Supply Chain Analysis and Design: Reducing Café Cost with LPP

This case study examines a university café's efforts to reduce the cost of its casserole dish using linear programming. The analysis focuses on optimizing the mix of potatoes and green beans, considering ingredient costs, nutritional requirements, and taste preferences. The café manager aims to minimize ingredient costs while adhering to constraints related to nutritional content (protein, iron, and vitamin C) and a minimum ratio of potatoes to green beans for palatability. The study formulates a linear programming model to determine the optimal quantities of each ingredient. The solution uses the solver tool in MS-Excel and presents the optimal ingredient mix and the minimum cost. The analysis explores the impact of changes in nutritional requirements (iron and vitamin C) and the price of green beans on the optimal solution. The findings highlight the sensitivity of the casserole cost to iron content and the importance of balancing cost reduction with nutritional and taste considerations. The study concludes with recommendations for the café manager based on the LPP analysis.

SUPPLY CHAIN
ANALYSIS AND
DESIGN
CASE STUDY: REDUCING CAFÉ
COST
STUDENT ID:
[Pick the date]

Linear Programming Model
INTRODUCTION
The given case highlights the situation of a university café which serves a dish named casserole
on Monday. The café manager aims to reduce costs for the next year and she proposes to cut the
costs by cutting down on the ingredient cost. This can be achieved towards buying ingredients
that are less expensive and have lower quality. Even though there are multiple ingredients for
this dish namely green beans, sautéed onions, cream of mushroom soup and boiled potatoes, but
the focus of the café manager is essentially on two ingredients namely potatoes and green beans.
These two ingredients have been selected owing to their greatest contribution to not only cost but
also nutrition and taste. The objective of this report is to present an analysis of the situation at
hand and thereby advice the manager on the ideal mix of green beans and potatoes which would
enable the café to reduce costs of the casserole dish while respecting the constraint.
ISSUES
Based on the information provided in the case study, it is apparent that the objective is to reduce
the cost of the casserole dish by determining the ideal quantity of green beans and potatoes that
must be required for making the dish. Even though the café manager is willing to compromise
on the quality of ingredients, but there are certain constraints or issues that need to be
accommodated while pursuing the endeavour of reducing the cost.
Constraint 1: Nutritional weekly requirements of the university must be fulfilled and it is
estimated that all the nutrition for the casserole dish may be attributed to only the green beans
and potatoes. As a result, there quantity can be reduced to such an extent that the guidelines of
the university are breached.
Constraint 2: The café manager is willing to compromise on the taste as according to her the
students anyways do not like the dish and consume the same only because on Monday, the
alternatives available are quite thin. However, the café cook highlights that in order for the dish
to be edible, it is essential that potato and green beans should be present in the dish in a
minimum ratio of 6:5. Any proportional lower than this would make the dish inedible and hence
would not be acceptable.
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Linear Programming Model
Hence, from the above discussion it is apparent that a Linear Programing Problem (LPP) would
be required in the given case to find solution to the given problem where there is the objective of
reducing cost and simultaneously there are two major constraints which need to be kept in mind
while aiming to achieve the objective. The formulation and solving of LPP has been discussed in
detail in the following section.
Formulation of Linear Programming Model
The aim is to determine the optimal solution for the linear programming problem at hand. This
includes the optimal cost of the main two ingredients (Green beans and Potatoes) or cost of
casserole in order to reduce the total cafe cost and hence enhance the economic viability.
The information and data summary regarding the price, nutrition requirement, casserole content,
total amount of casserole need to make in a week, ingredient content are highlighted below:
1. Jane Lim, the café manager purchases the potatoes and green beans from wholesaler at the
following prices.
Ingredients ¿ $ per pound (lb) ¿ $ per gram s (gm )
Potatoes 0.30 6.61 ×10−4
Green beans 0.95 2.094 ×10−3
2. The university has highlighted the following nutritional requirement for the weekly total
casserole. It is essential parameter that the weekly casserole must have the requisite content
of protein, iron and vitamin C or it would not be considered appropriate on nutritional
grounds.
Nutritional Requirement Grams/milligram Grams
Protein 160 grams 160
Iron 65 milligrams 0.065
Vitamin C 1000 milligrams 1.00
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Linear Programming Model
3. Jane had decided the minimum ratio of potatoes and green beans in the edible casserole in
order to maintain the edible taste of casserole.
Potatoes
¿ Beans
6
5 ∨1.2
4. The major aspect in regards to determine the cost of potatoes and green beans is the total
amount of casserole produced in one week. The minimum amount of casserole that needs to
be prepared in a week is shown below:
Minimum amount of casserole production 12kg 12000 g
5. The nutrient content of green beans and potatoes are given below:
In order to convert the oz into gram, the following relation has been taken into consideration.
*28.35 gram=1 oz
Nutrients content Potatoes Green Beans Potatoes Green Beans
Protein 1.4 g
100 g
5.9 g
10 oz
0.014 g 0.0208 g
Iron 0.255 mg
100 g
3.56 mg
10 oz
0.255∗10−5 g 1.25∗10−5 g
Vitamin C 11mg
100 g
29.75 mg
10 oz
11∗10−5 g 1.049∗10−4 g
Linear programing model
The various relevant variables need to be defined so that the objective function and relevant
constraints can be quantitatively described and captured.
Let assume that x1=amount of potato∈grams∧x2 amount of ¿ beans∈grams.
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Linear Programming Model
Objective function
min C= ( 6.61 ×10−4 ) x1 + ( 2.094 × 10−3 ) x2
C represents the cost of casserole ingredients and the objective is to minimise the same.
Subject to constraints
 Protein nutrient in the weekly casserole must be atleast 160g
0.014 x1 +0.0208 x2 ≥ 160
 Iron nutrient in the weekly casserole must be atleast 0.065g
( 0.255∗10−5 ) x1+ ( 1.25∗10−5 ) x2 ≥0.065
 Vitamin C content in the weekly casserole should be atleast 1000mg or 1 g
( 11∗10−5 ) x1 + ( 1.049∗10−4 ) x2 ≥1
 Minimum amount of weekly casserole that needs to be prepared is 12kg or 12000g
x1+ x2 ≥12000
 Ingredient ratio in casserole as specified by the café cook need to be adhered to
5 x1 ≥ 6 x2
5 x1−6 x2 ≥ 0
 Non- negativity constraints since the amount of green beans and potatoes cannot be negative
and hence necessarily have to be positive. However, there is no requirement of these being
integers only and hence no such constraint is levied.
x1 , x2 ≥ 0
Final Linear programing model
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Linear Programming Model
Objective function
min c= ( 6.61×10−4 ) x1+ ( 2.094 ×10−3 ) x2
Subject to constraints
0.014 x1 +0.0208 x2 ≥ 160
( 0.255∗10−5 ) x1+ ( 1.25∗10−5 ) x2 ≥0.065
( 11∗10−5 ) x1 + ( 1.049∗10−4 ) x2 ≥1
x1+ x2 ≥12000
5 x1 ≥ 6 x2
5 x1−6 x2 ≥ 0
x1 , x2 ≥ 0
Part A
The amount of green beans and amount of potatoes that needs to be purchased each week by
Jane for the preparation of casserole with the minimum ingredient costs after considering all the
given constraints are determined with the help of solver of the linear problem. The relevant
output that has been obtained from MS-Excel using the solver tool has been highlighted below.
Solver output
5

Linear Programming Model
Answer report
6

Linear Programming Model
Sensitivity report
Limit report
The optimal solution as obtained from the above solver output is summarised below.
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Linear Programming Model
Minimum cost of ingredient = $24.44
Amount of potatoes x1=10160.8 g∨10.16 kg
Amount of ¿ beans x2=8467.4 g∨8.47 kg
Part B
Student Union meets at the time of Health Awareness Week has found that the nutritional
requirement of casserole established by the university is not appropriate and thus, the following
changes have been in the nutritional requirement of casserole.
 Iron content - 100 mg∨0.1 gram
 Vitamin C content −500 mg∨0.5 gram
Hence, the linear programing model will be modified. Only the constraints that are based on iron
content and vitamin C content would change. The relevant output that has been obtained from
MS-Excel using the solver tool has been highlighted below.
Objective function
min c= ( 6.61×10−4 ) x1+ ( 2.094 ×10−3 ) x2
Subject to constraints
0.014 x1 +0.0208 x2 ≥ 160
( 0.255∗10−5 ) x1+ ( 1.25∗10−5 ) x2 ≥0.1
( 11∗10−5 ) x1 + ( 1.049∗10−4 ) x2 ≥ 0.5
x1+ x2 ≥12000
5 x1 ≥ 6 x2
5 x1−6 x2 ≥ 0
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Linear Programming Model
x1 , x2 ≥ 0
Solver output
Answer report
9

Linear Programming Model
Sensitivity report
Limit report
The optimal solution as obtained from the above solver output is summarised below.
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Linear Programming Model
Minimum cost of ingredient = $18.55
Amount of potatoes x1=7712.08 g∨7.71 kg
Amount of ¿ beans x2=8467.4 g∨8.47 kg
Part C
It has been observed by café manager Jane that the wholesaler does not have much amount of
green beans and hence, the price of green bean becomes higher. Hence, minimum cost and
optimal amount of ingredient based on the effect of increased price of green beans would be
determined by taking the modified constraint which is highlighted below:
Price of green beans = $ 1.20 per lb∨2.64∗10−3 per g
Only the objective function would be changed. The new linear programming problem is shown
below:
Objective function
min c= ( 6.61×10−4 ) x1+ ( 2.64∗10−3 ) x2
Subject to constraints
0.014 x1 +0.0208 x2 ≥ 160
( 0.255∗10−5 ) x1+ ( 1.25∗10−5 ) x2 ≥0.065
( 11∗10−5 ) x1 + ( 1.049∗10−4 ) x2 ≥1
x1+ x2 ≥12000
5 x1 ≥ 6 x2
5 x1−6 x2 ≥ 0
x1 , x2 ≥ 0
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Linear Programming Model
The relevant output that has been obtained from MS-Excel using the solver tool has been
highlighted below.
Solver output
Answer report
12

Linear Programming Model
Sensitivity report
13

Linear Programming Model
Limit report
The optimal solution as obtained from the above solver output is summarised below.
Minimum cost of ingredient = $29.07
Amount of potatoes x1=10160.8 g∨10.16 kg
Amount of ¿ beans x2=8467.40 g∨8.46 kg
Part D
The modified nutrimental requirement as per part b and the new price of green beans as per part
c would be taken into consideration to determine the optimal amount and minimum cost of
casserole for the café.
Objective function
min c= ( 6.61×10−4 ) x1+ ( 2.64∗10−3 ) x2
Subject to constraints
0.014 x1 +0.0208 x2 ≥ 160
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Linear Programming Model
( 0.255∗10−5 ) x1+ ( 1.25∗10−5 ) x2 ≥0.1
( 11∗10−5 ) x1 + ( 1.049∗10−4 ) x2 ≥ 0.5
x1+ x2 ≥12000
5 x1 ≥ 6 x2
5 x1−6 x2 ≥ 0
x1 , x2 ≥ 0
The relevant output that has been obtained from MS-Excel using the solver tool has been
highlighted below.
Answer report
Sensitivity report
15

Linear Programming Model
Limit report
The optimal solution as obtained from the above solver output is summarised below.
Minimum cost of ingredient = $22.06
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Linear Programming Model
Amount of potatoes x1=7712.08 g∨7.71 kg
Amount of ¿ beans x2=6426.73 g∨6.42 kg
Part E
From the above analysis, it is apparent that the cost of casserole is majorly influenced by the iron
nutrient. A key observation which provides support to the above conclusion can be drawn from
part (d) where it is clearly highlighted that doubling the iron nutrient would lead to increasing the
cost in a significant manner. This is not the case for other nutrients. Therefore the key nutrient to
focus on in relation to the casserole cost is iron.
Conclusion
Using the LPP model, the amount of potato and green beans has been determined so that the
objective of reducing cost is achieved without the breach of the constraints regarding nutrition
and taste. In the base case, the amount of potato requirement came out as 10.16 kg and green
beans required came out as 8.47 kg. Also, the case where the iron requirement is raised and
Vitamin C required in relaxed has been analysed using the LPP model. This has resulted in
alteration of the potato quantity to 7.71 kg while the beans quantity continues to remain the
same. Besides, using the given model, the condition where there is an increase in the price of
green beans has also been analysed which does not lead to much variation in the quantity of
ingredients but only impacts the minimum price. Based on the analyses combining both the
change in the nutritional requirement and rising price of green greens, it has comes to light that
there could be potentially significant change in the cost if the minimum requirement for iron is
raised.
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