This part requires students to solve using a linear-programming problem/an integer-programming problem/goal programming problem.
There is an objective function that has to be maximised or minimised, subject to a set of constraints. Students should clearly define the decision variables, define and explain the objective function and define and explain each of the constraints. In addition, the solutions must be clearly and fully explained.
In order that students and groups tackle unique problems, students are asked to consider their student id numbers. All ids are made up of seven digits of the form 3664255. The particular digits in your and your partner ids will appear at various places below. If, for example, your id is 3664255 and you see a reference to A, B, C for you this means 6, 6, 4. In addition, 2C means 24. Then, if, for example, your partner id is 4885255 and you see a reference to D, E, F, this means 8, 8, 5. Then, 2E means 28. If you work by yourself, A-F are your 6 last digit IDs.
For these parts, the report needs to address the followings.
The report should tell a story.
All variables should be clearly defined.
For each problem, define and briefly explain the objective function specified as an equation. For example, P = 3X1 + 5X2.
For each problem, define and briefly explain all of the constraints specified as equations or inequalities. For example, 4X1 + 13X2  100.
Present the solutions as tables.
Solver answer reports should not be in the main text of the report. If you want to include them, place them in an appendix.
Part A: Healthy Foods Incorporated (11)
Healthy Foods Incorporated is selecting the ingredients for a batch of Peanut Bars, Coconut Whirls and Fruit Logs. The following table shows the ingredients that are available and the costs per kilogram.
Ingredient Cost ($/Kgm) Available quantity (Kgm)
Milk chocolate 1.2A 1,500
Sugar 0.3B 300
Peanuts 1.8C 600
Coconut 0.9D 400
Milk solids 0.4E 200
Dried fruit 2.2F 400
Healthy Foods wants to determine the minimum cost mixture for an order of 1,000 kilograms of Peanut Bars, 800 kilograms of Coconut Whirls and 700 kilograms of Fruit Logs.
In Peanut Bars:
the weight of chocolate must be at least 50 percent of the total weight;
the weight of sugar should be no more than 10 percent of the total weight;
the weight of peanuts must be at least 30 percent of the total weight;
there should be no coconut;
the weight milk solids must be no less than 10 percent and no more than 20 percent the total weight; and
there should be no dried fruit.
In Coconut Whirls:
the weight of chocolate must be at least 40 percent the total weight;
the weight of sugar should be no more than 10 percent of the total weight;
there should be no peanuts;
the weight coconut must be at least 30 percent and no more than 40 percent of the total weight;
the weight milk solids must be no less than 10 percent and no more than 20 percent the total weight; and
there should be no dried fruit.
In Fruit Logs:
the weight of chocolate must be at least 50 percent of the total weight;
the weight of sugar should be no more than 10 percent of the total weight;
the weight of peanuts must be at least 10 percent and no more than 20 percent of the total weight;
the weight coconut must be at least 10 percent and no more than 20 percent of the total weight;
there should be no milk solids; and
the weight dried fruit must be at least 20 percent of the total weight.
Formulate the linear-programming problem. Solve it using Excel Solver. Produce a report concerning the model and the results, including the values of the slack and surplus variables.
How would the solution change if the price of milk chocolate was 50 percent less than its original value?
Part B: Rocket Cargo Incorporated (11)
Rocket Cargo Incorporated takes delivery 630 litre refrigerators at three warehouses supplied from five factories. The following table shows the transport costs in dollars per unit shipped between the factories and the warehouses. It also shows for the next planning period the stocks at each factory and the capacity to handle the product at each warehouse.
Warehouse
Factory W1 W2 W3 Stock
F1 1A 2E 3E 250
F2 1B 2F 3F 300
F3 2C 3F 3A 200
F4 1D 3A 3B 350
F5 2E 2B 2D 150
Capacity 300 400 500
Rocket delivers refrigerators from the three warehouses to seven retail businesses. The following table shows the transport costs in dollars per unit shipped between the warehouses and the retail businesses. It also shows the demand at each retail business.
Retail Business
Warehouse R1 R2 R3 R4 R5 R6 R7
W1 1A 2E 5A 4A 1A 2B 2B
W2 1B 2F 2B 3B 2B 1A 1A
W3 2C 2A 1C 4D 2E 2C 2C
Demand 190 180 220 170 210 240 180
Formulate this transhipment problem as a cost minimisation integer-programming problem. Solve it using Excel Solver. What is the optimal transhipment plan? Which if any of the factories will have stock left over, which if any of the warehouses will have unused capacity and which if any of the retail businesses will be undersupplied?
How will the solution change if there is penalty of $25 per unit short at retail business R4?
Part C: The Farmer’s American Bank of Leesburg (7)
The Farmer’s American Bank of Leesburg is planning to install a new computerised accounts system. Bank management has determined the activities required to complete the project, the precedence relationships of the activities, and activity time estimates, as shown in the following table:
Determine the expected project completion time and variance and determine the probability that the project will be completed in 40 weeks or less.
The manager plans to crash the project. The data is below.
Activity Activity Time (weeks) Activity Cost
Normal Crash Normal Crash
a 9 7 4800 6300
b 11 9 9100 15500
c 7 5 3000 4000
d 10 8 3600 5000
e 1 1 0 0
f 5 3 1500 2000
g 6 5 1800 2000
h 3 3 0 0
i 1 1 0 0
j 2 2 0 0
k 8 6 5000 7000
The normal activity times are considered to be deterministic and not probabilistic. Using the computer, crash the network to 26 weeks. Indicate how much it would cosy the bank and then indicate the critical path.
General Matters
This assignment is worth 30% of the total mark for the subject. This is a group assignment, which must be submitted in pairs of two students. Each member of the group will receive the same mark. However, students who do not cooperate or contribute equally towards the work of the group may be penalised and /or asked to make an individual submission for the whole assignments.
You are encouraged to submit your assignment using a word processor to improve your presentation. You will lose marks for bad presentation. English expression will also be counted within the marking scale. Report presentation is 1 mark.
Use A4 size paper and fasten assignment on the left-hand side. Do not use folder or plastic covers. Students must retain a copy of their assignment. Also remember plagiarism is a serious offence.