OMGT2287 Supply Chain Modelling and Design Assessment

Introduction

OMGT2287 Supply Chain Modelling and Design Assessment 3: Case 2

• PROBLEM FORMULATION:

According to the case given, Company B is a retailer of mobile phones in Australia that works 250 days in a year. The manager is determining a minimum-cost inventory plan for an upcoming phone to be launched in the market. The information regarding the annual demand, cost price and inventory cost are given as follows:

The objective is to solve this problem using a non-linear programming (NLP) model to determine the Economic Order Quantity (EOQ), total inventory cost, total ordering cost, inventory holding cost, transportation cost and the profit from the inventory plan.

• Decision Variables: The decision variable here is the order quantity (Q), i.e. in how much quantity the order should be placed in order to minimize the total inventory cost.
• Measurementunit of Decision variables: The decision variables will be unitless positive integers.

• Constraint Equations:

The decision variable i.e. the order quantity (Q) should always be greater than zero. The optimized decision variable, i.e. the optimum order quantity is actually called the Economic Order Quantity (EOQ).

There are two major components in the total inventory cost – Ordering cost and the Total inventory carrying cost. In the inventory carrying cost, there are further three major components – inventory holding cost, in-transit holding cost, and the freight cost.

[calculations shown in “analysis” part of “Data” sheet in the excel file]

• The ordering cost = (D/Q) * Co = (annual demand / Order quantity) * Cost per order
• The Inventory holding cost = (Q/2) * C_i * h_i1 = (Order quantity / 2) * Cost Price * (inventory holding cost as a % of cost price)
• The In-transit holding cost = (Q/2) * C_i * h_i2 = (Order quantity / 2) * Cost Price * (in-transit holding cost as a % of cost price)
• The total Inventory holding cost = Inventory holding cost + In-transit holding cost
• The Gross Weight (kg) (W) = {Net weight W₁ (gm) + Tare Weight W₂ (gm)} / 1000
• The Freight cost = Order Quantity * Gross Weight * Freight rate per kg* Orders per year
• Total cost for purchasing the phones = (D*C_i) = Annual demand * Cost Price
• Total cost [TC = D*C_i + D*C_o + (Q/2)*C_i*h_i] (including freight cost) = Ordering cost + Purchasing Cost + Total inventory holding cost + Freight cost

• Total Cost Price = D * C_i = Annual Demand * Cost price per item
• Total RRP (Recommended Retail Price) = D * Cr = Annual Demand * RRP per item
• Net Profit from Inventory Plan = Total RRP – Total Cost Price – Total Inventory Cost

• Objective function:

Minimize F,

• Where F = Total Cost = Ordering cost + Purchasing Cost + Total inventory holding cost + Freight cost

Explanation: Here the Objective function is to minimize the Total Cost (TC). Also, when the total inventory cost is minimum, the difference between the Ordering cost and the total inventory carrying cost (including freight cost) will be ideally zero. So, we have also tried to formulate another new objective function to minimize the difference between the Ordering cost and the total inventory carrying cost (including freight cost). Both gave the same result. “GRG Nonlinear” Optimization method has been used in both cases as the Ordering cost is inversely proportional to the decision variable (order quantity Q), which makes the objective function equation non-linear.

• PROBLEM SOLVING:

The problem has been solved in three different methods (2 Excel methods and one MATLAB method) as discussed below:

• Implementation of the excel model 1:

The non-linearity of the objective function makes it very difficult for excel solver even if you select the GRG Non-linear method. So, we have selected the detailed analysis method to find the optimal solution (i.e. minimizing the total inventory cost by varying the decision variable, the Order quantity Q).

[calculations shown in “Analysis” sheet in the excel file]

The decision variable (Order quantity) can only take the positive integer values and that is why we have selected its range to be 1, 2, 3 …. 150. The ordering cost, inventory holding cost, in-transit holding cost, total inventory holding cost, freight cost and the total inventory cost are calculated as the functions of the decision variables. [formulas mentioned above here at “Problem formulation” section]. The objective is to find the minimum total inventory cost and the corresponding order quantity value.

Also, the deviation from the minimum cost, and the net profit from the inventory plan (i.e. Total RRP – Total Cost Price – Total Inventory Cost) has been calculated for all the values of the decision variables. It has been observed that the Net Profit attains the maximum value corresponding to the order quantity having the minimum total inventory cost. Also, the difference between the Ordering cost and the total inventory carrying cost (including freight cost) attains the minimum value corresponding to the order quantity having the minimum total inventory cost.

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• SOLUTION:

The total inventory cost decreases sharply from Order quantity 0 to 14 and attains the minimum value ($ 9,666) for order quantity (Q = 14) and then again increases gradually with the increase in order quantity. The total cost (TC) also attains minimum value of $ 9,80,766 for Q = 14. The net profit increases steeply till Q = 14 and attains the maximum value $ 98,334 and then gradually decreases.

Also, the difference between the Ordering cost and the total inventory carrying cost (including freight cost) attains the minimum value $ 23.62, corresponding to the order quantity (Q = 14) having the minimum total inventory cost

• Implementation of the excel model 2:

This is using the excel solver to minimize the difference between the Ordering cost and the total inventory carrying cost.

• Setting up the excel solver:

The Objective function is to minimize cell G12 (difference between the Ordering cost and the total inventory carrying cost including freight cost). The decision variable is cell C3 (the order quantity) and the constraint equation are:

• the order quantity should be greater than 0 (i.e. cell C3 >= 1)
• the order quantity should be an integer

• SOLUTION:

The solution has come out the exact same as before, i.e. the total inventory cost attains the minimum value ($ 9,666) for order quantity (Q = 14). The total cost (TC) also attains minimum value of $ 9,80,766 for Q = 14.

• Implementation of the MATLAB model:

The non-linear optimization model is very suitable to solve in MATLAB. Here both the Objective function and the constraint function are required to write first in the MATLAB code editor, as shown below.

Then after saving both the functions, we need to go to APPS (in the top taskbar), then select Optimization tool and write the optimization function and the constraint equation as follows:

The final solution is the decision variable Q = X [1] = 14.

• Discussions:

• Economic order quantity for the phone in units = 14
• Economic order quantity for the phone in Kgs = 14 * 0.44 = 6.16 kg
• The total cost for purchasing the phones = $ 9,71,100.
• The total cost for ordering = $ 4,821.
• The total cost for holding the inventory = $ 1,133.
• The total cost for transportation = $ 2,956.
• The total cost for holding the phones during transit = $ 755.
• The total cost for this inventory plan = $ 9,666.
• The number of orders = 64 per year.
• Ordering point = 14 (No. of phones).
• The profit from this inventory plan = $ 98,334 (Annually).

[calculations shown in the “Data” sheet in the excel file]

The calculation for the ordering point and the yearly order plan is also shown here: