OMGT2287 Supply Chain Modelling and Design

Introduction

The growth and expansion of an enterprise depend critically on management decisions. Every organization has to make decisions on how to effectively allocate the available resources to various segments of the firm (Olayinka, et al. 2015). This is critical and necessary to minimize the operation costs of the firm. Resource allocation can be made via the use of quantitative techniques like linear programming. From the definition by Heizer and Render (2004), linear programming is a mathematical approach that is applied to rationalise many management decisions regarding the allocation of economic resources (Fagoyinbo, et al. 2011). This study is designed to examine a real-life application of linear programming in management decision-making.

The task is a case study of optimal decision-making in a non-profit organization (The Childcare Centre). The child-care centre is an organization that has been operating for over 20 years. The organization is located in Melbourne Australia and is managed by a local university to provide quality care for children. The firm has a total of 16 educators with additional educators supplied by an urgency nearby. Currently, the organization has 5 rooms where children are distributed based on age. Being that the wages and availability of the educators vary, the management of the firm needs to find a way to allocate the educators to the rooms to minimize the total wages paid.

Problem formulation

To formulate the problem as a linear programming issue, the first step is to define the decision variables, the objective function, and the constraints. The decision variables are the inputs that management wants to control. In this case, the decision variables are the name of educators to be allocated to each of the 5 rooms.  The aim is to distribute the educators in a way that minimized the total wages paid to them. The objective function is a mathematical equation that takes into account the decision variables. The target of the management was to minimize the objective function. In this case, the objective function is an equation that represents the total wages paid to the educators per day. The model constraints are the mathematical equations that incorporate decision variables to define the boundaries of the possible solution. The childcare case will have the following constraints:

Each educator can only be allocated to one room for the whole week.

The number of educators with diploma qualifications should be at least one for every 3 educators allocated to a room.

The ratio of educators allocated to a room and the number of children in the room be equal or greater than the minimum ratio set by the National Quality Framework.

LP model

Let  be tutor allocated to room   be the hourly rate of tutor and  the hours that a tutor is available per day.

Then the linear model is

Subject to

the constraint that ensures each permanent tutor is only allocated to one room for the whole week

the constraint that ensures urgency tutors are only contracted when there is a need

where Q is a matrix representing the ratio of tutors to educators per room

where D is a matrix of the ratio of diploma educators to the total number of educators in the room and T the matrix that represents the quality, ratio expected for diploma holders in relation to all the tutors in the room. The data was entered in excel and the model was formulated and solved using excel solver.

Problem-solving

1. Model implementation in excel
2. Solver set up

The set up of the linear model in the solver add in should be as presented in the table below (Goh, 2019).

Discussion

Optimal solution

For the firm to minimise the total wages paid to the tutors per day the number of tutors available on each room on each day should be as described in the table below.

The tutor’s allocated for each of the rooms during the week is as follows. Room 1 will have tutors 2, 3, 4 and 15, Room 2 to have tutors 1, 7, 8 and 16, Room 3 have tutors 5, 6, 10 and 13, Room 4 have tutors 11 and 12 while Room 5 have tutors 9 and 14. The available permanent tutors at the centre are adequate to meet the National Quality Framework (NQF) minimum ratio of educators to children. The centre will not have to contact any additional tutors from the urgency. The tutor room allocation will cost the urgency a total of $ 3,338.71 per day. This value will reduce for every day that a tutor assigned to a room is not available.

The number of children and room capacity in the centre is described in the table below.

The table shows that there is available space for new children as follows:  

The availability of space for new children is based on available room space. The center therefore may have to contact new tutors from the urgency to sustain the NQF ratio where arrival of new children demands so.

Educator 6 & 7 are on leave

Educators 6 and 7 are diploma holders if they will be on leave next week then the centre will have to contact two new tutors from the urgency at a rate of $51.77 per hour. The admission of the urgency tutors will increase the daily wages of the tutors to $3,698.68. The urgency tutors contacted should be replace tutor 6 and 7 in the various rooms they were previously allocated.

Recommendation

The Childcare centre should allocate the rooms to the permanent tutors as follows:

Room 1 to have tutors 2, 3, 4 and 15

Room 2 to have tutors 1, 7, 8 and 16

Room 3 to have tutors 5, 6, 10 and 13

Room 4 have tutors 11 and 12

Room 5 to have tutors 9 and 14

When tutor 6 and 7 will be on leave next week, the centre should hire two diploma qualified tutors from the urgency and send one to room 2 and the other to room 3.

References

Fagoyinbo, I.S., Akinbo, R.Y., Ajibode, I.A. and Olaniran, Y.O.A., 2011. Maximization of profit in manufacturing industries using linear programming techniques: Geepee Nigeria Limited. Mediterranean Journal of Social Sciences, 2(6), pp.97-97.

Goh, C., 2019. Solve problems with linear programming and Excel. [Online]
Available at: https://www.fm-magazine.com/issues/2019/feb/linear-programming-microsoft-excel.html
[Accessed 7 April 2021].

Hazier, J. & Render, B. (2004). Operations Management: Process and Value Chains 8th Edition, New Jersey, Prentice Hall

Olayinka, I., Olusegun, A.K., Kellikume, G. and Kayode, K., 2015. Entrepreneur decision making process and application of linear programming technique. European Journal of Business, Economics and Accountancy, 3(5), pp.231-238.

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