LINEAR PROGRAMMING

LINEAR PROGRAMMING

A firm produces ties using various materials. Their aim, as most firms in the economy, is to
maximize profit but they face some production constraints. Your task is to find the optimal
solution by applying linear programming using Excel. Your results need to show the values for
each decision variable, and the maximum profit subject to the constraints. The Excel sheet
need to be uploaded on the student portal, and a print screens of the pages need to be copied
on Word document (stating your name and student ID), and handed over in class on May 5th
2017.
DUBAI Ties is a firm, based in Deira, that produces four varieties of ties:
? Two are blends of polyester and cotton
? One is expensive all-silk
? One is all-polyester
The table on the below shows the cost and availability of the three materials used in the
production process:
MATERIAL COST PER YARD ($)
MATERIAL AVAILABLE
PER MONTH (YARDS)
Silk 24 1,200
Polyester 6 3,000
Cotton 9 1,600
The firm has contracts with several major department store chains to supply ties.
? Contracts require a minimum number of ties but may be increased if demand increases.
? Their goal is to maximize monthly profit given the decision variables stated below.
Decision variables:
X1 = number of all-silk ties produced per month
X2 = number all-polyester ties
X3 = number of blend 1 polyester-cotton ties
X4 = number of blend 2 silk-cotton ties
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DATA PROVIDED:
VARIETY
OF TIE
SELLING
PRICE
PER TIE
($)
MONTHLY
CONTRACT
MINIMUM
MONTHLY
DEMAND
MATERIAL
REQUIRED
PER TIE
(YARDS)
MATERIAL
REQUIREMENTS
All silk 19.24 5,000 7,000 0.125 100% silk
All polyester 8.70 10,000 14,000 0.08 100% polyester
Poly – cotton
blend 1 9.52 13,000 16,000 0.10
50% polyester –
50% cotton
Silk-cotton
blend 2 10.64 5,000 8,500 0.11
60% silk –
40% cotton
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PROBLEM 2: LINEAR PROGRAMMING USING GRAPHICAL REPRESENTATION
An ice cream company is involved in the production of two different tastes: Cherry and Kiwi.
Two resources needed to produce Cherry and Kiwi ice cream are Milk and Sugar.
The table below gives the required amount for each item:
ITEM UNITS/MILK KG/SUGAR
Cherry ice cream 4 5
Kiwi ice cream 6 2
The company has 200 units of Milk available and 150 Kg Sugar available. The revenue received
for each item produced (all production is sold) is AED 400 for a pack of Cherry Ice cream (2Kg)
and AED 600 for a pack of Kiwi Ice cream (2Kg).
i. Determine the Decision variables
ii. Formulate the Profit Objective Function
iii. Formulate the production constraints
iv. Determine the feasible region (graphically)
v. Draw at least three Iso-profit lines
vi. Determine the optimal solution.
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PROBLEM 3: Decision tree analysis
The Dubai Multi-Group (DMG) is set up to conduct soil explorations in order to ascertain
whether significant metal deposits that are worthy of further commercial exploitation are
present or not.
Current DMG has an option to purchase outright a parcel of land for £3m. If it purchases this
parcel of land then it will conduct a geological exploration of the land. Past experience indicates
that for the type of parcel of land under consideration geological explorations cost approximately
£1m and yield significant metal deposits as follows:
GOLD 0.05% chance
SILVER 0.2% chance
PLATINUM 1% chance
Only one of these three metals is ever found (if at all), i.e. there is no chance of finding two or
more of these metals and no chance of finding any other metal. If Platinum is found then the
parcel of land can be sold for £30m, if gold is found then the parcel of land can be sold for £250m
and if silver is found the parcel of land can be sold for £150m.
DMG can, if they wish, pay £750,000 for the right to conduct a three-day test exploration before
deciding whether to purchase the parcel of land or not. Such three-day test explorations can only
give a preliminary indication of whether significant metal deposits are present or not and past
experience indicates that three-day test explorations cost £250,000 and indicate that significant
metal deposits are present 50% of the time.
If the three-day test exploration indicates significant metal deposits then the chances of finding
Platinum, Gold and Silver increase to 3%, 2% and 1% respectively. If the three-day test
exploration fails to indicate significant metal deposits then the chances of finding Platinum, Gold
and Silver decrease to 0.75%, 0.04% and 0.175% respectively.
QUESTIONS:
1. Draw the decision tree, showing all values.
2. What would you recommend MDG should do and why?
3. Interestingly, information exists that a Sharjah company working in a related field to
DMG is prepared to pay half of all costs associated with this parcel of land in return for
half of all revenues. Under these circumstances what would you recommend DMG
should do and why?