Finding critical path and expected project completion time


Question 1

Melissa Bakery is preparing for the coming thanksgiving festival. Bakery plans to bake and sell its favourite cookies; butter cookies, chocolate cookies and almond cookies. A kilogram of butter cookies needs three cups of flour, one cup each of special ingredient and choc chip. A cup of special ingredient is added to five cups of flour together with three cups of choc chic to bake a kilogram of chocolate cookies. For baking a kilogram of almond cookies; Melissa needs four cups of flour, a cup of special ingredient and two cups of choc chip. Though, each day the bakery could only allocate at most 400 cups of flour, 100 cups of special ingredient and 210 cups of choc chip to bake the cookies. Melissa calculates a daily profit of RM10 for butter cookies, RM20 for chocolate cookies and RM15 for almond cookies. The bakery wants to maximize the daily profit.

a. Formulate the given problem as a linear programming problem.    
b. The following is the final simplex tableau for the above problem:

Cj                        10       20     15     0           0       0   
     Solution Mix       x1       x2     x3     S1        S2     S3      Quantity
10    x1                 1         0     1/2    3/4         0       0         37.5
0      S2                 0      -2/3     0     1/2         0      -1         5
20    x2                  0       1      1/2    1/4         0       1/3        57.5
        Zj                 10       20    15     25/2      20/3    0            m
    Cj-Zj                 0         0        0     -25/2    -20/3   0

i. Set up the initial simplex tableau for the above problem             

ii. How many kilograms of each cookie must be baked?               

iii. Find out the value of m                         

iv. Identify any ingredient which is not fully utilized. State the amount unused.

v. How will the optimum solution change if the RHS value for the first resource increases by 10 units?

Question 2

The Maju Supermarket stocks Munchies Cereal. Demand for Munchies is 4,000 boxes per year and the super market is open throughout the year. Each box costs $4 and it costs the store $60 per order of Munchies, and it costs $0.80 per box per year to keep the cereal in stock. Once an order for Munchies is placed, it takes 4 days to receive the order from a food distributor.

a. Determine the optimal order quantity.               

b. Compute the total inventory cost associated with the optimal order quantity.

c. What is the reorder point?   

d. What is the cycle time?                   

Question 3

a. A company has three factories A, B and C thast supply units to warehouses X, Y and Z every month. The capacities of the factories are 60, 70 and 80 units at A, B and C respectively. The needs of X, Y and Z per month are 50, 80 and 80 units respectively. Transportation costs per unit in ringgits are given in the following table. How many units must ship from each factory so that total cost is minimum? Use VAM method for initial solution and Stepping Stone method to get optimal solution.

                       Warehouses
Factories        X    Y    Z

A                   8    7    5
B                   6    8    9
C                   9    6    5

b. The Dean of the Faculty of Science at City Science University has decided to apply the Hungarian method in assigning lecturers to courses for the next semester. As a criterion for judging who must teach each course, the Dean reviews the past two years’ teaching evaluations (which were filled out by students).  As each of the four lecturers taught each of the four courses at one time or another during the two-year period, the Dean is able to record a course rating for each lecturer. These ratings are shown in the table below. Determine the best assignment of lecturers to courses to maximize the overall teaching rating.

Lecturer               Biology    Chemistry    Physics    Applied Sciences
 Nora Bee               90              65             95            40
Lee Along               70              60             80            75
Rama  Sundar         85              40             80            60
Charles Abby          55               80            65            55

Question 4

The project of building a backyard swimming pool consists of eight major activities and has to be finished within 19 weeks. Activities and related data are given in the following table:

Activity    Immediate predecessor    Activity time (weeks)
A                            -                                  3
B                            -                                  6
C                            A                                 2
D                           B,C                               5
E                           D                                  4
F                           E                                   3
G                          B,C                                9
H                          F,G                                 3

a. Draw a network diagram for this problem.           
b. Find out the critical path and the expected project completion time.

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