Open Journal of Energy Efficiency, 2013, 2, 139-142
http://dx.doi.org/10.4236/ojee.2013.23018 Published Online September 2013 (http://www.scirp.org/journal/ojee)
Simulation Using Sensitivity Analysis of a Product
Production Rate Optimization Model of a Plastic Industry
Mala Abba-Aji1, Vincent Ogwagwu2, Bukar Umar Musa3
1Department of Mechanical Engineering, University of Maiduguri, Maiduguri, Nigeria
2Federal University of Technology, Minna, Nigeria
3Department of Electrical and Electronics Engineering, University of Maiduguri, Maiduguri, Nigeria
Received January 1, 2013; revised March 5, 2013; accepted June 2, 2013
Copyright © 2013 Mala Abba-Aji et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This study analyzes the sensitivity analysis using shadow price of plastic products. This is based on a research carried
out to study optimization problem of BOPLAS, a plastic industry in Maiduguri, North eastern Nigeria. Simplex method
of Linear programming is employed to formulate the equations which were solved by using costenbol software. Sensi-
tivity analysis using shadow price reveals that the price of wash hand bowls is critical to the net benefit (profit) of the
company.
Keywords: Sensitivity Analysis; Simplex Method; Linear Programming; Optimization
1. Introduction
Shadow prices: the simplex-method provides more use-
ful information than just the optimal solution to a linear
programming problem. From the optimal tableau, the
value of each resource in terms of its contribution to
profits and overheads is determined. For the sensitivity
analysis, the net benefit (or cost) of adjusting the amount
of resource can also be determined. The relative value of
a resource with respect to the objective function in a lin-
ear programming problem is called its shadow price. It is
the amount of change in the objective function per unit
change in its right-hand side value.
2. Methodology
2.1. Injection Molding Process
Injection molding is one of the most important plastics
molding processes. It is carried out usually on horizontal
hydraulic press.
Granular thermoplastic materials are gravity fed from
a hopper into a pressure chamber ahead of a plunger.
The moving plunger causes the granular plastic to be
compressed and then forced through a heating cylinder to
palletize it. A torpedo shaped object in the centre of the
heating cylinder, assists uniform heating.
The palletized plastic is then injected through an injec-
tion nozzle at great pressure into the die cavity to form
the required component. The die is water-cooled; making
the injected plastic to freeze almost immediately the die
cavity is filled.
The plunger returns, and the mould open to eject the
formed material. The mold closes and the cycle is re-
peated.
In modern machines, as used in the company, the feed
plunger is replaced with a motor driven screw plasticizer.
It serves the function of both part-heating the plastic
granules by internal sheer and feeding it to the mould.
(Resistance heater bands are still used on the heating
cylinder). The screw–plasticizer helps to ensure that the
thermoplastic fed through the injector nozzle is main-
tained at a constant and uniform temperature and viscos-
ity.
The process requires the use of expensive dies, usually
called molds; thus its use has to be justified by large
production runs. The process is easily automated, and
cycle times of just a few seconds are common, making
injection molding the most widely used process for pro-
ducing plastic items. Also a wide range of shapes and
plastic materials can be molded [1].
2.2. Simplex-Method Algorithm
High customer demand of kettle, water jug, wash-hand
bowel, Big Bowel, medium bowel and small bowel was
observed within the period of August to February of every
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M. ABBA-AJI ET AL.
140
year; but the company is uncertain of allocating the opti-
mal proportion of the products.
Let A = kettle B = water jug C = wash hand bowel
D = big bowel E = medium bowel F = small bowel
G = parker H = Hanger
Let X1, X2, X3, X4, X5, be the proportions of prod-
ucts to be produced. These are decision variables of the
model, and h, H, Ф, d, e, be duration of injection, charge
and cooling of the various products respectively as
shown in Table 1. These durations; injection, cooling,
and charge time were recorded from the injection mold-
ing machine
Capacity; is the maximum time assigned to the injec-
tion molding machine through the function setting of a
mini computer attached to the machine. Only an experi-
enced machine operator could do this.
Contribution to profit;
Let: a, b, c, d, e, f and g be contribution to profits of
the products A, B, C, D, E, F and G respectively.
The contribution to profit and overhead per unit of
each product is determined. The company was uncertain
about how many of each product to produce in order to
maximize their profit. The simplex-method provides in-
formation more than just the optimal solution to linear
programming problem. The optimal tableau determines
the value of each resource in terms of its contribution to
profits and overhead. We can also determine the net
benefit (or cost) of adjusting the amount of resources.
The simplex equations can be written as;
Maximize – ax1 + bx2 + cx3 + dx4 + ex5
Subject to
Injection hIx1 + HIx2 + ФIx3 + dIx4 + eIx5 ≤ CI
Charge hcx1 + Hcx2 + Фcx3 + dc x4 + еcx5 ≤ CII
Cooling hgx1 + Hgx2 + Фgx3 + dgx4 + egx5 ≤ CIII [3]
x1, x2, x3, x4, x5 ≥ 0
Using Gauss Jordan Complete elimination method, se-
ries of tableau will be obtained, procedures of elimina-
tion being repeated until there are no further negative
Table 1. Resource and maximum capacities of products [2].
(a)
Resource type A B C D Capacity
Injection time hI H
I ФI d I C I
Charge Time hc H
c Фc d c C II
Cooling Time hg H
g Фg d g C III
(b)
Resource type E Capacity
Injection time eI C
I
Charge Time ec C
II
Cooling Time eg C
III
values in the last row i.e. the objective function row.
N
Sales; 230003059.97
Less Variable cost;
Materials; 7146900.85
Machine operator’s wages; 532,000
Diesel; 1,250,000
Metered power supply; 65,550
Overtime; 84,000 9 078450.8 5
Total contribution 13924609.12
Less fixed costs;
Accountant salary; 46,800
Courier service; 2650
Communication facilities; 16,500
Transportation; 84,000
Lubricants; 115,000 686,150
Profit 13238459.12 [2]
The contribution at any given level of sales can be
found by using the formula;
Contribution = sales × p/v ratio
where p = profit v = volume [3].
The proportion to be produced so as to maximize the
contribution to profit of each product and the cost impli-
cation of adjusting constraints could be achieved by
solving the linear programming model. From the analysis
above, the equation can be written as;
Maximize 40.98x1 + 25.62x2 + 2.65x3 + 2.61x4 +
4.23x5
Subject to
Injection 9.5x1 + 7.5x2 + 8x3 + 6x4 + 8x5 ≤ 15
Charge 11.3x1 + 9x2 + 10x3 + 6x4 + 8x5 ≤ 14
Cooling 15x1 + 5x2 + 6.5x3 + 6x4 + 8x5 ≤ 15
where x1, x
2, x3, x4, x5 ≥ 0 (non-negativity constraint)
[4,5]
3. Results and Discussions
A computer program, academic version software was
used to solve the generated equations and after ten itera-
tions obtained the following results;
Value of the objective function = 47.150, yield
x1 = 8280, x2 = 0.5159, x3 = 0.0, x4 = 0.0, x5 = 0.0
[4,6,7].
Assuming an incremental value of N 5 to each of the
five products of interest for five different values, keeping
other products constants, employing sensitivity analysis,
25 different simulations were carried out, which gave the
following results in Table 2.
When a shadow unit prices are used, with an incre-
ments of N 5.00 on the initial unit prices, significant
changes in the maximized profits of water jugs and wash
hand bowls were noticed.
The maximum contribution to profit will be obtained
when a shadow price of N 25 increments on the initial
unit price of wash hand bowls is used, yielding the value
of the objective function, p = 78.00.

M. ABBA-AJI ET AL.
141
Table 2. Computer programme d results for shadow pric e s.
S/No. PRODUCTS
Initial value plus
N 5
Initial value plus
N 10
Initial value plus
N 15
Initial value plus
N 20
Initial value plus
N 25
1 Water jugs 51.29 55.43 59.57 63.71 67.85
2 Wash hand bowls 49.73 55.40 63.18 70.00 78.00
3 Big bowls 47.15 47.15 47.15 47.15 47.15
4 Parker 47.15 47.15 47.15 52.76 64.42
5 Hanger 47.15 47.15 47.150 47.150 51.153
Table 3: Results showing the benefit of adjusting the constraint for wash hand bowls.
Products
Proportion
S/P per month
B/model (N)
S/P per month
A/model (N) C/p
Simplex results
(proportions)
based on C/p
Maximzed
profits (P)
Q/month
B/model
Optimum
Q/month
A/model
Net profits
(N)
X1
1) 785695.57
2) 833343.57
3) 877988.57
4) 922633.57
5) 967278.57
45.98
50.98
55.98
60.98
65.98
0.828
0.828
0.828
0.828
0.828
51.290
55.431
59.570
63.711
67.851
8929 A/model = 221507778.734
X2
1) 951537.6
2) 1045137.6
3) 1138737.6
4) 1232937.6
5) 1352937.6
2145257.28
30.62
35.62
40.62
45.62
50.62
0.516
1.556
1.556
1.556
1.556
49.730
55.409
63.187
70.000
78.000
18720 30,287.4 B/model = 220688459.1
Profit margin = 819319.634
The cycle time for wash hand bowls is 35.5 seconds.
The company is using 8-hours per day, quantity produced
in a month = 60 × 60 × 8/35.5 × 1.5556 × 24 = 30287.4
Units
Selling price per month = Quantity/month X Unit price
Selling price per month = 30287.4 × 70.83 = N 2,
145257.284
Substituting the selling price back into the profit state-
ment for the month of February, 2005: when the unit
volume for wash hand bowls V = 18,720, S/month = N
1325937.6 yielding total sales of N 230,453,059
N
Sales; 230453059.9
Less Variable cost; 9078450.85
Total contribution 221374,609.1
Less fixed costs; 686,150
Profit 2220688459.1
Using the optimum quantity or volume of wash hand
bowls V = 30287.4 Units and selling price of N 2,
145257.284,
N
Sales: 231272379.584
Less variable cost: 9078450.8 5
Total contribution: 222193928.734
Less fixed costs: 686,150
Profit: 221507778.734
From the results obtained after simulating the equa-
tions of the × linear programming using simplex method,
the value of the objective function, which was the profit
foregone was 47.150445528799 and the optimal propor-
tions of the products to be produced using injection
molding machine, based on their contribution to profits
are:
X1, proportion of water jugs to be produced = 0.8280
X2, proportion of wash hand bowls to be produced =
0.5159
X3, proportion of big bowls to be produced = 0.0
X4, proportion of packer to be produced = 0.0
X5, proportion of hanger to be produced = 0.0
Summary of the results present the net profit for water
jug and wash hand bowl are presented in Table 3.
4. Conclusions
Since the maximum time the company used was 8 hours
per day, and the cycle time for water jugs and wash hand
bowls are 36 units and 35.5 units, then the optimum
number of the two products to be produced per day will
now be, 966.15 units for water jugs and 1572.457 units
for wash hand bowls. The profit margin obtained was
N 25062868.41 per month [2,8].
This is a clear justification why the company needs to
emphasize the production of water jugs and wash hand

M. ABBA-AJI ET AL.
142
bowls as regard to injection molding machine. Further-
more, sensitivity analysis, using shadow price, revealed
that the price of wash hand bowls is critical to the net
benefit (profit) of the company. When the proposed unit
selling price N 60.5 is used for wash hand bowls, opti-
mum quantity of 30287 units will be produced yielding a
maximum net benefit of N 819, 319.634 per month. The
company needs reconsider the price of wash hand bowls
as regard to injection molding machine, and also concen-
trates on the other products being considered in this
analysis in order to improve their selling prices, taking
into cognizance, the quality of the products, customer
requirements and customer affordability.
In this paper, sophisticated cost model that requires the
use of design parameters to provide design alternatives
can be carried out.
Apart from the time constraint considered in this work,
temperature is another constraint that affects the produc-
tion of plastics. Further research can then be carried out
when temperature constraints from blow film molding
and extrusion units of the industry were obtained.
The procurement of raw materials is a major challenge
facing plastic industries in Nigeria. The government
should encourage petrochemical industries producing
plastic raw materials, like the one in Port Harcourt, to be
in full production. That will reduce cost of importing raw
materials from abroad. Also the foreign raw materials
have a very low melting point compared to the one pro-
duced in Nigeria. This is not pleasant for molding proc-
ess.
Finally, it is strongly recommend that Nigerian indus-
tries should adopt the modern operation research tech-
niques so that they can obtain optimum results and make
proper decisions.
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