Question 1 : ASSIGNMENT LP
A transportation company in Mandeville has five delivery vehicles to deliver products along five
delivery routes which have different costs according to the truck used. The estimated costs in United
States Dollars for each truck along each route are provided in the following table:
TRUCK ROUTE
A B C D E
1 1,010 960 600 840 6730
2 800 1,040 940 970 790
3 680 880 830 900 590
4 580 880 860 640 700
5 1,120 770 700 950 1,020
Required:
a) Use the Hungarian assignment method to determine which assembly line should be
assigned to assemble which computer so as to minimize the overall cost
[12 marks ]
b) Formulate the problem as an LP problem to be solved using software such as “excel
solver “
[18 marks ]
Question 2 : Inventory
A gourmet coffee shop in downtown San Fernando is open 200 days a year and sells an average
of 75 pounds of Blue Mountain Peak coffee beans a day. It purchases coffee beans for $15.00 per
a pound. The following information is available about operation of the sale of its coffee:
Order cost = $16 / order
Annual holding cost per unit = 20% of cost per unit
Desired cycle service level = 95%
Lead time = 4 days
Standard deviation of daily demand = 15 pounds per day
Current on-hand inventory is 350 pound, with no open orders or backorders.
a) What is the EOQ? [5 marks]
b) What is the number of order placed per year? [2 marks]
c) What would be the average time between orders? [2 marks]
d) What is the ROP? [3 marks]
e) An inventory withdrawal of 10 bags was just made. - Is it time to reorder?
[3 marks]
f) If the store currently uses a lot size of 500 pounds, determine the annual holding cost and
the annual ordering cost of this policy? [4 marks]
g) What would be the annual cost saved, if any, by shifting from the 500-pounds lot size to the
EOQ? [6 marks]
Question 1:
a) To use the Hungarian assignment method to determine which assembly line should be assigned to assemble which computer so as to minimize the overall cost, you can follow these steps:
Step 1: Create a cost matrix based on the given estimated costs for each truck along each route. In this case, the cost matrix would be:
1010 960 600 840 6730
800 1040 940 970 790
680 880 830 900 590
580 880 860 640 700
1120 770 700 950 1020
Step 2: Subtract the smallest value in each row from all the values in that row, and subtract the smallest value in each column from all the values in that column. This step is called row and column reduction. The resulting matrix would be:
150 0 0 150 6010
0 240 140 170 0
0 200 150 220 0
0 300 280 60 120
350 0 0 250 300
Step 3: Find the minimum number of lines needed to cover all the zeros in the reduced matrix. In this case, 5 lines are needed.
Step 4: Check for optimality by finding the smallest element not covered by any lines. Subtract this smallest element from all the uncovered elements, and add it to all the elements at the intersection of lines. Repeat this process until there are no uncovered zeros. The resulting matrix would be:
0 0 0 100 5860
0 160 60 90 0
50 200 0 170 0
0 250 230 10 80
300 0 0 150 200
Step 5: Assign the assembly lines to the trucks based on the final reduced matrix. In this case, the assignments would be:
Line 1: Route B
Line 2: Route D
Line 3: Route A
Line 4: Route C
Line 5: Route E
Therefore, the assembly lines should be assigned to the trucks as follows:
Truck 1: Route B
Truck 2: Route D
Truck 3: Route A
Truck 4: Route C
Truck 5: Route E
b) To formulate the problem as an LP problem to be solved using software such as "excel solver," you can follow these steps:
Step 1: Define the decision variables. In this case, you would define binary decision variables representing the assignment of each assembly line to each truck.
Step 2: Define the objective function. The objective is to minimize the overall cost. Therefore, you would define the objective function as the sum of the product of each decision variable and its corresponding cost.
Step 3: Define the constraints. The constraints would include ensuring that each truck is assigned to exactly one assembly line and each assembly line is assigned to exactly one truck.
Step 4: Set up the LP problem in the software, specifying the decision variables, objective function, and constraints. Use the solver function in the software to solve the LP problem and obtain the optimal solution.
Question 2:
a) To find the EOQ (Economic Order Quantity), you can use the following formula:
EOQ = √((2 * Order Cost * Annual Demand) / Holding Cost per Unit)
In this case, the order cost is $16, the annual demand is 75 pounds per day multiplied by 200 days, and the holding cost per unit is 20% of $15. Plug in these values into the formula to calculate the EOQ.
b) The number of orders placed per year can be calculated by dividing the annual demand by the EOQ. Use the value of the annual demand calculated in part a) and the EOQ obtained in part a) to calculate the number of orders.
c) The average time between orders can be calculated by dividing the number of working days in a year (200) by the number of orders placed per year. Use the value of the number of orders calculated in part b) to calculate the average time between orders.
d) The Reorder Point (ROP) is the level of inventory at which a new order should be placed. It can be calculated by multiplying the lead time by the average daily demand and adding a safety stock if required. In this case, the lead time is 4 days and the average daily demand can be calculated by dividing the annual demand by the number of working days in a year.
e) To determine if it is time to reorder after an inventory withdrawal, you need to compare the remaining inventory with the Reorder Point (ROP). If the remaining inventory is less than or equal to the ROP, it is time to reorder.
f) The annual holding cost can be calculated by multiplying the average inventory level by the holding cost per unit. The average inventory level can be calculated by dividing the EOQ by 2. The annual ordering cost can be calculated by multiplying the number of orders placed per year by the order cost.
g) To determine the annual cost saved by shifting from the 500-pound lot size to the EOQ, you can compare the annual holding cost and annual ordering cost of both policies. Subtract the annual holding cost and annual ordering cost of the EOQ policy from the annual holding cost and annual ordering cost of the 500-pound lot size policy to calculate the annual cost saved.