value of the objective function (std form) C(subB) (Trans) B(inv) b. For the final tableau (optimal basic feasible solution) also give the matrices D and B(inv), and the vector C(subB)(Trans)B(inv)D - C(subD)(Trans)
p.115 #3.14b,g 3.15, 3.16 (you found standard form in HW 2) Use Maple for matrix calculations - show the tableau at each step [probably copy/paste?] (don't copy out the arithmetic)
p 91 #3.1, 3.2, with the algebraic search method from Activity 12 10/7 [You already have the standard form of these - Probably want to use Maple for matrix manipulations]
p.91 #3.1, 3.2 p.110 #3.13a, b, f All with the simplex algorithm [Will want Maple for matrix manipulation]. Notice how the simplex algorithm produces the same steps as the search method in solving 3.1 and 3.2. Watch out in 3.13 a and b - Don't forget the pivot operation to clear out the bottom row below the pivot 1's before looking for entering or departing variables.
No written work due Recall miniproject is due 10/9
p.91 #3.1, 3.2, 3.3, 3.5 p.100 #3.6-3.10
p.83 #2.2(an object is in Aintersect B if and only if it is in A and is in B), 2.3, 2.5bcd (read "y" as "x2"), 2.8, 2.12, 2.13bcfhi, 2.18, 2.19 (Show that each extreme point of S1 is a point of S)
1.) p.39 #1.1-1.6 - Carry out steps 1-5 (the set-up, including identification of assumptions and definition of variables, conversion to standard form) of the Math Programming procedure on p 32 of the text. set-up dicsussed Tuesday, standard form on Thursday (9/11)
2.) p. 66 #1.51-1.53
No additional written work due on this date
In the text: p.3 #2, 3 ; p.9 #7, 9; p.18 # 15, 16bc; p.22 #19, 22; p.24 #24, 27
Complete the journal entries for 8/26 and 8/28
Complete (as a team) the exercises and Crticial Thinking questions from Activities 2 and 3 - hand in a team copy.